travelling wave effects

Travelling Wave

When something about the physical world changes, the information about that disturbance gradually moves outwards, away from the source, in every direction. As the information travels, it travels in the form of a wave. Sound to our ears, light to our eyes, and electromagnetic radiation to our mobile phones are all transported in the form of waves. A good visual example of the propagation of waves is the waves created on the surface of the water when a stone is dropped into a lake. In this article, we will be learning more about travelling waves.

Describing a Wave

A wave can be described as a disturbance in a medium that travels transferring momentum and energy without any net motion of the medium. A wave in which the positions of maximum and minimum amplitude travel through the medium is known as a travelling wave. To better understand a wave, let us think of the disturbance caused when we jump on a trampoline. When we jump on a trampoline, the downward push that we create at a point on the trampoline slightly moves the material next to it downward too.

When the created disturbance travels outward, the point at which our feet first hit the trampoline recovers moving outward because of the tension force in the trampoline and that moves the surrounding nearby materials outward too. This up and down motion gradually ripples out as it covers more area of the trampoline. And, this disturbance takes the shape of a wave.

Following are a few important points to remember about the wave:

• The high points in the wave are known as crests and the low points in the wave are known as troughs.
• The maximum distance of the disturbance of the wave from the mid-point to either the top of the crest or to the bottom of a trough is known as amplitude.
• The distance between two adjacent crests or two adjacent troughs is known as a wavelength and is denoted by 𝛌.
• The time interval of one complete vibration is known as a time period.
• The number of vibrations the wave undergoes in one second is known as a frequency.
• The relationship between the time period and frequency is given as follows:
• The speed of a wave is given by the equation

Different Types of Waves

Different types of waves exhibit distinct characteristics. These characteristics help us distinguish between wave types. The orientation of particle motion relative to the direction of wave propagation is one way the traveling waves are distinguished. Following are the different types of waves categorized based on the particle motion:

• Pulse Waves – A pulse wave is a wave comprising only one disturbance or only one crest that travels through the transmission medium.
• Continuous Waves – A continuous-wave is a waveform of constant amplitude and frequency.
• Transverse Waves – In a transverse wave, the motion of the particle is perpendicular to the direction of propagation of the wave.
• Longitudinal Waves – Longitudinal waves are the waves in which the motion of the particle is in the same direction as the propagation of the wave.

Although they are different, there is one property common between them and that is the transportation of energy. An object in simple harmonic motion has an energy of

Constructive and Destructive Interference

A phenomenon in which two waves superimpose to form a resultant wave of lower, greater, or the same amplitude is known as interference. Constructive and destructive interference occurs due to the interaction of waves that are correlated with each other either because of the same frequency or because they come from the same source. The interference effects can be observed in all types of waves such as gravity waves and light waves.

According to the principle of superposition of the waves , when two or more propagating waves of the same type are incidents on the same point, the resultant amplitude is equal to the vector sum of the amplitudes of the individual waves. When a crest of a wave meets a crest of another wave of the same frequency at the same point, then the resultant amplitude is the sum of the individual amplitudes. This type of interference is known as constructive interference. If a crest of a wave meets a trough of another wave, then the resulting amplitude is equal to the difference in the individual amplitudes and this is known as destructive interference.

Stay tuned to BYJU’S to learn more physics concepts with the help of interactive videos.

Watch the video and understand longitudinal and transverse waves in detail.

Frequently Asked Questions – FAQs

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13.2 Wave Properties: Speed, Amplitude, Frequency, and Period

Section learning objectives.

By the end of this section, you will be able to do the following:

• Define amplitude, frequency, period, wavelength, and velocity of a wave
• Relate wave frequency, period, wavelength, and velocity
• Solve problems involving wave properties

Teacher Support

The learning objectives in this section will help your students master the following standards:

• (B) investigate and analyze the characteristics of waves, including velocity, frequency, amplitude, and wavelength, and calculate using the relationship between wave speed, frequency, and wavelength;
• (D) investigate the behaviors of waves, including reflection, refraction, diffraction, interference, resonance, and the Doppler effect.

Section Key Terms

[BL] [OL] [AL] Review amplitude, period, and frequency for simple harmonic motion.

Wave Variables

In the chapter on motion in two dimensions, we defined the following variables to describe harmonic motion:

• Amplitude—maximum displacement from the equilibrium position of an object oscillating around such equilibrium position
• Frequency—number of events per unit of time
• Period—time it takes to complete one oscillation

For waves, these variables have the same basic meaning. However, it is helpful to word the definitions in a more specific way that applies directly to waves:

• Amplitude—distance between the resting position and the maximum displacement of the wave
• Frequency—number of waves passing by a specific point per second
• Period—time it takes for one wave cycle to complete

In addition to amplitude, frequency, and period, their wavelength and wave velocity also characterize waves. The wavelength λ λ is the distance between adjacent identical parts of a wave, parallel to the direction of propagation. The wave velocity v w v w is the speed at which the disturbance moves.

Tips For Success

Wave velocity is sometimes also called the propagation velocity or propagation speed because the disturbance propagates from one location to another.

Consider the periodic water wave in Figure 13.7 . Its wavelength is the distance from crest to crest or from trough to trough. The wavelength can also be thought of as the distance a wave has traveled after one complete cycle—or one period. The time for one complete up-and-down motion is the simple water wave’s period T . In the figure, the wave itself moves to the right with a wave velocity v w . Its amplitude X is the distance between the resting position and the maximum displacement—either the crest or the trough—of the wave. It is important to note that this movement of the wave is actually the disturbance moving to the right, not the water itself; otherwise, the bird would move to the right. Instead, the seagull bobs up and down in place as waves pass underneath, traveling a total distance of 2 X in one cycle. However, as mentioned in the text feature on surfing, actual ocean waves are more complex than this simplified example.

Watch Physics

Amplitude, period, frequency, and wavelength of periodic waves.

This video is a continuation of the video “Introduction to Waves” from the "Types of Waves" section. It discusses the properties of a periodic wave: amplitude, period, frequency, wavelength, and wave velocity.

The crest of a wave is sometimes also called the peak .

The Relationship between Wave Frequency, Period, Wavelength, and Velocity

Since wave frequency is the number of waves per second, and the period is essentially the number of seconds per wave, the relationship between frequency and period is

just as in the case of harmonic motion of an object. We can see from this relationship that a higher frequency means a shorter period. Recall that the unit for frequency is hertz (Hz), and that 1 Hz is one cycle—or one wave—per second.

The speed of propagation v w is the distance the wave travels in a given time, which is one wavelength in a time of one period. In equation form, it is written as

From this relationship, we see that in a medium where v w is constant, the higher the frequency, the smaller the wavelength. See Figure 13.8 .

[BL] For sound, a higher frequency corresponds to a higher pitch while a lower frequency corresponds to a lower pitch. Amplitude corresponds to the loudness of the sound.

[BL] [OL] Since sound at all frequencies has the same speed in air, a change in frequency means a change in wavelength.

[Figure Support] The same speaker is capable of reproducing both high- and low-frequency sounds. However, high frequencies have shorter wavelengths and are hence best reproduced by a speaker with a small, hard, and tight cone (tweeter), whereas lower frequencies are best reproduced by a large and soft cone (woofer).

These fundamental relationships hold true for all types of waves. As an example, for water waves, v w is the speed of a surface wave; for sound, v w is the speed of sound; and for visible light, v w is the speed of light. The amplitude X is completely independent of the speed of propagation v w and depends only on the amount of energy in the wave.

Waves in a Bowl

In this lab, you will take measurements to determine how the amplitude and the period of waves are affected by the transfer of energy from a cork dropped into the water. The cork initially has some potential energy when it is held above the water—the greater the height, the higher the potential energy. When it is dropped, such potential energy is converted to kinetic energy as the cork falls. When the cork hits the water, that energy travels through the water in waves.

• Large bowl or basin
• Cork (or ping pong ball)
• Measuring tape

Instructions

• Fill a large bowl or basin with water and wait for the water to settle so there are no ripples.
• Gently drop a cork into the middle of the bowl.
• Estimate the wavelength and the period of oscillation of the water wave that propagates away from the cork. You can estimate the period by counting the number of ripples from the center to the edge of the bowl while your partner times it. This information, combined with the bowl measurement, will give you the wavelength when the correct formula is used.
• Remove the cork from the bowl and wait for the water to settle again.
• Gently drop the cork at a height that is different from the first drop.
• Repeat Steps 3 to 5 to collect a second and third set of data, dropping the cork from different heights and recording the resulting wavelengths and periods.
• Interpret your results.
• No, only the amplitude is affected.
• Yes, the wavelength is affected.

Students can measure the bowl beforehand to help them make a better estimation of the wavelength.

Links To Physics

Geology: physics of seismic waves.

Geologists rely heavily on physics to study earthquakes since earthquakes involve several types of wave disturbances, including disturbance of Earth’s surface and pressure disturbances under the surface. Surface earthquake waves are similar to surface waves on water. The waves under Earth’s surface have both longitudinal and transverse components. The longitudinal waves in an earthquake are called pressure waves (P-waves) and the transverse waves are called shear waves (S-waves). These two types of waves propagate at different speeds, and the speed at which they travel depends on the rigidity of the medium through which they are traveling. During earthquakes, the speed of P-waves in granite is significantly higher than the speed of S-waves. Both components of earthquakes travel more slowly in less rigid materials, such as sediments. P-waves have speeds of 4 to 7 km/s, and S-waves have speeds of 2 to 5 km/s, but both are faster in more rigid materials. The P-wave gets progressively farther ahead of the S-wave as they travel through Earth’s crust. For that reason, the time difference between the P- and S-waves is used to determine the distance to their source, the epicenter of the earthquake.

We know from seismic waves produced by earthquakes that parts of the interior of Earth are liquid. Shear or transverse waves cannot travel through a liquid and are not transmitted through Earth’s core. In contrast, compression or longitudinal waves can pass through a liquid and they do go through the core.

All waves carry energy, and the energy of earthquake waves is easy to observe based on the amount of damage left behind after the ground has stopped moving. Earthquakes can shake whole cities to the ground, performing the work of thousands of wrecking balls. The amount of energy in a wave is related to its amplitude. Large-amplitude earthquakes produce large ground displacements and greater damage. As earthquake waves spread out, their amplitude decreases, so there is less damage the farther they get from the source.

Grasp Check

What is the relationship between the propagation speed, frequency, and wavelength of the S-waves in an earthquake?

• The relationship between the propagation speed, frequency, and wavelength is v w = f λ . v w = f λ .
• The relationship between the propagation speed, frequency, and wavelength is v w = λ f . v w = λ f .

Virtual Physics

Wave on a string.

In this animation, watch how a string vibrates in slow motion by choosing the Slow Motion setting. Select the No End and Manual options, and wiggle the end of the string to make waves yourself. Then switch to the Oscillate setting to generate waves automatically. Adjust the frequency and the amplitude of the oscillations to see what happens. Then experiment with adjusting the damping and the tension.

Which of the settings—amplitude, frequency, damping, or tension—changes the amplitude of the wave as it propagates? What does it do to the amplitude?

• Frequency; it decreases the amplitude of the wave as it propagates.
• Frequency; it increases the amplitude of the wave as it propagates.
• Damping; it decreases the amplitude of the wave as it propagates.
• Damping; it increases the amplitude of the wave as it propagates.

Solving Wave Problems

Worked example, calculate the velocity of wave propagation: gull in the ocean.

Calculate the wave velocity of the ocean wave in the previous figure if the distance between wave crests is 10.0 m and the time for a seagull to bob up and down is 5.00 s.

The values for the wavelength ( λ = 10.0   m ) ( λ = 10.0   m ) and the period ( T = 5.00 s) ( T = 5.00 s) are given and we are asked to find v w v w Therefore, we can use v w = λ T v w = λ T to find the wave velocity.

Enter the known values into v w = λ T v w = λ T

This slow speed seems reasonable for an ocean wave. Note that in the figure, the wave moves to the right at this speed, which is different from the varying speed at which the seagull bobs up and down.

Calculate the Period and the Wave Velocity of a Toy Spring

The woman in Figure 13.3 creates two waves every second by shaking the toy spring up and down. (a)What is the period of each wave? (b) If each wave travels 0.9 meters after one complete wave cycle, what is the velocity of wave propagation?

Strategy FOR (A)

To find the period, we solve for T = 1 f T = 1 f , given the value of the frequency ( f = 2 s − 1 ). ( f = 2 s − 1 ).

Enter the known value into T = 1 f T = 1 f

Strategy FOR (B)

Since one definition of wavelength is the distance a wave has traveled after one complete cycle—or one period—the values for the wavelength ( λ = 0.9   m ) ( λ = 0.9   m ) as well as the frequency are given. Therefore, we can use v w = f λ v w = f λ to find the wave velocity.

Enter the known values into v w = f λ v w = f λ

v w = f λ = ( 2  s −1 )(0 .9 m) = 1 .8 m/s . v w = f λ = ( 2  s −1 )(0 .9 m) = 1 .8 m/s .

We could have also used the equation v w = λ T v w = λ T to solve for the wave velocity since we already know the value of the period ( T = 0.5 s) ( T = 0.5 s) from our calculation in part (a), and we would come up with the same answer.

Practice Problems

The frequency of a wave is 10 Hz. What is its period?

• The period of the wave is 100 s.
• The period of the wave is 10 s.
• The period of the wave is 0.01 s.
• The period of the wave is 0.1 s.

What is the velocity of a wave whose wavelength is 2 m and whose frequency is 5 Hz?

Check Your Understanding

Use these questions to assess students’ achievement of the section’s Learning Objectives. If students are struggling with a specific objective, these questions will help identify such objective and direct them to the relevant content.

What is the amplitude of a wave?

• A quarter of the total height of the wave
• Half of the total height of the wave
• Two times the total height of the wave
• Four times the total height of the wave
• The wavelength is the distance between adjacent identical parts of a wave, parallel to the direction of propagation.
• The wavelength is the distance between adjacent identical parts of a wave, perpendicular to the direction of propagation.
• The wavelength is the distance between a crest and the adjacent trough of a wave, parallel to the direction of propagation.
• The wavelength is the distance between a crest and the adjacent trough of a wave, perpendicular to the direction of propagation.
• f = ( 1 T ) 2
• f = ( T ) 2

When is the wavelength directly proportional to the period of a wave?

• When the velocity of the wave is halved
• When the velocity of the wave is constant
• When the velocity of the wave is doubled
• When the velocity of the wave is tripled

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Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute Texas Education Agency (TEA). The original material is available at: https://www.texasgateway.org/book/tea-physics . Changes were made to the original material, including updates to art, structure, and other content updates.

Access for free at https://openstax.org/books/physics/pages/1-introduction

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16.1 Traveling Waves

Learning objectives.

By the end of this section, you will be able to:

• Describe the basic characteristics of wave motion
• Define the terms wavelength, amplitude, period, frequency, and wave speed
• Explain the difference between longitudinal and transverse waves, and give examples of each type
• List the different types of waves

We saw in Oscillations that oscillatory motion is an important type of behavior that can be used to model a wide range of physical phenomena. Oscillatory motion is also important because oscillations can generate waves, which are of fundamental importance in physics. Many of the terms and equations we studied in the chapter on oscillations apply equally well to wave motion ( Figure ).

Types of Waves

A wave is a disturbance that propagates, or moves from the place it was created. There are three basic types of waves: mechanical waves, electromagnetic waves, and matter waves.

Basic mechanical waves are governed by Newton’s laws and require a medium. A medium is the substance a mechanical waves propagates through, and the medium produces an elastic restoring force when it is deformed. Mechanical waves transfer energy and momentum, without transferring mass. Some examples of mechanical waves are water waves, sound waves, and seismic waves. The medium for water waves is water; for sound waves, the medium is usually air. (Sound waves can travel in other media as well; we will look at that in more detail in Sound .) For surface water waves, the disturbance occurs on the surface of the water, perhaps created by a rock thrown into a pond or by a swimmer splashing the surface repeatedly. For sound waves, the disturbance is a change in air pressure, perhaps created by the oscillating cone inside a speaker or a vibrating tuning fork. In both cases, the disturbance is the oscillation of the molecules of the fluid. In mechanical waves, energy and momentum transfer with the motion of the wave, whereas the mass oscillates around an equilibrium point. (We discuss this in Energy and Power of a Wave .) Earthquakes generate seismic waves from several types of disturbances, including the disturbance of Earth’s surface and pressure disturbances under the surface. Seismic waves travel through the solids and liquids that form Earth. In this chapter, we focus on mechanical waves.

Electromagnetic waves are associated with oscillations in electric and magnetic fields and do not require a medium. Examples include gamma rays, X-rays, ultraviolet waves, visible light, infrared waves, microwaves, and radio waves. Electromagnetic waves can travel through a vacuum at the speed of light, [latex]v=c=2.99792458\times {10}^{8}\,\text{m/s}.[/latex] For example, light from distant stars travels through the vacuum of space and reaches Earth. Electromagnetic waves have some characteristics that are similar to mechanical waves; they are covered in more detail in Electromagnetic Waves in volume 2 of this text.

Matter waves are a central part of the branch of physics known as quantum mechanics. These waves are associated with protons, electrons, neutrons, and other fundamental particles found in nature. The theory that all types of matter have wave-like properties was first proposed by Louis de Broglie in 1924. Matter waves are discussed in Photons and Matter Waves in the third volume of this text.

Mechanical Waves

Mechanical waves exhibit characteristics common to all waves, such as amplitude, wavelength, period, frequency, and energy. All wave characteristics can be described by a small set of underlying principles.

The simplest mechanical waves repeat themselves for several cycles and are associated with simple harmonic motion. These simple harmonic waves can be modeled using some combination of sine and cosine functions. For example, consider the simplified surface water wave that moves across the surface of water as illustrated in Figure . Unlike complex ocean waves, in surface water waves, the medium, in this case water, moves vertically, oscillating up and down, whereas the disturbance of the wave moves horizontally through the medium. In Figure , the waves causes a seagull to move up and down in simple harmonic motion as the wave crests and troughs (peaks and valleys) pass under the bird. The crest is the highest point of the wave, and the trough is the lowest part of the wave. The time for one complete oscillation of the up-and-down motion is the wave’s period T . The wave’s frequency is the number of waves that pass through a point per unit time and is equal to [latex]f=1\text{/}T.[/latex] The period can be expressed using any convenient unit of time but is usually measured in seconds; frequency is usually measured in hertz (Hz), where [latex]1\,{\text{Hz}=1\,\text{s}}^{-1}.[/latex]

The length of the wave is called the wavelength and is represented by the Greek letter lambda [latex](\lambda )[/latex], which is measured in any convenient unit of length, such as a centimeter or meter. The wavelength can be measured between any two similar points along the medium that have the same height and the same slope. In Figure , the wavelength is shown measured between two crests. As stated above, the period of the wave is equal to the time for one oscillation, but it is also equal to the time for one wavelength to pass through a point along the wave’s path.

The amplitude of the wave ( A ) is a measure of the maximum displacement of the medium from its equilibrium position. In the figure, the equilibrium position is indicated by the dotted line, which is the height of the water if there were no waves moving through it. In this case, the wave is symmetrical, the crest of the wave is a distance [latex]\text{+}A[/latex] above the equilibrium position, and the trough is a distance [latex]\text{−}A[/latex] below the equilibrium position. The units for the amplitude can be centimeters or meters, or any convenient unit of distance.

The water wave in the figure moves through the medium with a propagation velocity [latex]\mathbf{\overset{\to }{v}}.[/latex] The magnitude of the wave velocity is the distance the wave travels in a given time, which is one wavelength in the time of one period, and the wave speed is the magnitude of wave velocity. In equation form, this is

This fundamental relationship holds for all types of waves. For water waves, v is the speed of a surface wave; for sound, v is the speed of sound; and for visible light, v is the speed of light.

Transverse and Longitudinal Waves

We have seen that a simple mechanical wave consists of a periodic disturbance that propagates from one place to another through a medium. In Figure (a), the wave propagates in the horizontal direction, whereas the medium is disturbed in the vertical direction. Such a wave is called a transverse wave . In a transverse wave, the wave may propagate in any direction, but the disturbance of the medium is perpendicular to the direction of propagation. In contrast, in a longitudinal wave or compressional wave, the disturbance is parallel to the direction of propagation. Figure (b) shows an example of a longitudinal wave. The size of the disturbance is its amplitude A and is completely independent of the speed of propagation v .

A simple graphical representation of a section of the spring shown in Figure (b) is shown in Figure . Figure (a) shows the equilibrium position of the spring before any waves move down it. A point on the spring is marked with a blue dot. Figure (b) through (g) show snapshots of the spring taken one-quarter of a period apart, sometime after the end of` the spring is oscillated back and forth in the x -direction at a constant frequency. The disturbance of the wave is seen as the compressions and the expansions of the spring. Note that the blue dot oscillates around its equilibrium position a distance A , as the longitudinal wave moves in the positive x -direction with a constant speed. The distance A is the amplitude of the wave. The y -position of the dot does not change as the wave moves through the spring. The wavelength of the wave is measured in part (d). The wavelength depends on the speed of the wave and the frequency of the driving force.

Waves may be transverse, longitudinal, or a combination of the two. Examples of transverse waves are the waves on stringed instruments or surface waves on water, such as ripples moving on a pond. Sound waves in air and water are longitudinal. With sound waves, the disturbances are periodic variations in pressure that are transmitted in fluids. Fluids do not have appreciable shear strength, and for this reason, the sound waves in them are longitudinal waves. Sound in solids can have both longitudinal and transverse components, such as those in a seismic wave. Earthquakes generate seismic waves under Earth’s surface with both longitudinal and transverse components (called compressional or P-waves and shear or S-waves, respectively). The components of seismic waves have important individual characteristics—they propagate at different speeds, for example. Earthquakes also have surface waves that are similar to surface waves on water. Ocean waves also have both transverse and longitudinal components.

Wave on a String

A student takes a 30.00-m-long string and attaches one end to the wall in the physics lab. The student then holds the free end of the rope, keeping the tension constant in the rope. The student then begins to send waves down the string by moving the end of the string up and down with a frequency of 2.00 Hz. The maximum displacement of the end of the string is 20.00 cm. The first wave hits the lab wall 6.00 s after it was created. (a) What is the speed of the wave? (b) What is the period of the wave? (c) What is the wavelength of the wave?

• The speed of the wave can be derived by dividing the distance traveled by the time.
• The period of the wave is the inverse of the frequency of the driving force.
• The wavelength can be found from the speed and the period [latex]v=\lambda \text{/}T.[/latex]
• The first wave traveled 30.00 m in 6.00 s: [latex]v=\frac{30.00\,\text{m}}{6.00\,\text{s}}=5.00\frac{\text{m}}{\text{s}}.[/latex]
• The period is equal to the inverse of the frequency: [latex]T=\frac{1}{f}=\frac{1}{2.00\,{\text{s}}^{-1}}=0.50\,\text{s}.[/latex]
• The wavelength is equal to the velocity times the period: [latex]\lambda =vT=5.00\frac{\text{m}}{\text{s}}(0.50\,\text{s})=2.50\,\text{m}.[/latex]

Significance

The frequency of the wave produced by an oscillating driving force is equal to the frequency of the driving force.

Check Your Understanding

When a guitar string is plucked, the guitar string oscillates as a result of waves moving through the string. The vibrations of the string cause the air molecules to oscillate, forming sound waves. The frequency of the sound waves is equal to the frequency of the vibrating string. Is the wavelength of the sound wave always equal to the wavelength of the waves on the string?

The wavelength of the waves depends on the frequency and the velocity of the wave. The frequency of the sound wave is equal to the frequency of the wave on the string. The wavelengths of the sound waves and the waves on the string are equal only if the velocities of the waves are the same, which is not always the case. If the speed of the sound wave is different from the speed of the wave on the string, the wavelengths are different. This velocity of sound waves will be discussed in Sound .

Characteristics of a Wave

A transverse mechanical wave propagates in the positive x -direction through a spring (as shown in Figure (a)) with a constant wave speed, and the medium oscillates between [latex]\text{+}A[/latex] and [latex]\text{−}A[/latex] around an equilibrium position. The graph in Figure shows the height of the spring ( y ) versus the position ( x ), where the x -axis points in the direction of propagation. The figure shows the height of the spring versus the x -position at [latex]t=0.00\,\text{s}[/latex] as a dotted line and the wave at [latex]t=3.00\,\text{s}[/latex] as a solid line. (a) Determine the wavelength and amplitude of the wave. (b) Find the propagation velocity of the wave. (c) Calculate the period and frequency of the wave.

• The amplitude and wavelength can be determined from the graph.
• Since the velocity is constant, the velocity of the wave can be found by dividing the distance traveled by the wave by the time it took the wave to travel the distance.
• The period can be found from [latex]v=\frac{\lambda }{T}[/latex] and the frequency from [latex]f=\frac{1}{T}.[/latex]

• The distance the wave traveled from time [latex]t=0.00\,\text{s}[/latex] to time [latex]t=3.00\,\text{s}[/latex] can be seen in the graph. Consider the red arrow, which shows the distance the crest has moved in 3 s. The distance is [latex]8.00\,\text{cm}-2.00\,\text{cm}=6.00\,\text{cm}.[/latex] The velocity is [latex]v=\frac{\Delta x}{\Delta t}=\frac{8.00\,\text{cm}-2.00\,\text{cm}}{3.00\,\text{s}-0.00\,\text{s}}=2.00\,\text{cm/s}.[/latex]
• The period is [latex]T=\frac{\lambda }{v}=\frac{8.00\,\text{cm}}{2.00\,\text{cm/s}}=4.00\,\text{s}[/latex] and the frequency is [latex]f=\frac{1}{T}=\frac{1}{4.00\,\text{s}}=0.25\,\text{Hz}.[/latex]

Note that the wavelength can be found using any two successive identical points that repeat, having the same height and slope. You should choose two points that are most convenient. The displacement can also be found using any convenient point.

The propagation velocity of a transverse or longitudinal mechanical wave may be constant as the wave disturbance moves through the medium. Consider a transverse mechanical wave: Is the velocity of the medium also constant?

• A wave is a disturbance that moves from the point of origin with a wave velocity v .
• A wave has a wavelength [latex]\lambda[/latex], which is the distance between adjacent identical parts of the wave. Wave velocity and wavelength are related to the wave’s frequency and period by [latex]v=\frac{\lambda }{T}=\lambda f.[/latex]
• Mechanical waves are disturbances that move through a medium and are governed by Newton’s laws.
• Electromagnetic waves are disturbances in the electric and magnetic fields, and do not require a medium.
• Matter waves are a central part of quantum mechanics and are associated with protons, electrons, neutrons, and other fundamental particles found in nature.
• A transverse wave has a disturbance perpendicular to the wave’s direction of propagation, whereas a longitudinal wave has a disturbance parallel to its direction of propagation.

Conceptual Questions

Give one example of a transverse wave and one example of a longitudinal wave, being careful to note the relative directions of the disturbance and wave propagation in each.

A sinusoidal transverse wave has a wavelength of 2.80 m. It takes 0.10 s for a portion of the string at a position x to move from a maximum position of [latex]y=0.03\,\text{m}[/latex] to the equilibrium position [latex]y=0.[/latex] What are the period, frequency, and wave speed of the wave?

What is the difference between propagation speed and the frequency of a mechanical wave? Does one or both affect wavelength? If so, how?

Propagation speed is the speed of the wave propagating through the medium. If the wave speed is constant, the speed can be found by [latex]v=\frac{\lambda }{T}=\lambda f.[/latex] The frequency is the number of wave that pass a point per unit time. The wavelength is directly proportional to the wave speed and inversely proportional to the frequency.

Consider a stretched spring, such as a slinky. The stretched spring can support longitudinal waves and transverse waves. How can you produce transverse waves on the spring? How can you produce longitudinal waves on the spring?

Consider a wave produced on a stretched spring by holding one end and shaking it up and down. Does the wavelength depend on the distance you move your hand up and down?

No, the distance you move your hand up and down will determine the amplitude of the wave. The wavelength will depend on the frequency you move your hand up and down, and the speed of the wave through the spring.

A sinusoidal, transverse wave is produced on a stretched spring, having a period T . Each section of the spring moves perpendicular to the direction of propagation of the wave, in simple harmonic motion with an amplitude A . Does each section oscillate with the same period as the wave or a different period? If the amplitude of the transverse wave were doubled but the period stays the same, would your answer be the same?

An electromagnetic wave, such as light, does not require a medium. Can you think of an example that would support this claim?

Storms in the South Pacific can create waves that travel all the way to the California coast, 12,000 km away. How long does it take them to travel this distance if they travel at 15.0 m/s?

Waves on a swimming pool propagate at 0.75 m/s. You splash the water at one end of the pool and observe the wave go to the opposite end, reflect, and return in 30.00 s. How far away is the other end of the pool?

[latex]2d=vt\Rightarrow d=11.25\,\text{m}[/latex]

Wind gusts create ripples on the ocean that have a wavelength of 5.00 cm and propagate at 2.00 m/s. What is their frequency?

How many times a minute does a boat bob up and down on ocean waves that have a wavelength of 40.0 m and a propagation speed of 5.00 m/s?

Scouts at a camp shake the rope bridge they have just crossed and observe the wave crests to be 8.00 m apart. If they shake the bridge twice per second, what is the propagation speed of the waves?

What is the wavelength of the waves you create in a swimming pool if you splash your hand at a rate of 2.00 Hz and the waves propagate at a wave speed of 0.800 m/s?

[latex]v=f\lambda \Rightarrow \lambda =0.400\,\text{m}[/latex]

What is the wavelength of an earthquake that shakes you with a frequency of 10.0 Hz and gets to another city 84.0 km away in 12.0 s?

Radio waves transmitted through empty space at the speed of light [latex](v=c=3.00\times {10}^{8}\,\text{m/s})[/latex] by the Voyager spacecraft have a wavelength of 0.120 m. What is their frequency?

Your ear is capable of differentiating sounds that arrive at each ear just 0.34 ms apart, which is useful in determining where low frequency sound is originating from. (a) Suppose a low-frequency sound source is placed to the right of a person, whose ears are approximately 18 cm apart, and the speed of sound generated is 340 m/s. How long is the interval between when the sound arrives at the right ear and the sound arrives at the left ear? (b) Assume the same person was scuba diving and a low-frequency sound source was to the right of the scuba diver. How long is the interval between when the sound arrives at the right ear and the sound arrives at the left ear, if the speed of sound in water is 1500 m/s? (c) What is significant about the time interval of the two situations?

(a) Seismographs measure the arrival times of earthquakes with a precision of 0.100 s. To get the distance to the epicenter of the quake, geologists compare the arrival times of S- and P-waves, which travel at different speeds. If S- and P-waves travel at 4.00 and 7.20 km/s, respectively, in the region considered, how precisely can the distance to the source of the earthquake be determined? (b) Seismic waves from underground detonations of nuclear bombs can be used to locate the test site and detect violations of test bans. Discuss whether your answer to (a) implies a serious limit to such detection. (Note also that the uncertainty is greater if there is an uncertainty in the propagation speeds of the S- and P-waves.)

a. The P-waves outrun the S-waves by a speed of [latex]v=3.20\,\text{km/s;}[/latex] therefore, [latex]\Delta d=0.320\,\text{km}.[/latex] b. Since the uncertainty in the distance is less than a kilometer, our answer to part (a) does not seem to limit the detection of nuclear bomb detonations. However, if the velocities are uncertain, then the uncertainty in the distance would increase and could then make it difficult to identify the source of the seismic waves.

A Girl Scout is taking a 10.00-km hike to earn a merit badge. While on the hike, she sees a cliff some distance away. She wishes to estimate the time required to walk to the cliff. She knows that the speed of sound is approximately 343 meters per second. She yells and finds that the echo returns after approximately 2.00 seconds. If she can hike 1.00 km in 10 minutes, how long would it take her to reach the cliff?

A quality assurance engineer at a frying pan company is asked to qualify a new line of nonstick-coated frying pans. The coating needs to be 1.00 mm thick. One method to test the thickness is for the engineer to pick a percentage of the pans manufactured, strip off the coating, and measure the thickness using a micrometer. This method is a destructive testing method. Instead, the engineer decides that every frying pan will be tested using a nondestructive method. An ultrasonic transducer is used that produces sound waves with a frequency of [latex]f=25\,\text{kHz}.[/latex] The sound waves are sent through the coating and are reflected by the interface between the coating and the metal pan, and the time is recorded. The wavelength of the ultrasonic waves in the coating is 0.076 m. What should be the time recorded if the coating is the correct thickness (1.00 mm)?

16.1 Traveling Waves Copyright © 2016 by OpenStax. All Rights Reserved.

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Travelling Waves of Voltage & Currents in Circuits | Electrical Engineering

In this article we will discuss about:- 1. Introduction to Travelling Waves 2. Surge Impedance and Velocity of Propagation 3. Specifications 4. Reflection and Refraction5. Typical Cases of Line Terminations 6. Equivalent Circuit 7. Bifurcated Line 8. Reactive Termination 9. Successive Reflections, Bewley Lattice or Zigzag Diagram. 

Introduction to Travelling Waves :

A transmission line is a distributed parameter circuit and a distinguishing feature of such a circuit is its ability to support travelling waves of voltage and current. A circuit with distributed parameters has a finite velocity of electro­magnetic field propagation. In such a circuit the changes in voltage and current, owing to switching and lightning, do not occur simultaneously in all parts of the circuit but spread out in the form of travelling waves or surges.

When a transmission line is suddenly connected to a voltage source by closing of a switch, the whole of the line is not energized all at once (the voltage does not appear instan­taneously at the other end). This is due to presence of distributed constants (inductance and capacitance in a loss free line).

When switch S is closed, the inductance L 1 acts as an open circuit and C 1 as short circuit instantaneously. The same instant next section cannot be charged because the voltage across capacitor C 1 is zero. So unless the capacitor C 1 is charged to some value whatsoever, charging of the capacitor C 2 through L 2 is not possible which, of course, will take some finite time.

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The same argument applies to the third section, fourth section and so on. So we see that the voltage at the successive sections builds up gradually. This gradual build-up of voltage over the transmission line conductors can be regarded as though a voltage wave is travelling from one end to the other end and the gradual charging of the capacitances is due to associated current wave. The current wave, which is accompanied by a voltage wave sets up a magnetic field in the surrounding space.

At junctions and terminations these surges undergo reflections and refractions. In an extensive network with many lines and junctions, the number of travelling waves initiated by a single incident wave will mushroom at a considerable rate as the waves split and multiple reflections occur.

It is true that the total energy of the resultant waves cannot exceed the energy of the incident wave. However, it is possible for the voltage to build up at certain junctions due to reinforcing action of several waves. For a complete study of the phenomenon the use of Bewley lattice diagram or digital computer is necessary. In this article we will study some of the elementary aspects of wave propa­gation, reflections and refractions.

Surge Impedance and Velocity of Propagation :

The gradual establishment of line voltage can be considered as due to voltage wave travelling from the supply source end towards the far end, and the progressive charging of the line capacitances will account for the associated current wave.

Assume that in a very small time δt the conditions of a current I and a voltage E are established along a length δx of the line (Fig. 8.2). The emf E is balanced by the back emf generated by the magnetic flux which is produced by the current in this length of the line. The inductance of the length δx is Lδx, (L is inductance of line per unit length) so that the flux built up is IL δx and the back emf is the rate of build-up viz. IL δx/δt.

So we have E = [IL (δx/δt)] = ILv                           …(8.1)

where v is the velocity of propagation of wave.

The current I carries a charge I δt in the time δt, and this charge remains on the line to charge it up to the potential of E.

Since the capacitance of length δx of the line is C δx (C is the capacitance of the line per unit length), its charge is ECδx, so we have –

I δt = EC δx

or I = [EC (δx/δt)] = EC v                                          …(8.2)

The switching of an emf E on to the line results therefore in a wave of current I and velocity v where I and v are given by Eqs. (8.1) and (8.2).

Dividing Eq. (8.1) by Eq. (8.2), we have –

E/I = ILv/ECv = I/E . L/C

or E 2 /I 2 = L/C

or E/I = √L/C = Z n (say)                                            …(8.3)

Crest of the wave is the maximum amplitude of the wave and is usually expressed in kV or kA.

Front of the wave is the portion of the wave before crest and is expressed in time from beginning of the wave to the crest value in ms or µs. However, for waves having a slow initial rate of rise, i.e., a long toe, it is better to consider virtual front as determined by straight lines between 30% and 90% points [Fig. 8.3(b)]. The virtual front is 1.667 × t 1 . The extension of this straight line to X-axis gives O 1 i.e., the virtual zero.

Tail of the wave is the portion beyond the crest. It is expressed in time (µs) from beginning of the wave to the point where the wave has reduced to 50% of its value at crest. For waves having a slow initial rate of rise, i.e., long toe, the tail time is measured from O 1 to 50% value on tail [Fig. 8.3(b)],

4. Polarity:

Polarity is the polarity of crest voltage or current. A positive wave of 500 kV crest, 1 µs front and 25 µs tail will be represented as + 500/1.0/25.0.

A travelling wave can be represented mathematically in a number of ways. The simplest and most commonly used representation is the infinite rectangular or step wave illustrated in Fig. 8.4. Such a wave jumps suddenly from zero to full value and is maintained at that value there-after.

As this wave has front causing maximum gradients and sustained tail producing maximum oscillations in machine windings, it is most dangerous to apparatus/equipment. Hence, the analysis based on it is liable to err on the safer side.

Reflection and Refraction of Travelling Waves :

If a travelling wave arrives at a point where the impedance suddenly changes the wave is partly transmitted and partly reflected. Loading points, line-cable junctions and even faults constitute such discontinuities. Independent waves meeting along a line will combine in accordance with their polarity to provide different voltage and current levels at the meeting point.

It is convenient to adopt a standard sign convention, and in what follows, forward waves of current and voltage are given the same polarity. If the wave is being reflected the corresponding current and voltage waves are given opposite polarity. This may be illustrated by considering waves of current and voltage being transmitted along a line of characteristic impedance Z C terminated by an impedance Z (Fig. 8.5).

Let E and I represent the incident waves, E T and I T represent the transmitted (or refracted) waves and E R and I R the reflected waves. The state of affairs is illustrated in Fig. 8.6.

The following relations hold good for incident, transmitted and reflected voltage and current waves –  

E = I Z C                                            …(8.7a)

E T = I T Z                                         …(8.7b)

E R = – I R Z C                                    …(8.7c)

The negative sign in Eq. (8.7c) is because of the fact that E R and I R are travelling in the negative direction of x or backwards on the same line.

The transmitted voltage and current will be respectively the algebraic sum of incident and reflected voltage and current waves.

E T   = E + E R                                    …(8.8a)

I T = I + I R                                      …(8.8b)

Substituting the values of I, I R and I T from Eqs. (8.7a, b, c) in Eq. (8.8 b), we have –

E T /Z = [(E/Z C ) – (E R /Z C )]         …(8.9)

From Eqs. (8.8 a) and (8.9), we have –

E T /Z = [(E/Z C ) – (E T – E/Z C )]  

or E T x Z C /Z + E T = 2 E

or E T [1 + (Z C /Z)] = 2 E

or E T = {E [2Z/(Z + Z C )]}                                            …(8.10)

I T = (E T /Z) = [2 E/(Z + Z C )] = (2 Z C I/Z + Z C )         …(8.11)

E R = E T – E = {[2 ZE/(Z + Z C )] – E} = [E (Z – Z C )/(Z + Z C )]                              …(8.12)

or I R = (– E R /Z C )= [(– E/Z C )x (Z – Z C /Z + Z C )] = [I (Z C Z/Z C + Z)]                …(8.13)

The coefficients (2Z/Z + Z C )and (Z – Z C /Z + Z C )are called coefficients of refraction and reflection respectively.

It will be observed that the transmitted or refracted current and voltage always have positive polarity. The polarity of the reflected waves depends on the magnitude relationship between Z C and Z. If Z C > Z, the voltage wave is negative and the current wave positive, but vice-versa if Z > Z C .

Typical Cases of Line Terminations :

1. Short-Circuited Line :

If the line is short circuited at the receiving end, i.e., Z = 0, then the transmitted and reflected waves are given as – 

E T = 0                                    …(8.14 a)

I T = 2 I                                  …(8.14 b)

E R = – E                                …(8.14 c)

I R = I                                     …(8.14 d)

The unique characteristic of the short circuit is that voltage across it is zero. When an incident voltage wave E arrives a short circuit, the reflected voltage wave must be – E to satisfy the condition that the voltage across the short circuit is zero.

The waves are shown in Fig. 8.7:

2. Open-Circuited Line :

If the line is open circuited at the receiving end, i.e., Z is infinite, the transmitted and reflected waves are given as – 

E T = 2 E (doubling effect)                              …(8.15 a)

I T = 0                                                                       …(8.15b)

E R = E                                                                      …(8.15c)

I R = –  I                                                                   …(8.15 d)

An open circuit at the end of a line demands that the current at that point is always zero. Thus when an incident current wave I arrives at the open circuit, a reflected wave equal to – I is at once initiated to satisfy the boundary condition. The waves are shown in Fig. 8.8.

3. Line Terminated by an Impedance Equal to Surge Impedance :

If the line is terminated with an impedance equal to surge impedance, i.e., Z = Z C , we have –

E T = E                                                     …(8.16 a)

I T = I                                                      …(8.16 b)

E R = 0                                                    …(8.16 c)

I R = 0                                                    …(8.16 d)

It means that the line is correctly terminated and there will be no reflection and E T and I T will be equal respectively to E and I.

4. Line Connected to a Cable :

A wave travelling over the line and entering the cable, as shown in Fig. 8.9, looks into a different impedance and, therefore, it suffers reflection and refraction at the junction.

The refracted or transmitted voltage is given as –

E T = 2 Z 2 E/Z 1 + Z 2                             …(8.17)

The other waves can be had by employing the relations of Eqs. (8.11), (8.12) and (8.13). The surge impedances of the overhead line and cable are approximately 500 Ω and 50 Ω respectively. With these values it can be seen that the voltage entering the cable will be –

E T = {E x [(2 x 50)/(50 + 500)]} = 2/11 E

or roughly 20% of the incident voltage.

This is the reason that an overhead line is terminated near a station by connecting the station equipment to the overhead line through a short length of underground cable. Besides the reduction in magnitude of the voltage wave, it also reduces the steepness of the wave. It is because of capacitance of the cable.

The reduction in steepness is very important because it is one of the factors for reducing the voltage distribution along the equipment windings. In connecting the overhead line to a station equipment through a cable the important point to be remembered is that the length of the cable should not be shorter than the expected length of the wave otherwise successive reflections at the junction may cause piling up of voltage and the voltage at the junction may attain the value of incident voltage.

Equivalent Circuit for Travelling Wave Studies :

An equivalent circuit for studies of travelling waves can be easily developed by making use of Thevenin’s theorem that provides a mathematical technique for replacing a two terminal network by a voltage source V T , and resistance R T connected in series.

The voltage source V T , called the Thevenin’s equivalent voltage, is the open-circuit voltage that appears across the load terminals when the load is removed or disconnected and resistance R T , called the Thevenin’s equivalent resistance, is equal to the resistance of the network looking back into the load terminals.

A line with a surge impedance Z C is shown in Fig. 8.10(a). A surge wave E travels on the line towards the terminals XX’. The open-circuit voltage across the terminals XX’ is 2 E and the impedance seen from the terminals XX’, with the sending end of the line short circuited is Z C . Thus the circuit given in Fig. 8.10(a) can be replaced by the equivalent circuit shown in Fig. 8.10(b).

From the equivalent circuit shown –  

Current through Z, I T = [2 E/(Z + Z C )]                  …(8.18)

and voltage across Z, E T = [2 EZ/(Z + Z C )]          …(8.19)

The above results are the same as Eqs. (8.11) and (8.10). The values of E R and I R can be determined from Eqs. 8.8(a) and (b). The equivalent circuit is very useful for the studies of travelling waves and can be employed for resistive as well as reactive terminations.

Bifurcated Line :

Let a line of natural impedance Z C bifurcate into two branches of natural impedances Z 1 and Z 2 . As far as the voltage wave is concerned, the refracted (transmitted) portion will be the same for both branches as they are in parallel but the refracted currents will be different as Z 1 ≠ Z 2 .

Let the incident wave be (E, I) travelling to the right, the reflected wave (E R , I R ) travelling to the left and the transmitted waves (E T , I T1 ) and (E T , I T2 ) travelling towards the right, as shown in Fig. 8.11(a).

The analysis is exactly similar to that of a line terminated by impedance Z and the desired results can be had by using Eqs. (8.10), (8.11), (8.12), (8.13) and (8.8a) and (8.8b). Equivalent circuit is shown in Fig. 8.11 (b).

From equivalent circuit shown in Fig. 8.11 (b), we have –  

Z = [Z 1 Z 2 /(Z 1 + Z 2 )]                                                                            …(8.20)

E T = {[2 E/(Z 1 Z 2 )/(Z 1 + Z 2 )] + Z C x [Z 1 Z 2 /(Z 1 + Z 2 )]}                …(8.21)

I T1 = E T /Z 1                                                                                              …(8.22)

and I T2 = E T /Z 2                                                                                      …(8.23)

E R = E T – E                                                                                             …(8.24)

I R = I T1 + I T2 – I                                                                                    …(8.25)

The above procedure can be extended for the analysis of a line branching off into any number of lines.

Reactive Termination :

1. Reflection from Terminal Inductance :

Let the line be terminated with an inductance L, as illustrated in Fig. 8.12.

The voltage across the inductance is given as –  

e T = e L = [L (di L /dt)]                                              …(8.26)

Since from Eqs. (8.86), (8.7a) and (8.7c)

i L = i T = i + i R = [(e/Z C ) – (e R /Z C )]                     …(8.27)

Substituting value of i L from Eq. (8.27) in Eq. (8.26), we have –

e L = [L (d/dt)] [(e/Z C ) – (e R /Z C )]                     …(8.28)

and since from Eq. (8.8a)

e L = e + e R   

So e + e R = [L (d/dt)] [(e/Z C ) – (e R /Z C )]

or {– e + [(L/Z C )(de/dt)]}= {e R + [(L/Z C )(de R /dt)]}                       …(8.29)                                                       

Let the incident wave e be of constant value E, so that its time derivative is zero. So Eq. (8.29) becomes –

[(L/Z C )(de R /dt)] = – (E + e R )

or [de R /(E + e R )] = (– Z C /L)dt                                     …(8.30)

Integrating both sides of above Eq. (8.30), we have –  

log e (E + e R ) = (– Z C /L)t + A                                        …(8.31)

where A is a constant of integration which can be evaluated from initial condition i.e., when t = 0, i L = 0, because when the wave arrives at the inductance terminal the inductance L does not carry any current, the whole of it is reflected. It means that when t = 0, e R = E and therefore

log e (E + E) = 0 + A

or A = log e 2 E

Substituting A = log e 2 E in Eq. (8.31), we have –

log e (E + e R ) = (– Z C /L)t + log e 2 E

or log e [(E + e R )/2 E] = (– Z C /L)t

Also when the potential difference across the inductance is zero, the current passing though it will be twice that of incident current (re­fer to Eq. 8.14b)

Again e L = e + e R   [refer to Eq. (8.8a)]

= E + E (2 e (– ZC/L)t – 1) = 2 Ee (– ZC/L)t                      …(8.33)       

Current through inductance,  i L = [(E/Z C )– (e R /Z C )]

The capacitance termination is of more practical importance owing to the paradox that for lightning waves a transformer acts as a capacitance rather than as an inductance. The lightning surge arrives at a transformer as a travelling wave, and the front of the wave is so steep and rises so suddenly to a maximum, that there is practically no time for current to start to flow through the large inductance of the transformer winding.

However, there is some small capacitance in the transformer (between turns and between winding and the core), and the transformer’s reaction to the lightning surge is governed largely by this capacitance, rather than by inductance.

3. Reflection from a Line Terminated with a Parallel Combination of Capacitance and Resistance:

When a travelling wave meets a termination composed of parallel combination of a capacitance C and resistance R, as illustrated in Fig. 8.16, the problem is the same as that illustrated in Fig. 8.15, except that Z in Eqs. (8.10) and (8.11) must be replaced by

The voltage at the termination is thus

It is usual to specify the condition of the wave front by stating the time the wave takes to increase from 10 to 90 percent of its value and multiplying by 1.25. If the wave attains x of its value in time t.

Now, the increment of the voltage at the receiving end due to any reflection is twice the amplitude of the incident wave, because of the reflection without change of sign. Also, the final voltage at this end is the sum to infinity of all such increments.

Thus, in the above example, it is –  

2 (0.754 – 0.4287 + 0.2437 – …).

It is simpler to express the series generally in term of α, thus,  

At the receiving end the increment of voltage is the sum of the incident and reflected waves at each reflection, so that the ultimate voltage at this point is the sum to infinity of the series.

The above transmission line can be represented by its equivalent circuit having L and C distributed over the whole line as shown below.

When the switch S is closed, the voltage at the load end does not appear immediately at the load end. As soon as the Switch S is closed, inductance L1 acts as open circuit and capacitance C1 acts as short circuit. Therefore as far as the capacitor C1 is not charged to some value, the charging of C2 through L2 is not possible. This means that charging of C2 through L2 will take some finite time. Similar reasoning applies to the other successive sections. Thus we see that whenever switch S is closed, there is a gradual voltage build up from the source end to the load end over the transmission line. This gradual voltage build up can be thought of due to a voltage wave travelling from one end to the other and the gradual charging of capacitor through inductor is due to current wave.

Let us understand this voltage and current wave in different wave by taking one analogy. Replace the switch S by water valve and each section of inductor and capacitor by a water tank as shown below.

When the water valve is opened, tank A will first fill up to the level of interconnection between the tanks due to flow of water from the valve. This water flow is the flow of current through the inductor L. And the tank level is nothing but the voltage developed across the shunt capacitor C. Once the tank A is filled up, tank B will start filling and when it is also filled up to the level, tank C will start to fill. This means that the filling of tank C will be complete after some finite time and not immediately. Thus the flow of water from tank A to tank C can be assumed as a wave propagating from tank A to tank C. Similarly, the tank levels can also be thought of a wave moving from tank A to tank C. Hope you understood the phenomenon of travelling wave on transmission line by this simple analogy.

Till now, we understood that there are current and voltage wave travelling over the line. Now we want to get the relationship between these two waves.

Relationship between Voltage and Current Wave:

Let the voltage wave and current wave travels a distance x in time t. Therefore the inductance and capacitance of line up to distance x will be Lx and Cx respectively. Let this wave travels a distance dx in time dt.

Since line is assumed to be lossless, whatever is the value of voltage wave and current wave at the beginning, the same will be at any time t. This means that, the magnitude of voltage and current wave at time t will be V and I respectively.

Hence the stored charge in shunt capacitance Q = VCx

and the flux in the series inductance Ø = ILx

But I = dQ/dt

         = CVdx/dt

But dx/dt = velocity of travelling wave = ν (say)

Therefore, I = CVν …….(1)

The voltage developed across the shunt capacitance,

⇒V = ILν ……….(2)

Dividing equation (1) and (2), we get

V / I = IL / CV

(V/I) 2 = L/C

V/I = √(L/C)

The above expression is the ratio of voltage and current having the dimension of impedance. Therefore it is called Surge Impedance. Note that Surge Impedance is the square root of ratio of series inductance L per unit length of line and shunt capacitance C per unit length of line. This simply means that this value will remain constant for a given transmission line. This value will not change due to change in length of line. The value of surge impedance for a typical transmission line is around 400 Ohm and that for a cable is around 40 ohm. Notice that the value of surge impedance for cable is less than that of transmission line. This is due to the higher value of capacitance of cable compared to the transmission line.

Velocity of Travelling Wave:

To get velocity of travelling wave, multiply (1) and (2) as below.

VI = (CVν) x (LIν)

ν = √(1/LC)   ……..(3)

The above expression is the velocity of travelling wave. Since L and C are per unit values, the velocity of travelling wave is constant. For overhead line the values of L and C are given as

L = 2×10 -7 ln(d/r) Henry / m

C = 2πε / ln (d/r)

From (3), the velocity of travelling wave for overhead line

ν = 1 / [{2×10 -7 ln(d/r)}{ 2πε / ln (d/r)}] 1/2

   = 1/[4πεx10 -7 ] 1/2

The permittivity of air, ε = (1/36π) x 10 -9

ν = 1 / [4πx(1/36π) x 10 -9 x10 -7 ] 1/2

   = 3 x10 8 m/sec.

From the above expression, we can have following conclusions:

The velocity of travelling wave for a lossless line is equal to the speed of light.

Since the cable core is surrounded by insulations and sheath , its relative permittivity ε r >1 and hence ε = ε 0 ε r > ε 0 (permittivity of air). Therefore the speed of travelling wave on cable is less than that of transmission line.

3 thoughts on “Travelling Wave on Transmission Line”

Great Content! Really Insightful for the students. Didn’t Know That “The velocity of travelling wave for a lossless line is equal to the speed of light”

very well explained. thanks for sharing.

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Traveling Wave Effects in Microwave Quantum Photonics

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DMT alters cortical travelling waves

Andrea alamia.

1 Cerco, CNRS Université de Toulouse, Toulouse, France

Christopher Timmermann

2 Computational, Cognitive and Clinical Neuroscience Laboratory (C3NL), Faculty of Medicine, Imperial College, London, United Kingdom

3 Centre for Psychedelic Research, Division of Psychiatry, Department of Brain Sciences, Imperial College London, London, United Kingdom

David J Nutt

Rufin vanrullen.

4 Artificial and Natural Intelligence Toulouse Institute (ANITI), Toulouse, France

Robin L Carhart-Harris

Associated data.

Alamia A. 2020. DMT alters cortical travelling waves. Open Science Framework. [ CrossRef ]

The data and the code to perform the analysis are available at : https://osf.io/wujgp/ .

The following dataset was generated:

Psychedelic drugs are potent modulators of conscious states and therefore powerful tools for investigating their neurobiology. N,N, Dimethyltryptamine (DMT) can rapidly induce an extremely immersive state of consciousness characterized by vivid and elaborate visual imagery. Here, we investigated the electrophysiological correlates of the DMT-induced altered state from a pool of participants receiving DMT and (separately) placebo (saline) while instructed to keep their eyes closed. Consistent with our hypotheses, results revealed a spatio-temporal pattern of cortical activation (i.e. travelling waves) similar to that elicited by visual stimulation. Moreover, the typical top-down alpha-band rhythms of closed-eyes rest were significantly decreased, while the bottom-up forward wave was significantly increased. These results support a recent model proposing that psychedelics reduce the ‘precision-weighting of priors’, thus altering the balance of top-down versus bottom-up information passing. The robust hypothesis-confirming nature of these findings imply the discovery of an important mechanistic principle underpinning psychedelic-induced altered states.

Introduction

N,N, Dimethyltryptamine (DMT) is a mixed serotonin receptor agonist that occurs endogenously in several organisms ( Christian et al., 1977 ; Nichols, 2016 ) including humans ( Smythies et al., 1979 ), albeit in trace concentrations. DMT, which is a classic psychedelic drug, is also taken exogenously by humans to alter the quality of their consciousness. For example, synthesized compound is smoked or injected but it has also been used more traditionally in ceremonial contexts (e.g. in Amerindian rituals). When ingested orally, DMT is metabolized in the gastrointestinal (GI) system before reaching the brain. Its consumption has most traditionally occurred via drinking ‘Ayahuasca’, a brew composed of plant-based DMT and β –carbolines (monoamine oxidize inhibitors), which inhibit the GI breakdown of the DMT ( Buckholtz and Boggan, 1977 ). Modern scientific research has mostly focused on intravenously injected DMT. Administered in this way, DMT’s subjective effects have a rapid onset, reaching peak intensity after about 2–5 min and subsiding thereafter, with negligible effects felt after about 30 min ( Strassman, 2001 ; Strassman, 1995a ; Timmermann et al., 2019 ).

Previous electrophysiological studies investigating changes in spontaneous (resting state) brain function elicited by ayahuasca have reported consistent broadband decreases in oscillatory power ( Riba et al., 2002 ; Timmermann et al., 2019 ), while others have noted that the most marked decreases occur in α-band oscillations (8–12 Hz) ( Schenberg et al., 2015 ). Alpha decreases correlated inversely with the intensity of ayahuasca-induced visual hallucinations ( Valle et al., 2016 ) and are arguably the most reliable neurophysiological signature of the psychedelic state identified to-date ( Muthukumaraswamy et al., 2013 ) – with increased signal diversity or entropy being another particularly reliable biomarker ( Schartner et al., 2017 ). In the first EEG study of the effects of pure DMT on on-going brain activity, marked decrease in the α and β (13–30 Hz) band power was observed as well as increase in signal diversity ( Timmermann et al., 2019 ). Increase in lower frequency band power (δ = 0.5–4 Hz and θ = 4–7 Hz) also became evident when the signal was decomposed into its oscillatory component. Decreased alpha power and increased signal diversity correlated most strongly with DMT’s subjective effects – consolidating the view that these are principal signatures of the DMT state, if not the psychedelic state more broadly.

Focusing attention onto normal brain function, outside of the context of psychoactive drugs, electrophysiological recordings in cortical regions reveal distinct spatio-temporal dynamics during visual perception, which differ considerably from those observed during closed-eyes restfulness. It is possible to describe these dynamics as oscillatory ‘travelling waves’, i.e. fronts of rhythmic activity which propagate across regions in the cortical visual hierarchy ( Lozano-Soldevilla and VanRullen, 2019 ; Muller et al., 2014 ; Sato et al., 2012 ). Recent results showed that travelling waves can spread from occipital to frontal regions during visual perception, reflecting the forward bottom-up flow of information from lower to higher regions. Conversely, top-down propagation from higher to lower regions appears to predominate during quiet restfulness ( Alamia and VanRullen, 2019 ; Halgren et al., 2019 ; Pang et al., 2020 ).

Taken together these results compel us to ask how travelling waves may be affected by DMT, particularly given their association with predictive coding ( Alamia and VanRullen, 2019 ; Friston, 2019 ) and a recent predictive coding inspired hypothesis on the action of psychedelics (‘REBUS’) – which posits decreased top-down processing and increased bottom-up signal passing under these compounds ( Carhart-Harris and Friston, 2019 ). Moreover, DMT lends itself particularly well to the testing of this hypothesis as its visual effects are so pronounced. Given that visual perception is associated with an increasing in forward travelling waves and eyes-closed visual imagery under DMT can feel as if one is ‘seeing with eyes shut’ ( de Araujo et al., 2012 ) – does a consistent increase in forward travelling waves under DMT account for this phenomenon?

Here we sought to address these questions by quantifying the amount and direction of travelling waves in a sample of healthy participants who received DMT intravenously, during eyes-closed conditions. We hypothesized that DMT acts by disrupting the normal physiological balance between top-down and bottom-up information flow, in favour of the latter ( Carhart-Harris and Friston, 2019 ). Moreover, we ask: does this effect correlate with the vivid ‘visionary’ component of the DMT experience? Providing evidence in favour of this hypothesis would indicate that forward travelling waves do play a crucial role in conscious visual experience, irrespective of the presence of actual photic stimulation.

Quantifying travelling waves

As demonstrated by both theoretical and experimental evidence ( Nunez, 2000 ; Nunez and Srinivasan, 2014 ; Nunez and Srinivasan, 2009 ), in most systems, including the human brain, travelling waves occur in groups (or packets) over some range of spatial wavelengths having multiple spatial and temporal frequencies. Given any configurations of electrodes, only parts of these packets can be successfully detected, i.e. waves shorter than the spatial extent of the array, and waves longer than twice the electrode separation distance (Nyquist criterion in space). In scalp recordings, the shorter waves may be mostly removed by volume conduction. As a consequence, waves recorded directly from the cortex emphasize shorter waves than the scalp recorded waves. Specifically, in the case of small cortical arrays, the overlap between cortical and scalp data may be minimal, and the estimated wave properties (including propagation direction) may differ. Additionally, it is important to consider that when waves are travelling in multiple directions at nearly the same time in ‘closed’ systems (e.g. the cortical/white matter), waves either damp out or interfere with each other to form standing waves (e.g. alpha waves travelling both forward and backward). It is reasonable to assume that the behaviour of these properties will relate to global brain and mind states, and be sensitive to state-altering psychoactive drugs ( Nunez, 2000 ; Nunez and Srinivasan, 2014 ; Nunez and Srinivasan, 2009 ).

Practically, we measure the waves’ amount and direction with a method devised in our previous studies ( Alamia and VanRullen, 2019 ; Pang et al., 2020 ). We slide a one-second time-window over the EEG signals (with 0.5 s overlap). For each time-window, we generate a 2D map (time/electrodes) by stacking the signals from five central mid-line electrodes (Oz to FCz, see Figure 1 ). For each map, we then compute a 2D-FFT, in which the upper- and lower-left quadrant represent the power of forward (FW) and backward (BW) travelling waves, respectively (since the 2D-FFT is symmetrical around the origin, the lower- and upper-right quadrants contain the same information). From both quadrants we extracted the maximum values, representing the raw amount of FW and BW waves in that time-window. Next, we performed the same procedure after having shuffled the electrodes’ order, thereby disrupting spatial information (including the waves’ directionality) while retaining the same overall spectral power. In other words, the surrogate measures reflect the amount of waves expected solely due to the temporal fluctuations of the signal. After having computed the maximum values for the FW and BW waves of the surrogate 2D-FFT spectra one hundred times (and averaging the 100 values), we compute the net amount of FW and BW waves in decibel (dB), by applying the following formula:

From each 1 s EEG epoch we extract a 2D-map, obtained by stacking signals from five midline electrodes. For each map we compute a 2D-FFT in which the maximum values in the upper- and lower-left quadrants represent respectively the amount of forward (FW – in blue) and backward (BW – in red) waves. For each map, we also compute surrogate values by shuffling the electrodes’ order 100 times, so as to retain temporal fluctuations while disrupting the spatial structure of the signals (including any travelling waves). Eventually, we compute the wave strength in decibel (dB) by combining the real and the surrogate values.

where W represents the maximum value extracted for each quadrant (i.e. forward FW or backward BW), and Wss the respective surrogate value. Importantly, this value – expressed in decibel – represents the net amount of waves against the null distribution. In other words, it is informative to compare this value to zero, to assess the significance of waves. On the other hand, a direct comparison between FW and BW waves in each time-bin is not readily interpretable, as it is possible to simultaneously record waves propagating in both directions—as observed during visual stimulation epochs (see below). In addition, it’s important to note that our waves’ analysis focuses on the sensor level, as source projections presents a number of important limitations, such as impairing long-range connections, as well as smearing of signals due to scalp interference ( Alexander et al., 2019 ; Freeman and Barrie, 2000 ; Nunez, 1974 ).

Does DMT influence travelling waves?

After defining our measure of the waves’ amount and direction, we investigated whether the intake of DMT alters the cortical pattern of travelling waves. Participants underwent two sessions in which they were injected with either placebo or DMT (see Materials and methods for details). Importantly, during all of the experiments, participants rested in a semi-supine position, with their eyes closed. EEG recordings were collected 5 min prior to drug administration and up to 20 min after. The left column of Figure 2A shows the amount of BW and FW waves in the 5 min preceding and following drug injection (either placebo or DMT). Consistent with previous observations on independent data ( Alamia and VanRullen, 2019 ), during quiet closed-eyes restfulness a significant amount of BW waves spread from higher to lower regions (as confirmed by a Bayesian t-test against zero for both DMT and Placebo conditions, BFs 10  >>100, error <0.01%, 95% Credible Intervals (CI) DMT: [0.221, 0.637], Placebo: [0.273, 0.666]), whereas no significant waves propagate in the opposite FW direction (Bayesian t-test against zero: BFs 10  <0.15, error <0.01%; 95% CI DMT: [−0.424, 0.088], Placebo: [−0.372, 0.110]). However, after DMT injection, the cortical pattern changed drastically: the amount of BW waves decreased (but remaining significantly above zero – BFs 10  = 12.6, 95% CI: [0.057, 0.322]), whereas the amount of FW waves increased significantly above zero (BF 10  = 5.4 95% CI: [0.027, 0.336]). These results, obtained by comparing the amount of waves before and after injection (pre-post factor) of Placebo or DMT (drug factor), were confirmed by two Bayesian ANOVA performed separately on BW and FW waves (all factors including interactions reported BFs 10  >>100, error <2%), and were not confounded by differences in dosage (see Materials and methods and Figure 2—figure supplement 1 ). A power analysis comparing DMT and Placebo conditions after infusion for both FW and BW direction revealed values above 90% (FW case: μ DMT =0.19, μ PLACEBO = -0.20 and σ = 0.29 yields to power equals to 0.9168; BW case: μ DMT = 0.18, μ PLACEBO = 0.51 and σ = 0.25 gives power equals to 0.9205; in both cases, we considered a type I error rate of 5%).

( A ) In the left panels the net amount of FW (blue, upper panel) and BW (red, lower panel) waves is represented pre- and post-DMT infusion. While BW waves are always present, FW waves only rise significantly above zero after DMT injection, despite participants having closed eyes. Asterisks denote values significantly different than zero, or between conditions. The panels to the right describe the minute-by-minute changes in the net amount of waves. Asterisks denote FDR-corrected p-values for amount of waves significantly different than zero. ( B ) Comparison between the waves’ temporal evolution after DMT injection (left panel) and with or without visual stimulation (right panel, from a different experiment in which participants, with open eyes, either watched a visual stimulus or a blank screen  Pang et al., 2020 ). Remarkably, the waves’ temporal profiles are very similar in the two conditions, for both FW and BW. ( C ) Comparison between changes in absolute power (as extracted from the 2D-FFT, that is FW and BW in Figure 1 ) due to DMT, placebo and visual stimulation. Remarkably, true photic visual stimulation and eyes-closed DMT induce comparably large reductions in absolute power. In fact, the effect with DMT appears to be even more pronounced (formal contrast not appropriate). Note that in the previous panels the changes in the net amount of waves were reported in dB, and occurred irrespective of the global power changes measured in panel C.

Figure 2—figure supplement 1.

Each line represents a different subject, whereas mean ± standard deviations are represented for each dosage, pre/post infusion for BW (red) and FW (blue) waves. Irrespective of the dosage, the amount of BW waves decreased after DMT infusion, whereas FW waves increased consistently for each subject.

Figure 2—figure supplement 2.

The difference in the pre-post DMT infusion observed along the sagittal line of electrodes (i.e. the one chosen for the first analysis, as reported in Figure 2A of the manuscript) is replicated considering another series of electrodes running from occipital to frontal regions between hemisphere, specifically from electrode P4 to F3 in the FW direction (diag1, Bayesian t-test BF = 4.059, error = 0.002%), and from P3 to F4 (diag2, Bayesian t-test BF = 4.848, error = 0.0001%). Interestingly, DMT induces a similar increase in FW waves, but less of a decrease in the BW direction (diag1 BW: BF = 1.948, error = 0.006%; diag1 BW: BF = 1.567, error = 0.002%). We also investigated a coronal line of electrodes, revealing waves travelling in a leftward and rightward direction above chance level (i.e. larger than 0 dB), but in-line with our hypothesis this pattern was not altered by DMT infusion (for both leftward and rightward waves BF <0.4, error ~0.02%). The bottom panel shows changes in absolute power (as extracted from the 2D-FFT, i.e. FW and BW in Figure 1 ) in each lines of electrodes. Due to DMT, we observed overall a large reduction in absolute power, in-line with previous results.

In order to explore different propagation axes than the midline, we ran the same analysis on one array of electrodes running from posterior right to anterior left regions, and one from posterior left to anterior right ones: in both cases we obtained similar results as for the midline electrodes, i.e. an increase and a decrease of FW and BW waves, respectively, following DMT infusion (see Figure 2—figure supplement 2 ). This suggests that the dominant natural propagation spread of travelling waves is along the axis that connects the furthest posterior and frontal recording channels. As a control, we additionally demonstrated that waves propagating from leftward to rightward regions (and vice versa), were not affected by DMT (see Figure 2—figure supplement 2 ). Besides, in-line with previous work on travelling waves ( Alexander et al., 2013 ; Alexander et al., 2006 ), an additional analysis based on relative phases of the alpha band-pass signals over all channels, confirmed the same results, with DMT indeed disrupting the typical top-down propagation of alpha-band waves. Furthermore, we ran a more temporally precise analysis, on a minute-by-minute scale, testing the amount of FW and BW waves in the two conditions, as shown in the right panels of Figure 2A . in-line with previous studies ( Strassman, 1995a ; Strassman, 1994 ; Timmermann et al., 2019 ), the changes in cortical dynamics appeared rapidly after intravenous DMT injection, and began to fade after about 10 min. Confirming our previous analysis, we observed an increase in FW waves (asterisks in the upper-right panel of Figure 2A show FDR-corrected significant p-values when testing against zero) and a decrease in BW waves, which, nonetheless, remained above zero (all FDR-corrected p-values<0.05). To our initial surprise, the dynamics elicited by DMT injection were remarkably reminiscent of those observed in another study, in which healthy participants alternated visual stimulation with periods of blank screen, without any drug manipulation ( Pang et al., 2020 ). Although a direct comparison is not statistically possible (because the two studies involved distinct subject groups and different EEG recording setups), we indirectly investigated the similarities between these two scenarios.

Comparison with perceptual stimulation

We recently showed that FW travelling waves increase during visual stimulation, whereas BW waves decrease, in-line with their putative functional role in information transmission ( Pang et al., 2020 ). In Figure 2B , for the sake of comparison, we contrast the cortical dynamics induced by DMT (left panel) with the results of our previous study (right panel Pang et al., 2020 ), in which participants perceived a visual stimulus (label ‘ON’) or stared at a dark screen (label ‘OFF’). Remarkably, mutatis mutandis, both FW and BW waves share a similar profile across the two conditions, increasing and decreasing respectively following DMT injection or visual stimulation. If we consider the absolute (maximum) power values derived from the 2D-FFT of each map (i.e. before estimating the surrogates and the waves’ net amount in decibel) as an estimate of spectral power, we can read the results reported in Figure 2C as an overall decrease in oscillatory power following DMT injection, more specifically in the frequency band with the highest power values (i.e. alpha band, but see next paragraph) ( Muthukumaraswamy et al., 2013 ; Riba et al., 2002 ; Schenberg et al., 2015 ; Timmermann et al., 2019 ). Such decrease in oscillatory power is also matched by a similar decrease induced by visual stimulation (all Bayesian t-test BFs 10  >>100). These results demonstrate that, despite participants having their eyes-closed throughout, DMT produces spatio-temporal dynamics similar to those elicited by true visual stimulation. These results therefore shed light on the neural mechanisms involved in DMT-induced visionary phenomena.

Does DMT influence the frequency of travelling waves?

Previous studies showed that DMT alters specific frequency bands (e.g. alpha-band Schenberg et al., 2015 ), mostly by decreasing overall oscillatory power ( Riba et al., 2002 ; Timmermann et al., 2019 ). Here, we investigated whether DMT influences not only the waves’ direction but also their frequency spectrum. We compared the frequencies of the maximum peaks extracted from the 2D-FFT (see Figure 1 ) before and after DMT or Placebo injection. Before infusion, both FW and BW waves had a strong alpha-range oscillatory rhythm ( Figure 3A , labeled ‘PRE’). Remarkably, following DMT injection, the waves’ spectrum changed drastically, with a significant reduction in the alpha-band, coupled with an increase in the delta and theta bands, for both FW (δ-band: BF 10  = 391.16, θ-band: BF 10  = 19.23, α-band: BF 10  = 16.04, β-band: BF 10  = 0.64; all errors < 0.001%) and BW waves (δ-band: BF 10  = 82.56, θ-band: BF 10  = 30.58, α-band: BF 10  = 549.54, β-band: BF 10  = 1.43; all errors < 0.005%). This result corroborates a previous analysis performed on EEG recordings from the same dataset ( Timmermann et al., 2019 ) as well as independent data pertaining to O-Phosphoryl-4-hydroxy-N,N-DMT (psilocybin), a related compound ( Muthukumaraswamy et al., 2013 ). Moreover, we investigated how DMT influences the amount of waves at each frequency.

( A ) Left and right panels show the waves’ frequencies computed from the maximum value from each quadrant in the 2D-FFT map for FW and BW waves, pre- and post-infusion. The histogram reflects the average between participants of the number of 1 s time-windows having a wave peak at the corresponding frequency. Notably, DMT significantly reduces α and β band oscillations, while enhancing δ and θ. Asterisks denote significant differences between DMT and Placebo conditions. ( B ) The upper panels show the amount of waves computed at each frequency of the 2D-FFT (i.e. not considering the maximum power per quadrant as in ( A ), but considering it for each frequency), for FW and BW waves, pre- and post-infusion. As shown in previous analysis, DMT induces an overall decrease of spectral power, especially in the alpha band BW waves, with the notable exception of an increase in FW waves in the alpha range.

As shown in Figure 3B , and in agreement with previous analyses, DMT induces an overall reduction in the amount of waves at each frequency, specifically in the alpha-band BW waves, but with the notable exception in the FW alpha band, in which DMT induces an increase in the waves’ direction.

What’s the relationship between FW and BW waves?

From the left panel of Figure 2B , it seems that during the first minutes after DMT injection, both FW and BW waves are simultaneously present in the brain. In an attempt to understand the overall relationship between FW and BW waves, we focused on the minutes when both BW and FW waves were significantly larger than 0 (minutes 2 to 5 after DMT injection, see Figure 2A ). On these data we performed a moment-by-moment correlation between their respective net amount (as measured in decibel – see Figure 1 ). We found a clear and significant negative relationship (Bayesian t-test against zero, pre-DMT BF 10  = 393.1, error <0.0001%, 95% CI: [−0.448,–0.212]; Post-DMT BF 10  = 381.9, error <0.0001%, 95% CI: [−0.479,–0.226]), very consistent across participants and irrespective of DMT injection (difference between pre- and post-, Bayesian t-test BF 10  = 0.225; error<0.02% Figure 4 , first panel). This result demonstrates that, in general, FW waves tend to be weaker whenever BW waves are stronger, and vice versa. In other words, FW and BW remain present after drug injection, sum to a consistent total amount, and remain inversely related; it is only the ratio of contribution from each that changes after DMT (i.e. less BW, more FW waves).

There is a negative correlation between the net amount of FW and BW waves, which is not influenced by the ingestion of DMT (left panel). The middle and the right panel show the relationship for a typical subject pre- and post-DMT injection.

Is there a correlation between waves and subjective reports?

We investigated whether changes in travelling waves under DMT correlated with the subjective effects of the drug. Specifically, for 20 min after DMT injection participants provided an intensity rating every minute and, when subjective effects faded, participants filled various questionnaires that addressed different aspects of the experience (see Timmermann et al., 2019 for details). First, we found a robust correlation between minute-by-minute intensity rates and the amplitude of the waves, as shown in the first panel of Figure 5 . This result reveals that the developing intensity of the drug’s subjective effects and changes in the amplitude of waves correlate positively (FW) or negatively (BW) across time, both peaking a few minutes after drug injection. Second, treating each time point independently, we again correlated intensity ratings with the amount of each wave type, across subjects. The middle panel of Figure 5 shows a clear trend for the correlation coefficients over time. Despite the limited number of data-points (n = 12), the correlation coefficients reach high values (~0.4), implying that, around the moment where the drug had its maximal effect (2–5 min after injection), those subjects who reported the most intense effects were also those who had the strongest travelling waves in the FW direction, and the weakest waves in the BW direction. Finally, we correlated the amount of FW and BW waves with ratings focused specifically on visual imagery: remarkably, ratings of all of the relevant questionnaire items correlated strongly with the increased amount of FW waves under DMT. As the same relationship was not apparent for the BW waves, this consolidates the view that visionary experiences under DMT correspond to higher amounts of FW waves in particular. Taken together with previous results from visual stimulation experiments independent of DMT ( Pang et al., 2020 ), these data strongly support the principle that cortical travelling waves (and increased FW waves in particular) correlate with the conscious visual experiences, whether induced exogenously (via direct visual stimulation) or endogenously (visionary or hallucinatory experiences).

The first panel shows the correlation between intensity rate and waves amplitude across time-points. Each dot represents a one-minute time-bin from DMT injection, the x-axis reflects the average intensity rating across subjects, and the y-axis indicates the average strength of BW or FW waves across subjects (both correlations p<0.0001). The middle panel shows the correlation coefficients across participants, obtained by correlating the intensity ratings and the waves’ amount separately for each time point. Solid lines show when the amount of waves is significantly larger than zero (always for BW waves, few minutes after DMT injections for FW waves – see Figure 2A ). However, given the limited statistical power (N = 12), and proper correction for multiple testing, correlations did not reach significance at any time point. The last panel shows the correlation coefficients between the visual imagery specific ratings provided at the end of the experiment (i.e. Visual Analogue Scale, see methods) and the net amount of waves (measured when both BW and FW were significantly different than zero, i.e. from minutes 2 to 5): for all 20 items in the questionnaire there was a positive trend between the amount of FW waves and the intensity of visual imagery, as confirmed by a Bayesian t-test against zero (BF for FW waves >> 100). We did not observe this effect in the BW waves (BF = 0.41).

In this study we investigated the effects of the classic serotonergic psychedelic drug DMT on cortical spatio-temporal dynamics typically described as travelling waves ( Muller et al., 2018 ). We analysed EEG signals recorded from 13 participants who kept their eyes closed while receiving drug. Results revealed that, compared with consistent eyes-closed conditions under placebo, eyes-closed DMT is associated with striking changes in cortical dynamics, which are remarkably similar to those observed during actual eyes-open visual stimulation ( Alamia and VanRullen, 2019 ; Pang et al., 2020 ). Specifically, we observed a reduction in BW waves, and increase in FW ones, as well as an overall decrease in α band (8–12 Hz) oscillatory frequencies ( Timmermann et al., 2019 ). Moreover, increases in the amount of FW waves correlated positively with real-time ratings of the subjective intensity of the drug experience as well as post-hoc ratings of visual imagery, suggesting a clear relationship between travelling waves and a distinct and novel type of conscious experience.

Relation to previous findings

Initiated by the discovery of mescaline, and catalysed by the discovery of LSD, Western medicine has explored the scientific value and therapeutic potential of psychedelic compounds for over a century ( Carhart-Harris, 2018 ; Schoen, 1964 ; Strassman, 1995b ). DMT has been evoking particular interest in recent decades, with new studies into its basic pharmacology ( Dean et al., 2019 ), endogenous function ( Barker et al., 2012 ) and effects on cortical activity in rats ( Artigas et al., 2016 ; Riga et al., 2014 ) and humans ( Daumann et al., 2010 ; de Araujo et al., 2012 ; Valle et al., 2016 ). There has been a surprising dearth of resting-state human neuroimaging studies involving pure DMT ( Palhano-Fontes et al., 2015 ; Timmermann et al., 2019 ) which, given its profound and basic effects on conscious awareness, could be viewed as a scientific oversight.

Previous work involving ayahuasca and BOLD fMRI found increased visual cortex BOLD signal under the drug vs placebo while participants engaged in an eyes-closed imagery task – a result that was interpreted as consistent with the ‘visionary’ effects of ayahuasca ( de Araujo et al., 2012 ). Despite some initial debate ( Bartolomeo, 2008 ), it is now generally accepted that occipital cortex becomes activated during visual imagery ( Fulford et al., 2018 ; Pearson, 2019 ). Placing these findings into the context of previous work demonstrating increased FW travelling waves during direct visual perception ( Alamia and VanRullen, 2019 ; Pang et al., 2020 ), our present findings of increased FW waves under DMT correlating with visionary experiences lend significant support to the notion that DMT/ayahuasca – and perhaps other psychedelics – engage the visual apparatus in a fashion that is consistent with actual exogenously driven visual perception. Future work could extend this principle to other apparently endogenous generated visionary experiences such as dream visions and other hallucinatory states. We would hypothesize a consistent favouring of FW waves during these states. If consistent mechanisms were also found to underpin hallucinatory experiences in other sensory modalities – such as the auditory one, a basic principle underlying sensory hallucinations might be established.

Pharmacological considerations

As a classic serotonergic psychedelic drug, DMT’s signature psychological effects are likely mediated by stimulation of the serotonin 2A (5-HT2A) receptor subtype. As with all other classic psychedelics ( Nichols, 2016 ) the 5-HT2A receptor has been found to be essential for the full signature psychological and brain effects of Ayahuasca ( Valle et al., 2016 ). In addition to its role in mediating altered perceptual experiences under psychedelics, the 5-HT2A receptor has also been linked to visual hallucinations in neurological disorders, with a 5-HT2A receptor inverse agonist having been licensed for hallucinations and delusions in Parkinson’s disease with additional evidence for its efficacy in reducing consistent symptoms in Alzheimer’s disease ( Ballard et al., 2018 ). Until recently, a systems level mechanistic account of the role of 5-HT2A receptor agonism in visionary or hallucinatory experiences has, however, been lacking.

Predictive coding and psychedelics

There is a wealth of evidence that Bayesian or predictive mechanisms play a fundamental role in cognitive and perceptual processing ( den Ouden et al., 2012 ; Kok and De Lange, 2015 ) and our understanding of the functional architecture underlying such processing is continually being updated ( Alamia and VanRullen, 2019 ; Friston, 2018 ). According to predictive coding ( Huang and Rao, 2011 ), the brain strives to be a model of its environment. More specifically, based on the assumption that the cortex is a hierarchical system – message passing from higher cortical levels is proposed to encode predictions about the activity of lower levels. This mechanism is interrupted when predictions are contradicted by the lower-level activity (‘prediction error’) – in which case, information passes up the cortical hierarchy where it can update predictions. Predictive coding has recently served as a guiding framework for explaining the psychological and functional brain effects of psychedelic compounds ( Carhart-Harris and Friston, 2019 ; Pink-Hashkes et al., 2017 ). According to one model ( Carhart-Harris and Friston, 2019 ), psychedelics decrease the precision- weighting of top-down priors, thereby liberating bottom-up information flow. Various aspects of the multi-level action of psychedelics are consistent with this model, such as the induction of asynchronous neuronal discharge rates in cortical layer 5 ( Celada et al., 2008 ), reduced alpha oscillations ( Carhart-Harris et al., 2016 ; Muthukumaraswamy et al., 2013 ) increased signal complexity ( Schartner et al., 2017 ; Timmermann et al., 2019 ) and the breakdown of large-scale intrinsic networks ( Carhart-Harris et al., 2016 ).

Recent empirically supported modelling work has lent support to assumptions that top-down predictions and bottom-up prediction-errors are encoded in the direction of propagating cortical travelling waves ( Alamia and VanRullen, 2019 ). Specifically, these simulations demonstrated that a minimal predictive coding model implementing biologically plausible constraints (i.e. temporal delays in the communication between regions and time constants) generates alpha-band travelling waves, which propagate from frontal to occipital regions and vice versa, depending on the ‘cognitive states’ of the model (input-driven vs. prior-driven), as confirmed by EEG data in healthy participants (in that case, processing visual stimuli vs. closed-eyes resting state).

The view that predictive coding could be the underlying principle explaining both the propagation of alpha-band travelling waves and the neural changes induced by psychedelics opened-up a tantalizing opportunity for testing assumptions both about the nature of travelling waves and how they should be modulated by psychedelics ( Carhart-Harris and Friston, 2019 ). Although we are restricted to speculation by the lack of direct experimental manipulation of top-down and bottom-up sensory inputs, our prior assumptions were so emphatically endorsed by the data, including how propagation-shifts related to subjective experience, that, in-line with prior hypotheses and motivations for the analyses, we were persuaded to infer about both the functional relevance of cortical travelling waves and brain action of psychedelics. Additional studies manipulating bottom-up and top-down analysis of sensory inputs with alternative perceptual designs will be required to confirm the relation between predictive coding, alpha-band oscillatory travelling waves and psychedelics states. Moreover, future studies can now be envisioned to examine how these assumptions translate to other phenomena such as non-drug induced visionary and hallucinatory states.

The present analyses were applied to the first EEG data on the effects of DMT on human resting- state brain activity. In-line with a specific prior hypothesis, clear evidence was found of a shift in cortical travelling waves away from the normal basal predominance of backward waves and towards the predominance of forward waves – remarkably similar to what has been observed during eyes-open visual stimulation. Moreover, the increases in forward waves correlated positively with both the general intensity of DMT’s subjective effects, as well as its more specific effects on eyes-closed visual imagery. These findings have specific and broad implications: for the brain mechanisms underlying the DMT/psychedelic state as well as conscious visual perception more fundamentally.

Materials and methods

Participants and experimental procedure.

In this study we analysed a dataset presented in a previous publication ( Timmermann et al., 2019 ), to address a very different scientific question using another analytical approach. Consequently, the information reported in this and the next paragraphs overlaps with the previous study (to which we refer the reader for additional details). Thirteen participants took part in this study (six females, age 34.4 ± 9.1 SD), sample size was chosen based on prior EEG and MEG studies and effect sizes with similar compounds. All participants provided written informed consent, and the study was approved by the National Research Ethics (NRES) Committee London – Brent and the Health Research Authority. The study was conducted in-line with the Declaration of Helsinki and the National Health Service Research Governance Framework.

Participants were carefully screened before joining the experiments. A medical doctor conducted physical examination, electrocardiogram, blood pressure and routine blood tests. A successful psychiatric interview was necessary to join the experiment. Other exclusion criteria were (1) under 18 years of age, (2) having no previous experience with psychedelic drugs, (3) history of diagnosed psychiatric illnesses, (4) excessive use of alcohol (more than 40 units per week). The day before the experiment a urine and pregnancy test (when applicable) were performed, together with a breathalyzer test.

Participants attended two sessions, in the first one, they received placebo, while DMT was administered in the second session. We employed a fixed-order, single blind design considering that psychedelics have been shown to induce lasting psychological changes ( Maclean et al., 2011 ), which could have led to confounding effects on the following session if DMT had been administered in the first session. Additionally, we aimed at ensuring familiarity with the research environment and the study team before providing the psychedelics compound. Given the lack of human data with DMT, progressively increasing doses were provided to different participants (four different doses were used: 7, 14, 18 and 20 mg, to 3, 4, 1, and 5 successive participants, respectively). EEG signals were recorded before and up to 20 min after drug delivery. Participants rested in a semi-supine position with their eyes closed during the duration of the whole experiment. The eyes-closed instruction was confirmed by visual inspection of the participants during dosing. At each minute, participants provided an intensity rating, while blood samples were taken at given time-points (the same for placebo and DMT conditions) via a cannula inserted in the participants’ arm. One day after the DMT session, participants reported their subjective experience completing an interview composed of several questionnaires (see Timmermann et al., 2019  for details). In this study we focused on the Visual Analogue Scale values.

EEG preprocessing

EEG signals were recorded using a 32-channels Brainproduct EEG system sampling at 1000 Hz. A high-pass filter at 0.1 Hz and an anti-aliasing low-pass filter at 450 Hz were applied before applying a band-pass filter at 1–45 Hz. Epochs with artifacts were manually removed upon visual inspection. Independent-component analysis (ICA) was performed and components corresponding to eye-movement and cardiac-related artifacts were removed from the EEG signal. The data were re-referenced to the average of all electrodes. All the preprocessing was carried out using the Fieldtrip toolbox ( Oostenveld et al., 2011 ), while the following analysis were run using custom scripts in MATLAB.

Waves quantification

We epoched the preprocessed EEG signals in 1 s windows, sliding with a step of 500 ms (see Figure 1 ). For each time-window, we then arranged a 2D time-electrode map composed of five electrodes (i.e. Oz, POz, Pz, Cz, FCz). From each map we computed the 2D Fast Fourier Transform (2DFFT – Figure 1 ), from which we extracted the maximum value in the upper and lower quadrants, representing respectively the power of forward (FW) and backward (BW) waves. We also performed the same procedure 100 times after having randomised the electrodes’ order: the surrogate 2D-FFT spectrum has the same temporal frequency content overall, but the spatial information is disrupted, and the information about the wave directionality is lost. In such a manner we obtained the null or surrogate measures, namely FWss and BWss, whose values are the average over the 100 repetitions (see Figure 1 ). Eventually, we computed the actual amount of waves in decibel (dB), considering the log-ratio between the actual and the surrogate values:

It is worth noting that this value represents the amount of significant waves against the null distribution, that is against the hypothesis of having no FW or BW waves.

Statistical analysis

All the analyses regarding the EEG signals were performed in MATLAB. Bayesian analyses were run in JASP ( Team, 2018 ). We ran separate Bayesian ANOVA for FW and BW conditions, and we considered as factors the time of injection (pre-post, see Figure 2A ) and drug condition (DMT vs Placebo). Subjects were considered to account for random factors. Regarding the minute-by-minute analysis ( Figure 2A , right panels), we performed a t-test at each time-bin against zero, and we corrected all the p-values according to the False Discovery Rate ( Benjamini and Yekutieli, 2005 ). All data and code to perform the analysis are available at https://osf.io/wujgp/ .

Acknowledgements

We dedicate this paper to the memory of Jordi Riba , a gracious man and pioneering psychedelic researcher.

Funding Statement

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Contributor Information

Virginie van Wassenhove, CEA, DRF/I2BM, NeuroSpin; INSERM, U992, Cognitive Neuroimaging Unit, France.

Timothy E Behrens, University of Oxford, United Kingdom.

Funding Information

This paper was supported by the following grants:

• Alexander Mosley Charitable Trust to Robin L Carhart-Harris.
• Ad Astra Chandaria Foundation to Robin L Carhart-Harris.
• CRCNS ANR-NSF ANR-19-NEUC-0004 to Rufin VanRullen.
• ANITI (Artificial and Natural Intelligence Toulouse Institute) Research Chair ANR-19-PI3A-0004 to Rufin VanRullen.
• Comision Nacional de Investigacion Cientifica y Tecnologica de Chile to Christopher Timmermann.

Additional information

No competing interests declared.

Software, Formal analysis, Visualization, Methodology, Writing - original draft.

Conceptualization, Data curation, Formal analysis, Methodology, Writing - review and editing.

Conceptualization, Data curation, Supervision, Funding acquisition.

Formal analysis, Supervision, Funding acquisition, Methodology, Writing - review and editing.

Conceptualization, Data curation, Supervision, Funding acquisition, Project administration, Writing - review and editing.

Human subjects: All participants provided written informed consent, and the study was approved by the National Research Ethics (NRES) Committee London - Brent and the Health Research Authority (16/LO/0897). The study was conducted in line with the Declaration of Helsinki and the National Health Service Research Governance Framework.

Additional files

Transparent reporting form, data availability.

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• eLife. 2020; 9: e59784.

Decision letter

David murray alexander.

KU Leuven, Belgium

In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.

Acceptance summary:

In this study, Alamia and colleagues describe the effect of N,N, Dimethyltryptamine DMT on the resting-state dynamics of α traveling waves. DMT is a serotonergic psychedelic drug that elicits vivid hallucinations. The authors tested whether DMT provoke a relative change in strength of forward- and backward- α traveling waves recorded with non-invasive electroencephalography. Specifically, the hypothesis was that traveling waves under DMT may show features of visual perception in the absence of photic stimulation. Indeed, following DMT consumption and during eyes-closed resting state, the authors report an increase of forward-traveling waves, an α power increase (comparable to photic stimulation) and of low-frequency components in the low-range spectrum. The implication of traveling waves are discussed in relation to predictive coding.

Decision letter after peer review:

Thank you for submitting your article "DMT alters cortical travelling waves" for consideration by eLife . Your article has been reviewed by three peer reviewers, one of whom is a member of our Board of Reviewing Editors, and the evaluation has been overseen by Timothy Behrens as the Senior Editor. The following individual involved in review of your submission has agreed to reveal their identity: David Murray Alexander (Reviewer #2).

The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.

We would like to draw your attention to changes in our revision policy that we have made in response to COVID-19 (https://elifesciences.org/articles/57162). Specifically, when editors judge that a submitted work as a whole belongs in eLife but that some conclusions require a modest amount of additional new data, as they do with your paper, we are asking that the manuscript be revised to either limit claims to those supported by data in hand, or to explicitly state that the relevant conclusions require additional supporting data.

Our expectation is that the authors will eventually carry out the additional experiments and report on how they affect the relevant conclusions either in a preprint on bioRxiv or medRxiv, or if appropriate, as a Research Advance in eLife , either of which would be linked to the original paper.

Alamia and colleagues describe the effect of N,N, Dimethyltryptamine DMT on human resting-state dynamics recorded with non-invasive EEG. DMT is a serotonergic psychedelic drug that elicits vivid hallucinations. The authors propose that DMT provokes a relative change in strength of forward- and backward- α [~10 Hz] traveling waves, indicative of bottom-up and top-down propagations of information. In three main analyses of EEG recordings following the administration of DMT, the authors report an increase of forward-traveling waves, α power (comparable to photic stimulation) and low-frequency components in the low-range spectrum.

Revisions for this paper:

I highlight three main issues that need to be addressed in a revised paper.

1) All three reviewers highlight specific points re. the limitations of the current analysis (e.g. a prior choice of electrodes) and the choice for quantifications of the traveling waves. The authors should clarify their rationale underlying their analytical choices. As suggested by reviewer 3, a section dedicated to "Quantifying travelling waves" may be helpful.

2) All three reviewers raised substantial concerns about the interpretability of scalp level recordings in relation to the generators of the signals as well as the inference that can be made on the traveling pattern. Reviewer 1 raised concerns about the full spectral changes that can be seen and reviewer 3 suggested the possibility of interference patterns.

3) Reviewers 1 and 2 consider the predictive coding hypothesis a far-stretched inference of current results, and thus needs to be refined.

Reviewer #1:

In this study, Alamia and colleagues describe the effect of N,N, Dimethyltryptamine DMT on the resting-state dynamics of α traveling waves. DMT is a serotonergic psychedelic drug that elicits vivid hallucinations. The authors herein propose that DMT provokes a relative change in strength of forward- and backward- α traveling waves recorded with non-invasive EEG. The authors hypothesized that traveling waves under DMT may show features of visual perception, namely an increase of forward going (occipital to frontal) traveling waves during eyes-closed resting-state thus independently of photic inputs. With three main analyses, the authors observe that following DMT, forward-traveling waves increase, α power increase (comparable to photic stimulation) and an overall increase of low-frequency components in the low-range spectrum.

I have several major concerns regarding the reliability of the analysis and subsequent interpretations. One is that while the authors quantified traveling waves, they also report a radical change in the overall spectral fingerprinting of the EEG (Figure 3) and clearly showing that α is largely suppressed following DMT thus largely diminishing the reliability and pertinence of focusing on α traveling wave while low-frequency are largely boosted. Second, the authors report consistent inverse relationships between FW and BW waves, which may simply result from moving dipoles that generate the signals; in light of this, the inverse relation in Figure 5 between FW/BW is not surprising. Finally, would a simpler measure of decrease in α power and increase of low spectral power reveal similar correlations to behavior?

Reviewer #2:

The authors present a sensor level analysis of traveling waves in the EEG, during dosage with DMT or saline. The work is a strong contribution to the study of cortical traveling waves (TWs) due to the pharmacological manipulation, which helps our understanding of the causal role of TWs. This contribution is bolstered by being able to draw parallels with the effects of visual stimulation (or not) during rest, reported in a contemporaneous manuscript. The manuscript will be of general interest to readers of eLife . The manuscript is crisply written.

The conclusions follow from the analysis.

1) The quantitative methods to analyse TWs are rather week, being focussed on direction of travel along the anterior-posterior axis. While these methods are sufficient to support the main conclusions, more could be teased from the experiment by following recent developments in TW quantification.

For example, using peak tracing along the lines of Massimini et al., 2004, would enable the detailed paths of the waves to be traced over the whole recording array, and velocity to be estimated.

Methods exist to estimate the spatial frequency of the waves on the scalp, as well as the proportion of traveling vs. standing waves (Alexander et al., 2016). Likewise, other directional components of the wave trajectory can be assessed by using PCA to create a spatial basis for the waves (Alexander et al., 2006, 2009, 2013).

It seems possible that important features of the data have been missed by limiting the analysis to electrodes FCz to Oz. For example, what if DMT influence and visual stimulation share a common primary direction, as is found, but DMT waves are more left posterior to right anterior and visual stimulation is more right posterior to left anterior (or vice versa)?

2) The sections on predictive coding are only tenuously supported by the data. In particular, I can see no discussion on how directional differences in the α band may be significant in this regard. What about situations where anterior-posterior differences are found in the δ band (Alexander et al., 2006; 2009)? Or if directional results were in another band? Because of the lack of specificity to this discussion, and the lack of explicit tests of this theoretical framework, I suggest these concepts be given a more appropriate weighting (less).

3) An obvious objection to the analysis is that it is sensor level. The authors need to address their reasons for doing this e.g. that source projections destroy real long-range correlations as well as blurring by the scalp and other tissues. See Nunez, 1974; Nunez et al., 1997; Freeman et al., 2000; 2003 and Alexander et al., 2019, for a summary of these issues.

Reviewer #3:

This study of the effects of the drug DMT on the direction and occurrence of EEG traveling waves seems generally plausible to me, although some important aspects are not discussed. I don't have major criticisms. However, as one who has studied EEG traveling and standing waves for many years, I worry that those unfamiliar with EEG wave phenomena may misinterpret some of these results given their partial dependence on the specific experimental methods employed. While I have not read previous papers by these authors that may fill in some of the important gaps, I list below some ideas that any reader interested in EEG waves and their neuro-scientific interpretation must be aware of. A summary paragraph in the section "Quantifying travelling waves" is recommended concerning the following basic concepts that must be understood if the results are to be interpreted correctly.

1) In all but the simplest systems, traveling waves occur in groups (packets) over some range of spatial wavelengths (multiple spatial frequencies, k). This is to be expected in brains, based on both theory and experiment (see Nunez and Srinivasan, 2006; 2014; Nunez, 2000).

2) Any experimental electrode array will be sensitive to only parts of these wave packets, e.g. only waves shorter than the spatial extent of the array and waves longer than twice the electrode separation distance (Nyquist criterion in space) can be resolved. In scalp recordings, the shorter waves may be mostly removed by volume conduction.

3) As a consequence of #2, waves recorded directly from the cortex (as indicated in several recent studies) will emphasize shorter waves than the scalp recorded waves. In the case of small cortical arrays, the ECoG overlap with scalp data may be minimal. Thus, the different estimated wave properties (including propagation direction) need not agree.

4) When waves are traveling in multiple directions at nearly the same time in "closed" systems (e.g., the cortical/white matter), there are only two possible results. Either the waves must damp out or they combine (interfere) to form standing waves (e.g. α waves traveling both forward and backward). One expects that the actual behavior depends on brain state, including the influence of drugs (see Nunez and Srinivasan, 2006; 2014; Nunez, 2000).

Author response

Revisions for this paper: I highlight three main issues that need to be addressed in a revised paper. 1) All three reviewers highlight specific points re. the limitations of the current analysis (e.g. a prior choice of electrodes) and the choice for quantifications of the traveling waves. The authors should clarify their rationale underlying their analytical choices. As suggested by reviewer 3, a section dedicated to "Quantifying travelling waves" may be helpful.

We have now acknowledged the limitations of the current analyses, and we performed several additional steps to improve on our previous approach. As explained in detail in what follows (and in the revised manuscript) we included a new analysis considering different lines of electrodes, spanning from posterior to anterior, but also left to right brain regions. We also performed an additional complementary analysis based on the suggestions of reviewer 2, showing similar results as our original analysis. Finally, as suggested by reviewer 3, we integrated the current “Quantifying travelling waves” paragraph with his generous suggestions, improving the readability of the manuscript to a non-specialized audience. We believe that these modifications will satisfy both the editor and all the reviewers.

We agree with the concerns raised by both reviewers, and we have included an additional analysis (now Figure 3B in the revised manuscript) that addresses specifically these concerns. Regarding the spectral changes, our new analysis avoids choosing a priori a single frequency band (i.e. the one corresponding to the global maximum in the 2D-FFT) but instead analyze the changes in all the spectrum. This novel approach, besides confirming our previous results, provides a fuller view of the overall changes in the spectral pattern induced by DMT. Concerning the source generators of the travelling waves pattern, we discuss this point in the revised manuscript, arguing that a sensor analysis is more appropriate in this case because it circumvents some limitations related specifically to source analysis (e.g. source projections impair long-range connections). Finally, as shown in Figure 4, backward and forward waves were negatively correlated on a trial by trial basis, which will tend to limit the possibility suggested by reviewer 3 of having interference patterns (resulting in standing waves).

We addressed carefully this point by giving overall less weight to the Predictive Coding hypothesis in the Discussion, as suggested by both reviewers. Additionally, as suggested specifically by reviewer 2, we clarify in the revised Discussion the link between Predictive Coding, α oscillations and travelling waves, and the motivation behind our original hypothesis and the relationship with the present results. More precisely, we previously demonstrated how a model based on Predictive Coding principles and implementing biologically plausible constraints gives rise to α-band travelling waves, whose direction of propagation depends on the “cognitive” state of the model/subject (FW during visual stimulation, BW during closed-eyes resting state). Then, starting from the premise that psychedelics disrupt prior distributions encoded in hierarchically high-level properties of brain function (Carhart-Harris and Friston, 2019), we formulated the specific hypothesis that DMT could specifically disrupt α-band travelling waves, enhancing their feed-forward propagation while decreasing the feed-back direction. All in all, we found it remarkable that such a specific hypothesis received such clear support in the data. However, we understand that interpretations can always be queried and more work is needed to scrutinize the one we offer based on our specific prior hypothesis. We have now rephrased our interpretation substantially in the revised version of the manuscript, to soften our conclusions and emphasize the need for more research.

Reviewer #1: In this study, Alamia and colleagues describe the effect of N,N, Dimethyltryptamine DMT on the resting-state dynamics of α traveling waves. DMT is a serotonergic psychedelic drug that elicits vivid hallucinations. The authors herein propose that DMT provokes a relative change in strength of forward- and backward- α traveling waves recorded with non-invasive EEG. The authors hypothesized that traveling waves under DMT may show features of visual perception, namely an increase of forward going (occipital to frontal) traveling waves during eyes-closed resting-state thus independently of photic inputs. With three main analyses, the authors observe that following DMT, forward-traveling waves increase, α power increase (comparable to photic stimulation) and an overall increase of low-frequency components in the low-range spectrum. I have several major concerns regarding the reliability of the analysis and subsequent interpretations. One is that while the authors quantified traveling waves, they also report a radical change in the overall spectral fingerprinting of the EEG (Figure 3) and clearly showing that α is largely suppressed following DMT thus largely diminishing the reliability and pertinence of focusing on α traveling wave while low-frequency are largely boosted. Second, the authors report consistent inverse relationships between FW and BW waves, which may simply result from moving dipoles that generate the signals; in light of this, the inverse relation in Figure 5 between FW/BW is not surprising. Finally, would a simpler measure of decrease in α power and increase of low spectral power reveal similar correlations to behavior?

We thank the reviewer for raising these important considerations. Regarding the relationship between the spectral changes in the EEG and the changes in the amount of waves, we performed an additional analysis quantifying these changes as a function of each frequency (i.e. extracting in the 2D-FFT the maximum value separately for each frequency). The first row of Author response image 1 (integrated in the revised version of the manuscript as figure 3B) shows the amount of FW and BW waves (in dB –as compared to the surrogate distribution) before and after DMT or Placebo infusion (i.e. Pre and Post). The second row shows the difference in the amount of waves between DMT and Placebo for each frequency. Interestingly the largest changes occur in the α band frequency for both forward and backward waves, even though after correcting for multiple comparison we found a significant reduction only in the BW α band. This analysis shows that, as suggested by the reviewer, the changes in the spectral fingerprint of the EEG do influence the waves’ propagation in several frequencies, but the largest changes systematically occur in the α band. This additional analysis has been introduced in the Results section (paragraph: “Does DMT influence the frequency of travelling waves?”)

The lower panels show the difference between DMT- and placebo-induced waves for each condition. As shown in the previous analysis, DMT induces an overall decrease of the waves’ amplitude, especially pronounced (and significant) in the α band BW waves, with the notable exception of FW waves in the α range, where an DMT-induced increase is observed.

Regarding the relationship between FW and BW waves, our analysis –shown in Figure 4 of the manuscript- suggests that after DMT –or during photic visual stimulation- FW and BW waves do not co-occur simultaneously, but tend to alternate temporally, as revealed by the negative correlation. We agree with the reviewer that moving dipoles could be responsible for the generation of these signals, as reported in an in-depth analysis of a previous paper investigating the source localization of similar waves patterns (Lozano-Soldevilla and VanRullen, 2019). In line with a similar comment from reviewer 2, we added in the manuscript a reference to source vs sensors analysis (“In addition, it is important to note that our waves’ analysis focuses on the sensor level, as source projections present a number of important limitations, such as impairing long-range connections, as well as smearing of signals due to scalp interference (Alexander et al., 2019; Freeman and Barrie, 2000; Nunez, 1974)”). Lastly, a previous analysis on the same dataset (Timmermann et al., 2019) identified a correlation between theta- and δ-band spectral power changes and subjective behavior, but not for changes in the α range. This suggests that the correlation reported in Figure 5 is not a direct consequence of changes in the EEG spectral power.

Reviewer #2: The authors present a sensor level analysis of traveling waves in the EEG, during dosage with DMT or saline. The work is a strong contribution to the study of cortical traveling waves (TWs) due to the pharmacological manipulation, which helps our understanding of the causal role of TWs. This contribution is bolstered by being able to draw parallels with the effects of visual stimulation (or not) during rest, reported in a contemporaneous manuscript. The manuscript will be of general interest to readers of eLife. The manuscript is crisply written. The conclusions follow from the analysis. Concerns: 1) The quantitative methods to analyse TWs are rather week, being focussed on direction of travel along the anterior-posterior axis. While these methods are sufficient to support the main conclusions, more could be teased from the experiment by following recent developments in TW quantification. For example, using peak tracing along the lines of Massimini et al., 2004, would enable the detailed paths of the waves to be traced over the whole recording array, and velocity to be estimated. Methods exist to estimate the spatial frequency of the waves on the scalp, as well as the proportion of traveling vs. standing waves (Alexander et al., 2016). Likewise, other directional components of the wave trajectory can be assessed by using PCA to create a spatial basis for the waves (Alexander et al., 2006, 2009, 2013).

We thank the reviewer for his useful suggestions. As correctly noticed, our current method to detect travelling waves focuses exclusively on the Anterior-Posterior axis, in line with our previous studies (Alamia and VanRullen, 2019; Lozano-Soldevilla and VanRullen, 2019; Pang et al., 2020). However, we agree that more can be inferred from the data from other electrodes (see next point and comment to reviewer 1 for waves’ quantification on other axes). We applied a method similar to Alexander et al., 2006, 2009 and 2016, thus considering the phase of the signal over the entire array of electrodes. Specifically, we computed the phase of the α band-pass filtered signals pre- and post- DMT infusion, and referenced it to the central electrode Cz. The relative phase thus describes the propagation of the wave as compared to this electrodes: positive lags (in yellow in the Author response image 2 ) characterize earlier components, whereas negative lags (in blue) are associated with signals lagging behind. Author response image 2 summarizes the results in all conditions.

Reassuringly, the pattern of results confirms the disruption of the top-down flow, counterbalanced by a bottom-up component, specifically after the infusion of DMT, in line with our original analysis.

Interestingly, we observed that after placebo, the typical top-down propagation of α-band waves remains unaltered, whereas after DMT, waves propagate both FW and BW, as revealed by an overall phase distribution around zero. Overall these results confirmed the one obtained with the 2D-FFT approach. We opted for keeping the latter for consistency with our previous studies (but we mentioned this result in the revised manuscript along with the references)

“Besides, in line with previous work on travelling waves (Alexander et al., 2013, 2006), an additional analysis based on relative phases of the α band-pass signals over all channels confirmed the same results, with DMT disrupting the typical top-down propagation of α-band waves (not shown).”

We agree with the reviewer that the approach used by (Massimini et al., 2004), would allow to identify the detailed path of the waves, and potentially their velocity. However, in their work, Massimini and colleagues targeted slow 1Hz waves (the signal was actually low-pass filtered at 4Hz); for each cycle waves were tracked based on the localization of the main (negative) peak whose voltage was below a threshold of -80 V. This approach, which provides reliable results for low-frequency waves, may present some non-trivial additional limitations when applied to higher frequencies. Specifically, the identification of each peak/cycle may not be straightforward for higher frequencies (e.g. α band); in Massimini et al. the time window used to identify the peak spanned between +/-800ms to the earliest peak, but such a window should be proportionally much shorter to isolate single peaks in the α range, and thus be increasingly more sensitive to jittered noise. We therefore favored the 2D-FFT approach, which –despite its own limitations- seemed more suitable to describe waves with higher temporal frequencies. Finally, regarding the waves speed, it is possible to estimate their velocity from the 2D-FFT, considering both spatial and temporal frequencies as shown in our previous study (Alamia and VanRullen, 2019). The reported results are consistent with the speed recorded for cortical waves (macroscopic scale, speed ~1.5 – 2.0 m/s (Muller et al., 2018)).

We agree with the reviewer that the choice of the midline electrodes supports the main conclusion but prevents a broader view on the waves’ dynamic at the cortical level. Accordingly, and in line with the concerns of reviewer 1, we explored different lines of electrodes, to identify other axes of propagation. As shown in Figure 2—figure supplement 2, comparing PRE and POST DMT infusion reveals an increase of waves propagating from posterior to anterior regions considering an array of electrodes arranged from right posterior to left anterior (diag1 in Figure 2—figure supplement 2, Bayesian t-test BF=4.059, error=0.002%) and from left posterior to right anterior (diag2 in Figure 2—figure supplement 2, Bayesian t-test BF=4.848, error=0.0001%), similarly to the results obtained considering the main posterior-anterior axis (Bayesian t-test BF=5.4, error=0.001%). Additionally, we revealed a significant amount of waves (larger than 0 dB) propagating along the coronal line of electrodes (i.e. leftward and rightward), but those waves were not influenced by DMT infusion (for both leftward and rightward waves BF<0.4, error~0.02%). We included these analyses in the Results section and as Figure 2—figure supplement 2.

“In order to explore different propagation axes than the midline, we ran the same analysis on one array of electrodes running from posterior right to anterior left regions, and one from posterior left to anterior right ones: in both cases we obtained similar results as for the midline electrodes, that is, an increase and a decrease of FW and BW waves respectively following DMT infusion (see Figure 2—figure supplement 2).This suggests that the waves’ propagation spread to most posterior and frontal recording channels As a control, we additionally demonstrated that waves propagating from leftward to rightward regions (and vice versa) were not affected by DMT, as predicted by our hypothesis (see Figure 2—figure supplement 2).”

We agree and thank the reviewer for pointing out this shortcoming in the Discussion. The focus on the α-band originates from our previous study (Alamia and VanRullen, 2019) in which we demonstrated how a minimal Predictive Coding model implementing biologically plausible constraints (i.e. temporal delays in the communication between regions and time constants) generates α-band travelling waves whose direction of propagation is matched by experimental data. This result was the starting hypothesis that motivated the investigation of α-band travelling waves after DMT-infusion, under the hypothesis of the REBUS model (psychedelics disrupt prior distributions in higher brain regions- Carhart-Harris and Friston, 2019). In the revised version of the manuscript, we clarify the link between Predictive Coding, α oscillations and travelling waves. However, we agree with this reviewer (and reviewer 1 and the editor) that the Predictive Coding interpretation may not be directly but only indirectly supported by the current data, and so we have rephrased the relevant section and substantially softened it in the revised version of the manuscript, in accordance with this valid point.

“Specifically, these simulations demonstrated that a minimal Predictive Coding model implementing biologically plausible constraints (i.e. temporal delays in the communication between regions and time constants) generates α-band travelling waves, which propagate from frontal to occipital regions and vice versa, depending on the “cognitive states” of the model (input-driven vs prior-driven), as confirmed by EEG data in healthy participants (in that case, processing visual stimuli vs. closed-eyes resting state). The view that Predictive Coding could be the underlying principle explaining both the propagation of α-band travelling waves, and the neural changes induced by psychedelics opened-up a tantalizing opportunity for testing assumptions both about the nature of travelling waves and how they should be modulated by psychedelics (Carhart-Harris and Friston, 2019). Although we are restricted to speculation by the lack of direct experimental manipulation of top-down and bottom-up sensory inputs, our prior assumptions were so emphatically endorsed by the data, including how propagation-shifts related to subjective experience, that, in-line with prior hypotheses and motivations for the analyses, we were persuaded to infer about both the functional relevance of cortical travelling waves and brain action of psychedelics. Additional studies manipulating bottom-up and top-down analysis of sensory inputs with alternative perceptual designs will be required in order to confirm the relation between Predictive Coding, α-band oscillatory travelling waves and psychedelics states. Moreover, future studies can now be envisioned to examine how these assumptions translate to other phenomena such as non-drug induced visionary and hallucinatory states.”

We thank the reviewer for the useful reminder. We included in the “Quantifying the waves” paragraph of the revised manuscript a few sentence addressing this issue:

“In addition, it’s important to note that our waves’ analysis focuses at the sensor level, as source projections presents few important limitations such as impairing long-range connections, as well as smearing the signals due to the scalp inference (Alexander et al., 2019; Freeman and Barrie, 2000; Nunez, 1974)”

Reviewer #3: This study of the effects of the drug DMT on the direction and occurrence of EEG traveling waves seems generally plausible to me, although some important aspects are not discussed. I don't have major criticisms. However, as one who has studied EEG traveling and standing waves for many years, I worry that those unfamiliar with EEG wave phenomena may misinterpret some of these results given their partial dependence on the specific experimental methods employed. While I have not read previous papers by these authors that may fill in some of the important gaps, I list below some ideas that any reader interested in EEG waves and their neuro-scientific interpretation must be aware of. A summary paragraph in the section "Quantifying travelling waves" is recommended concerning the following basic concepts that must be understood if the results are to be interpreted correctly. 1) In all but the simplest systems, traveling waves occur in groups (packets) over some range of spatial wavelengths (multiple spatial frequencies, k). This is to be expected in brains, based on both theory and experiment (see Nunez and Srinivasan, 2006; 2014; Nunez, 2000). 2) Any experimental electrode array will be sensitive to only parts of these wave packets, e.g. only waves shorter than the spatial extent of the array and waves longer than twice the electrode separation distance (Nyquist criterion in space) can be resolved. In scalp recordings, the shorter waves may be mostly removed by volume conduction. 3) As a consequence of #2, waves recorded directly from the cortex (as indicated in several recent studies) will emphasize shorter waves than the scalp recorded waves. In the case of small cortical arrays, the ECoG overlap with scalp data may be minimal. Thus, the different estimated wave properties (including propagation direction) need not agree. 4) When waves are traveling in multiple directions at nearly the same time in "closed" systems (e.g., the cortical/white matter), there are only two possible results. Either the waves must damp out or they combine (interfere) to form standing waves (e.g. α waves traveling both forward and backward). One expects that the actual behavior depends on brain state, including the influence of drugs (see Nunez and Srinivasan, 2006; 2014; Nunez, 2000).

We are very grateful to the reviewer for her/his overall positive opinion on our work, and her/his useful suggestions. As recommended, we integrated in the “Quantifying travelling waves” paragraph all the points listed above, with the corresponding references. We believe such changes improved the readability of the paper for those unfamiliar with EEG analysis, and hopefully will be satisfying and adequate for the reviewer.

“As demonstrated by both theoretical and experimental evidence (Nunez, 2000; Nunez and Srinivasan, 2014, 2009), in most systems, including the human brain, traveling waves occur in groups (or packets) over some range of spatial wavelengths having multiple spatial and temporal frequencies. Given any configurations of electrodes, only parts of these packets can be successfully detected, i.e. waves shorter than the spatial extent of the array, and waves longer than twice the electrode separation distance (Nyquist criterion in space). In scalp recordings, the shorter waves may be mostly removed by volume conduction. As a consequence, waves recorded directly from the cortex emphasize shorter waves than the scalp recorded waves. Specifically, in the case of small cortical arrays, the overlap between cortical and scalp data may be minimal, and the estimated wave properties (including propagation direction) may differ. Additionally, it is important to consider that when waves are traveling in multiple directions at nearly the same time in "closed" systems (e.g., the cortical/white matter), waves either damp out or interfere with each other to form standing waves (e.g. α waves traveling both forward and backward). It is reasonable to assume that the behavior of these properties will relate to global brain and mind states, and be sensitive to state-altering psychoactive drugs (Nunez, 2000; Nunez and Srinivasan, 2014, 2009).”

Massimini M, Huber R, Ferrarelli F, Hill S, Tononi G. 2004. The sleep slow oscillation as a traveling wave. J Neurosci 24 :6862–6870. doi:10.1523/JNEUROSCI.1318-04.2004

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But the question remains: Why would Israel order the blasts now?

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As the war on Hamas in Gaza reaches its one-year mark, the Israel Defense Forces now aim to push Hamas’ larger, more powerful allied organization in Lebanon away from its northern border, Carl said.

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The Hezbollah airstrikes have forced many civilians living in the northernmost territories of Israel to flee.

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On Monday – just a day before roughly 2,800 people were injured in Mossad’s head-turning pager attack – the Israeli security cabinet approved “returning the residents of the north securely to their homes” as a formal war objective.

The move marked “the first time that northern Israel is officially included in Israel’s stated war objectives,” according to a joint report by the American Enterprise Institute’s Critical Threats Project and the Institute for the Study of War. 

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For example, Israeli Prime Minister Benjamin Netanyahu said last week that a political settlement alone will not return displaced citizens to northern Israel and that Israel is “preparing for a broad campaign” to this effect.

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It comes as the Israeli Defense Forces have severely weakened Hamas in Gaza, leaving open an uncertain future for the remainder of the war in the Palestinian-controlled territory.

Exactly what comes next remains, but the pager operation shows that Israel was already devoting resources to combating Hezbollah long before the war against Hamas began to wind down, Carl said.

“Of course, we don’t know whether Israel is going to launch a major military offensive into Lebanon anytime soon … but nonetheless, we should understand the pager attack and then the more recent attack that happened today in this broader context of this Israeli desire to push Hezbollah away.”

Blasts’ effects

Regardless of whether more attacks are soon to come, Israel’s recent blasts have already become “Hezbollah’s worst nightmare,” adjunct fellow at The Foundation for Defense of Democracies Seth J. Frantzman told The Post.

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“The nature and scale of the attack also likely stoked confusion and shock among some Hezbollah members,” according to a joint report by the American Enterprise Institute’s Critical Threats Project and the Institute for the Study of War.

“These effects could drive general paranoia within Hezbollah as well, given that Israel has demonstrated repeatedly in recent months how deeply it has infiltrated Iranian and Iranian-backed networks.”

The group shifted to using pagers out of concern over Israel’s ability to hack and track more modern technology with capabilities such as GPS. Now, Hezbollah must regroup to come up with a new communications tactic that may not be as fast and effective.

“The group is already concerned that its own internal landlines are compromised and it prefers to stay away from cell phones, leaving it with fewer options to coordinate its terrorist activities,” Frantzman said.

A second attack on Wednesday targeting other types of devices , such as hand-held radios key for Hezbollah’s communications, will likely cause further concern and paranoia among the terrorists, Frantzman said.

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“The types of devices impacted appear to be from a variety of technologies,” he said. “This kind of secondary strike will certainly cause Hezbollah to have fear and push the group into greater internal chaos.”

What happens now?

After back-to-back attacks, other Iranian-backed groups could also begin to question whether their communication devices have been sabotaged, too.

“There could be a domino affect now among Iranian-backed proxies, thrown into chaos by this attack and forced to rely on new methods to coordinate in the wake of the exploding pagers,” Frantzman said.

While Hezbollah has vowed to take revenge on Israel, a quick counterattack could spell trouble for the terror group.

“The attack will have a long-term psychological blow for Hezbollah because the organization will feel it has been penetrated and suffered a defeat in the eyes of the region,” Frantzman said. “This may cause the group to lash out, but if it lashes out it may do so in haste and make mistakes.”

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• Published: 18 September 2024

Three-dimensional wave breaking

• M. L. McAllister   ORCID: orcid.org/0000-0002-5142-3172 1 ,
• S. Draycott   ORCID: orcid.org/0000-0002-7372-980X 2 ,
• R. Calvert 3 ,
• T. Davey   ORCID: orcid.org/0000-0002-3298-1873 3 ,
• F. Dias   ORCID: orcid.org/0000-0002-5123-4929 4 , 5 &
• T. S. van den Bremer   ORCID: orcid.org/0000-0001-6154-3357 1 , 6  

Nature volume  633 ,  pages 601–607 ( 2024 ) Cite this article

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• Fluid dynamics
• Physical oceanography

Although a ubiquitous natural phenomenon, the onset and subsequent process of surface wave breaking are not fully understood. Breaking affects how steep waves become and drives air–sea exchanges 1 . Most seminal and state-of-the-art research on breaking is underpinned by the assumption of two-dimensionality, although ocean waves are three dimensional. We present experimental results that assess how three-dimensionality affects breaking, without putting limits on the direction of travel of the waves. We show that the breaking-onset steepness of the most directionally spread case is double that of its unidirectional counterpart. We identify three breaking regimes. As directional spreading increases, horizontally overturning ‘travelling-wave breaking’ (I), which forms the basis of two-dimensional breaking, is replaced by vertically jetting ‘standing-wave breaking’ (II). In between, ‘travelling-standing-wave breaking’ (III) is characterized by the formation of vertical jets along a fast-moving crest. The mechanisms in each regime determine how breaking limits steepness and affects subsequent air–sea exchanges. Unlike in two dimensions, three-dimensional wave-breaking onset does not limit how steep waves may become, and we produce directionally spread waves 80% steeper than at breaking onset and four times steeper than equivalent two-dimensional waves at their breaking onset. Our observations challenge the validity of state-of-the-art methods used to calculate energy dissipation and to design offshore structures in highly directionally spread seas.

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Wave breaking continues to be at the forefront of ocean wave research 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . Although it is a widely observable and ubiquitous natural phenomenon, the onset and subsequent process of wave breaking are not fully understood. Alongside not being fully understood, interest in wave breaking is also driven by the central role it plays in key oceanographic and air–sea interaction processes that, in turn, impact the world’s climate 1 , 10 . As waves become very large (steep), breaking occurs, initiating an irreversible turbulent process. The breaking process is the main mechanism for dissipating wave energy in the ocean and affects the transfer of mass, momentum, energy and heat between the air and the sea. Understanding how and when energy is dissipated is crucial to the accurate modelling of ocean waves 11 and is one of the most pressing unresolved issues in wave forecasting 12 . Uncertainty also remains in how breaking waves affect the production of sea spray and bubble-mediated gas exchange, both key factors in climate modelling 1 . Wave breaking is thought to limit the size that waves can grow to, making it an important factor in the formation of extreme or rogue waves 8 . Breaking waves also constitute the most severe loading conditions for offshore structures 13 .

This lack of understanding of wave breaking is a result of challenges associated with modelling breaking waves both numerically and experimentally. To fully resolve wave breaking numerically requires computationally costly, high-fidelity models such as direct numerical simulations of the Navier–Stokes equations (for example, ref. 14 ). Experimentally, although not without its complexities, producing breaking waves is somewhat simpler and quicker (for example, ref. 15 ). However, quantitatively measuring even simple, visually observable properties, such as surface elevation (at high spatial resolution), in the laboratory can be highly challenging, not to mention properties that are invisible to the naked eye, such as fluid kinematics. Both approaches to studying wave breaking (numerical and experimental) have, as a result of these computational and experimental limitations, been carried out primarily for two-dimensional (2D) conditions (in which waves propagate in only a single direction, sometimes referred to as unidirectional or long-crested waves). Thus, although the oceans are clearly three dimensional (3D), an assumption of two-dimensionality underpins most seminal and state-of-the-art research on wave breaking (for example, refs. 15 , 16 ). This assumption constitutes one of the major shortcomings in our current understanding of wave breaking.

Understanding when waves will break, wave-breaking onset, is the first step in fully understanding ocean wave breaking. Following a kinematic description of wave breaking (in which breaking occurs when the horizontal fluid velocity u at the crest of a wave is equal to the crest speed C ), Stokes 17 first proposed a limiting waveform above which waves may become no steeper and breaking will occur. This limit for 2D periodic (monochromatic) progressive waves propagating on deep water occurs at a steepness of k H /2 = 0.44, where k is the wavenumber and H is the wave height. Waves in the ocean are not monochromatic but comprise a spectrum of interacting wave components of different frequencies and directions. The shape and bandwidth of the spectra that underlie a given set of waves can cause the steepness at which wave breaking occurs to vary significantly, with breaking occurring at values of steepness above and below k H /2 = 0.44 (refs. 18 , 19 ).

Progress has been made in recent studies using dynamic 20 , 21 , 22 , energetic 16 and slope-based 19 signatures of breaking. How effective these methods of detecting wave-breaking inception and onset may be in directionally spread conditions remains unknown. Furthermore, the parameters used in these methods are defined locally as a property of individual waves, meaning that they may be used to diagnose but not necessarily predict the onset of wave breaking.

The situation in which two monochromatic waves cross at an angle Δ θ is the most simple example of directionally spread 3D waves. This canonical problem, as reviewed in ref. 23 , illustrates how the mechanism of wave breaking changes for 3D waves, due to a transition from purely travelling waves (Δ θ  = 0°; ref. 17 ) to purely standing waves (Δ θ  = 180°; ref. 24 ). Numerical studies have found that, as the crossing angle is increased, the almost-highest wave increases in steepness until a crossing angle of Δ θ  ≈ 136°, after which the steepness begins to fall, increasing again at an angle of Δ θ  ≈ 160° (refs. 25 , 26 ). This non-monotonic behaviour, as waves transition from travelling to standing, illustrates the interplay between the two different mechanisms by which wave breaking occurs for standing and travelling waves. For travelling waves, spilling or overturning at the wave crest is the mechanism of breaking onset, which occurs when the fluid velocity approaches the crest speed. For monochromatic standing waves, breaking occurs when the wave crest forms a jet and undergoes freefall so that it becomes unstable.

Early experimental studies involving directional spreading showed that introducing small amounts of directionality affects the resulting shape, severity and onset of wave breaking 27 , 28 , 29 , 30 . Johannessen and Swan 29 performed a systematic study of wave-breaking onset for focused wave groups. They observed a monotonic increase of the maximum crest amplitude with directional spreading. In these experiments 27 , 28 , 29 , 30 , the waves were narrowly spread about a single mean direction. In other words, their directional distributions were unimodal or correspond to what may be termed ‘following’ sea states. In the oceans, complex weather conditions can result in ‘crossing’ sea states, with a directional distribution that may be bimodal or even multi-modal. These directional conditions commonly occur when different weather systems combine, for example, when the wind and swell waves are travelling in different directions. An analysis of hindcast data for the Mediterranean Sea from 1979 to 2015 showed that 39% of the spectra were bimodal 31 . Complex highly spread directional spectra are also known to occur during extreme wind conditions such as cyclones 32 . Crossing sea states have been linked to the formation of extreme waves 33 , 34 , 35 . Previous experimental work has shown that wave-breaking phenomena and their effect on extreme-wave formation can become very different in highly spread conditions compared to 2D or narrowly spread conditions 36 , 37 . Crossing sea states represent the most probable conditions that can create highly directionally spread waves in the oceans. Crossing conditions were necessary to recreate the Draupner rogue wave in laboratory experiments 36 , as amplitude-limiting wave breaking made replicating at scale the surface elevation measured in the field impossible in less spread conditions, with important implications for oceanic rogue waves in general 8 .

This paper presents new experimental results for the effect of directional spreading on wave-breaking onset. The experiments were carried out in a unique circular wave tank capable of generating waves that travel in all directions, thus removing any limitation on the direction of travel of the waves. We identified the point at which wave-breaking onset occurs for focused wave groups with equal peak frequencies on deep water with a range of representative directional distributions. We then measured these with a purpose-built high-density wave gauge array to capture the spatial structure of the surface and, thus, the local slope of the waves at breaking onset. Focused wave groups were chosen for two reasons. First, they can be used as deterministic representations of extreme waves in random seas 38 , 39 and are, therefore, commonly used in laboratory and numerical studies of breaking waves 28 , 29 , 40 . Second, focused wave groups enable an explicit examination of the effect of directional spreading on the physical mechanisms involved in wave breaking. This effect can be obscured in statistical averages in a study using random seas, which would be further hampered by the inevitable effect of reflections in laboratory experiments with large directional spreading.

Based on these experimental results, we provide a general parameterization for 3D wave-breaking onset that can be used in wave forecasting and offshore engineering alike. The ability to generate omnidirectional waves paired with measurements from the high-density gauge array allowed us to examine the full range of 3D wave-breaking phenomena and capture the physical mechanisms of wave breaking. We classify the physical mechanisms into three regimes. Finally, we explored a new type of behaviour beyond breaking onset that exists only for highly directionally spread waves where, unlike in 2D, wave crests are no longer limited by breaking.

In our experiments, we varied the directional spreading using the two parameters spreading width σ θ and crossing angle Δ θ , where σ θ corresponds to the standard deviation of a wrapped normal distribution and Δ θ to the angle between the mean directions of two superimposed wrapped normal distributions (values of σ θ and Δ θ shown in Fig. 1f ). For each directional distribution, we iterated the input steepness α 0 of the wave group to find the value of α 0 at which wave-breaking onset occurs and thus obtain the maximally steep non-breaking wave group. The input steepness α 0 was calculated as a 0 k p , where a 0 is the sum of discrete wave components and k p is the peak wavenumber. We then measured this maximally steep non-breaking wave group using the high-density wave gauge array (see Fig. 1 for example measurements). The images recorded by the camera closely resemble the surface measurements obtained using the array (Fig. 1 ).

Figure 1d (top) shows how the amplitude of the steepest non-breaking waves increased as the spreading width σ θ was increased. The shapes of the wave groups in time appear similar (as would be expected following linear wave theory). The same was observed as the crossing angle Δ θ was increased (Fig. 1d , bottom). The corresponding frequency spectra were also similar until around 1.5 times the spectral peak, when the spectra that correspond to wave groups with the narrowest directional spreading exhibited a slightly fatter tail (Fig. 1e , top). Apart from at low frequencies (where differences can be explained by subharmonic bound waves 41 ), the spectra that correspond to the crossing wave groups show very little variation (Fig. 1e , top). These observations alongside the temporal symmetry of the wave groups may suggest that only the least directionally spread waves were significantly affected by nonlinear focusing, which is consistent with observations in refs. 29 , 36 , 37 . Examining our plots of surface elevations in space (Fig. 1a–c,g–i ) and time (Fig. 1d ) leads to quite different outcomes. In time, the different wave groups appear quite similar, but in space they appear very different. This demonstrates the need to measure the surface elevation in both space and time, as obtaining estimates of spatial surface properties, such as steepness, from the time domain will be very misleading for 3D waves (see also ref. 42 ).

Wave-breaking onset

We estimated both the global steepness and the local slope at the breaking-onset threshold. The global steepness S is a measure of the steepness based on a sum of spectral components S  = ∑ a n k n , where a n and k n are the amplitudes and wavenumbers of discrete wave components (see Methods for a review of different measures of steepness). Increasing σ θ from 0° to 50° for following (unimodal) directionally spread wave groups (Δ θ  = 0°) caused the threshold values of steepness \({S}_{{\rm{M}}}^{\star }\) to double (Fig. 2a ), where the subscript M denotes values measured in the wave tank and the superscript ⋆ denotes variables associated with the breaking-onset threshold (the steepest non-breaking waves we created). Increasing the crossing angle for two crossing wave groups also caused the threshold values of steepness to increase. They doubled at an angle of around 90° (compared to Δ θ  = 0°), at which point the breaking-onset steepness appeared to plateau (Fig. 2b ).

In Fig. 2c , the surface of \({S}_{{\rm{M}}}^{\star }\) is smooth and demonstrates how the breaking-onset steepness increased with the overall degree of spreading. In our experiments, we created following (Δ θ  = 0) and crossing (Δ θ  ≠ 0) wave groups, which have unimodal and bimodal directional spreading distributions, respectively. Directional spectra in the ocean may be multi-modal in direction, and an obvious distinction between following and crossing seas cannot always be made. It is, therefore, unlikely that all sea conditions may be parameterized by spreading width σ θ and Δ θ in the same manner as in our experiments. The effects of spreading ( σ θ ) and crossing (Δ θ ) are not unique (Fig. 2c ), and both are simply different measures of the overall degree of spreading of a given sea state. To try to provide a general parameterization of the 3D wave-breaking onset, we introduced two single-parameter measures of spreading: Ω 0 and Ω 1 .

Both Ω 0 and Ω 1 have values of 0 for unidirectional waves and increase as waves become more spread. The integral measure of spreading Ω 0 is a measure of the degree to which waves are standing (as opposed to travelling). It is simple to calculate based on the directional spectrum and is already output by the phase-averaged wave forecasting models WAVEWATCH III (ref. 43 ) and ECWAM (ref. 44 ). One drawback is that Ω 0 has the same value for 2D and axisymmetric standing waves, which have been shown to have different limiting slopes (1 and 0.7071, respectively 24 , 45 ). The phase-resolved measure of spreading Ω 1 is a measure of how directionality affects the (linearly predicted) local slope of individual waves (the latter has been linked to breaking onset 19 ), which is more complex to calculate. See Methods for a precise mathematical definition of both measures. Figure 3a,d illustrates how these two parameters vary with σ θ and Δ θ . The remaining panels of Fig. 3 demonstrate that both measures of spreading, Ω 0 and Ω 1 , can be used to uniquely parameterize the global steepness S ⋆ and the maximum local slope ∣ ∇ η ∣ ⋆ , which is measured using the high-density gauge array associated with 3D wave-breaking onset.

The maximum values of the measured global steepness are reasonably well described as a function of Ω 0 but do not collapse exactly onto a single curve (Fig. 3c ). However, from Fig. 3b it appears that the maximum local surface slope ∣ ∇ η ∣ ⋆ could potentially collapse onto a single curve when plotted as a function of Ω 0 (excluding one large outlying value for σ θ  = 0° and Δ θ  = 135°, where it was very difficult to distinguish the onset of breaking). The measured values of ∣ ∇ η ∣ ⋆ that we present are most probably lower than the actual maximum local slope owing to the finite resolution of our wave gauge array (and because we consider the absolute value of slope). This provides the rationale for shifting the fitted curve vertically (black dotted line; see Extended Data Table 2 for the parametric fitting coefficients) so that it is equal to \(\tan (3{0}^{^\circ })\) at Ω 0  = 0 (resulting in the black dashed line). When shifted like this, the range of surface slopes that the fit describes lies between \(\tan (3{0}^{^\circ })\) and \(\tan (4{5}^{^\circ })\) , which are the limiting slopes of monochromatic travelling and standing waves, respectively. For 2D numerical simulations, McAllister et al. 19 showed that wave-breaking onset occurred at a critical value of the local slope of \(\tan (3{0}^{^\circ })\) . Thus, the results in Fig. 3b suggest that the maximum local slope for 3D waves lies between these two limits, \(\tan (3{0}^{^\circ })\) and \(\tan (4{5}^{^\circ })\) .

In phase-averaged forecasts 43 , the local properties of individual waves are not resolved, but breaking must still be predicted. Therefore, parameters that are derived from measurements of individual waves may be used to diagnose but not predict the onset of wave breaking. Unlike global steepness, the local slope is a diagnostic not a predictive wave-breaking parameter. Linking the global breaking-onset steepness \({S}_{{\rm{M}}}^{\star }\) to a single-parameter measure of spreading will lead to a breaking-onset parameterization that has the potential to be used predictively (Fig. 3c,f ). The global steepness at which breaking onset occurs appears to be a linear function of Ω 1 (see Extended Data Table 2 for the parametric fitting coefficients). A simple parameterization like this, which links a property of the directional spectrum to the global steepness at which wave-breaking onset will occur, has the potential to predict when waves will break in the ocean in a stochastic, phase-averaged framework. However, Ω 1 is not necessarily a parameter that can be calculated quickly enough as a part of such large-scale models. Although not perfect, the fit in Fig. 3c may also describe the leading-order effect that increasing the directional spreading has on breaking-onset steepness as a function of Ω 0 , a parameter that is simpler to calculate. Also note that large values of Ω 0 , which correspond to near-axisymmetric conditions where this parameterization is less accurate, do not occur in the ocean in practice.

Wave-breaking mechanisms

As the overall degree of spreading was varied, the way that the resulting wave groups broke changed (Fig. 4 ). In narrowly spread conditions, as breaking is initiated, the crest of the wave overturns with a gentle spilling to more severe plunging motion of a horizontal jet of fluid. These observations of wave breaking have a strong resemblance to the familiar type of breaking that occurs in 2D, which we term ‘travelling-wave breaking’ (Fig. 4 , type I). For finite spreading, ‘localized travelling-wave breaking’ occurs. As the degree of spreading is increased for a unimodal spreading distribution, the breaking becomes increasingly localized in the lateral dimension ( y axis). As spreading is increased further, an axisymmetric standing wave begins to form. The initial breaking then occurs in the form of vertical jet 37 , which we term ‘standing-wave breaking’ (Fig. 4 , type II). Note that standing-wave breaking can be 2D or axisymmetric.

Illustrations of the three different wave-breaking phenomena: type I overturning ‘travelling-wave breaking’, type II vertical-jet forming ‘standing-wave breaking’ and type III ‘travelling-standing-wave breaking’. In type III, a near-vertical-jet emanates from a fast-moving ridge that forms as the crossing wave crests constructively interfere. Corresponding images were captured during experiments.

For crossing wave groups at low crossing angles, the waves break in a localized travelling manner like that of narrowly spread unimodal groups. As the crossing angle is increased, a fast-moving ridge of fluid forms along the mean direction of propagation as the crossing wave groups constructively interfere. Breaking occurs atop this ridge of fluid (Fig. 4 , middle), which we term ‘travelling-standing-wave breaking’ (Fig. 4 , type III). This form of breaking, which occurs in a similar manner to waves impacting on a wall or 2D standing waves 46 , is strikingly different from travelling-wave breaking. Additionally, travelling-standing-wave breaking results in qualitatively different mechanisms of air entrainment and mixing. Inertial dissipation models, such as the one presented in ref. 40 , which consider properties such as the height over which a plunging jet falls to estimate potential energy loss, will not be readily applicable to this type of wave breaking. Additionally, the vertical jetting motion we observe poses a risk to offshore vessels and structures and has implications for the wave-amplitude-limiting effect attributed to wave breaking (‘Post-breaking-onset behaviour’).

We have annotated Fig. 2 to indicate the regimes (in terms of the degree of directional spreading) in which these different breaking phenomena occur. The transition from localized travelling to travelling-standing-wave breaking is gradual, and the regimes can be defined only approximately, as denoted by the grey shaded area in Fig. 2b . The transition to travelling-standing-wave breaking is associated with a plateau in the global steepness at which the breaking onset occurs (Fig. 2c ).

Post-breaking-onset behaviour

Stokes 17 first derived the limiting form of a 2D progressive wave on deep water, implying that the steepness of unforced monochromatic waves has an upper limit ( k H /2 = 0.44). Waves with a steepness at or close to this limit are unstable and will break, and breaking is the process that provides an upper limit to wave height. When waves are 3D, the surface kinematics and breaking phenomena change, and the concept of a limiting waveform may no longer be viable 36 , 37 , 47 . To examine how directional spreading affects the extent to which wave breaking provides an upper limit to wave steepness, we performed further experiments in which we increased the steepness input to the wavemakers to 112.5, 125 and 150% of the values we observed wave-breaking onset at (denoted by \({\alpha }_{0}^{\star }\) , where α 0 is the input steepness) for a subset of four directional distributions ( σ θ  = 20° and Δ θ  = 0°, 45°, 90° and 135°).

As the input steepness of the waves was increased beyond the breaking threshold, Fig. 5a shows that the maximum amplitude measured in the wave tank increased beyond the breaking-onset threshold for all of the directional spreading conditions we tested. For the less spread cases (Δ θ  = 0°, 45° and 90°), the measured amplitude increased slightly less than the increase in the input amplitude. For Δ θ  = 135°, the measured amplitude was linearly proportional to the input amplitude up until \({a}_{0}/{a}_{0}^{\star }=1.25\) . At \({a}_{0}/{a}_{0}^{\star }=1.5\) , the measured amplitude a M reached a value almost 80% greater than the threshold \({a}_{{\rm{M}}}^{\star }\) . These results are consistent with experiments in which the amplitude of asymmetric waves created at a similar scale was limited only by the stability of the jets that formed 37 . The surfaces at the times of maximum surface elevation for these experiments (Δ θ  = 135°) correspond to the purple markers in Fig. 5a . Figure 5b–e shows the extent to which the maximum surface elevation continued to increase post-wave-breaking onset. These surfaces also show the formation of a steep ridge along the mean direction of propagation ( y  = 0), which we associate with travelling-standing-wave breaking. Note that the maximum surface elevations in Fig. 5 may have occurred after the onset of breaking and thus be measurements of breaking jets. Post-breaking onset, the gauges may measure a mixture of water and entrained air. In such cases, the overall resistance measured by the wave gauges will be higher due to the presence of air, and thus, the measured surface elevations may be lower than the highest point reached by a whitecap. If the free surface reaches elevations greater than those at breaking onset, this has important implications for the design of offshore structures regardless of whether the waves are pre- or post-breaking onset.

a , Post-breaking-onset behaviour. The graph shows the measured wave amplitude as a function of input amplitude, both normalized by values at the breaking onset, for experiments carried out at 112.5, 125 and 150% of the input breaking-onset steepness for wave groups with directional width σ θ  = 20° and crossing angles Δ θ  = 0°, 45°, 90° or 135°. The black dashed line is \({a}_{{\rm{M}}}/{a}_{{\rm{M}}}^{\star }={a}_{0}/{a}_{0}^{\star }\) . b – e , Surface elevation (side on, viewed from the x – z plane) at the times of maximum surface elevation, which correspond to the purple markers in a for σ θ  = 20° and Δ θ  = 135°. Error bars correspond to ±1 standard deviation ( Methods ).

The experimental results in this paper provide a general diagnostic (in terms of local slope) and predictive (in terms of global steepness) parameterization of 3D wave-breaking onset for deep-water ocean waves that is valid for all degrees of directional spreading. We significantly expand upon existing quantitative observations of 3D breaking waves. The local slope functions well as a diagnostic breaking-onset threshold parameter for 3D waves, as for the 2D numerical results in ref. 19 . The parameterization can be used to improve phase-averaged wave models, such as WAVEWATCH III 43 and ECWAM 44 and design codes for offshore structures.

The mechanism of wave breaking is radically altered by directional spreading, and this mechanism has implications for the amount of air entrained and energy dissipated, which should be the focus of future research. Improvements to the energy-dissipation terms in phase-averaged wave models, which are based on the assumption of two-dimensionality or weak directional spreading, could make use of the parameterization of 3D wave-breaking onset provided. We showed that 3D waves start to break and thus dissipate only at greater steepnesses, but future work should also aim to quantify the effect that directional spreading has on the dissipation itself.

Unlike in 2D, waves can exceed the onset steepness at which wave breaking first occurs. This has significant implications for understanding extreme-wave formation, which in turn has implications for the design of offshore structures, as wave breaking typically curtails crest-height exceedance probability distributions. The vertical jetting-type breaking behaviour observed for highly directionally spread seas is of special significance to offshore structure design, as wave-in-deck loads can be catastrophic. Designs that minimize the probability of wave-in-deck loads may come at a considerable cost 48 .

Our experiments were designed to address the fundamental problem of how three-dimensionality affects wave breaking. In doing so, we have made several choices, some of which affect the generality of our findings. We have not examined the effects of water depth or spectral shape (bandwidth), both of which are known to affect breaking onset. We expect their effects to be independent of those reported here. Crossing sea conditions can occur when wind and swell systems with different peak frequencies exist simultaneously, whereas the crossing wave groups in our experiments have the same frequencies. Therefore, we did not assess the effect of bimodality in frequency. Additionally, steep ocean waves are often accompanied by strong winds, ocean currents and rain. In windy conditions, long and much shorter waves interact, and long waves can modify the dynamics of the short waves 49 . Our results did not take into account these effects. Our approach of considering wave breaking in isolation of the above effects is supported by the results in ref. 50 , which suggest that dominant-wave breaking (at scales corresponding to the peak of the spectrum) is primarily driven by the properties of the waves themselves.

Finally, waves in the ocean are random, but our experiments used deterministic, focused, wave groups. Focused wave groups are commonly used to represent breaking and, thus, extreme waves in laboratory and numerical studies 28 , 29 , 40 , following a theoretical framework based on Gaussian random seas 38 , 39 . The use of wave groups is, therefore, unlikely to limit the generality of our findings. A new approach for predicting extreme-wave probabilities in highly directionally spread seas should take account of the greater breaking-onset steepness, as identified in this paper, but also of the effect of directionally spreading on the probabilities of wave crests (and thus breaking), which was not studied here. Such an approach should be validated in future work with new experiments using highly directionally spread random seas.

Experimental set-up

The experiments were carried out at the FloWave Ocean Energy Research Facility at the University of Edinburgh. The facility has a 2-m-deep, 25-m-diameter, circular wave tank surrounded by 168 flap-type wavemakers. The tank’s circular geometry can create waves travelling in any direction. The wavemakers are operated using a force-feedback control strategy. In this mode of operation, the wavemakers generate and also absorb waves to mitigate the build-up of reflections in the tank.

The surface elevation was measured in the tank using resistive wave gauges. To perform measurements with a high spatial resolution, we developed an 8 × 8 array of wire wave gauges spaced at 0.1 m intervals (array A). The array covers an area of 0.7 m by 0.7 m. Experiments were repeated with the array positioned in six different locations to achieve an effective measurement area spanning from x  = −1.4 to 2.8 m and y  = 0 to 0.7 m, where ( x  = 0,  y  = 0) is the centre of the circular wave tank (Extended Data Fig. 1 ). We also used a linear array of rigid drop-down wave gauges (array B) to measure the surface elevation over a larger proportion of the tank from x  = −8 to 6 m in two locations where y  = 0 and y  = −0.7 (Extended Data Fig. 1 ). Measurements from array B were used to confirm that the waves were focused in the region covered by array A, to validate measurements from the newly developed wire wave gauges and, when positioned at y  = −0.7 m, to test the symmetry of the waves, which we later assumed when plotting surfaces. The gauges were cleaned and calibrated at the start of each day of testing. Videos of the experiments were recorded using two cameras positioned at the side of the wave tank (Extended Data Fig. 1 ).

Experimental matrix

To produce breaking waves, we generated steep focused wave groups using linear dispersive focusing. Inputs to the wave tank were defined using linear wave theory. The desired surface elevation

is constructed as a linear summation of wave components with N discrete frequencies that propagate in M discrete directions with frequency ω n and wavenumber \({k}_{n}=| {{\bf{k}}}_{m,n}| =| [{k}_{n}\cos ({\theta }_{m}),{k}_{n}\sin ({\theta }_{m})]| \) that obey the linear dispersion relation \({\omega }_{n}^{2}=g{k}_{n}\tanh ({k}_{n}h)\) where g is the gravitational acceleration and  h is water depth. The direction θ  = 0 corresponds to waves that propagate in the positive x direction. z is positive in the upwards direction, with z  = 0 corresponding to the still-water level, and t is time. The phases ψ m , n are defined such that all components are in phase at the desired focus time ( t  = 0) and position ( x  = 0,  y  = 0). Defining the phase in this manner creates a focused wave group, assuming linear dispersive focusing. The duration of each experiment T defines the resolution of the discrete frequencies ω n  = 2π n / T used in equation ( 1 ). We set this to 64 s and set the number of discrete directions to M  = 144. We left 10 min of settling time between each experiment to allow any unabsorbed reflections in the tank to dissipate, which ensured that each experiment was carried out in as close to quiescent conditions as possible.

We defined the amplitude spectrum of the wave groups we created using the JONSWAP spectrum:

The parametric form of the JONSWAP spectrum in equation ( 2 ) generally corresponds to an energy spectrum (and, thus, the amplitude of discrete wave components \({a}_{n}\propto \sqrt{E({f}_{n})}\) . Instead, we set the amplitude spectrum to be proportional to the JONSWAP spectrum itself, a n   ∝   E ( f n ), as this gives the correct shape of extreme (and, thus, breaking) waves in an underlying random Gaussian sea 38 , 39 . In equation ( 2 ), f  =  ω /2π is frequency, and we set the peak frequency f p  = 0.75 Hz. Here, σ  = 0.9 and 0.7 for f  <  f p and f  >  f p , respectively. We chose a JONSWAP spectrum as this spectral shape represents well typical ocean conditions. We set γ  = 1 to give a broad underlying spectrum. A broad spectrum results in a wave group that is well dispersed and less steep at the wavemakers, which minimizes the errors associated with generating linear waves. Moreover, our preliminary experiments found that wave groups with broad underlying spectra exhibited only a single breaking crest, whereas focused wave groups based on narrower spectra were more likely to break several times.

We defined the directional spreading of the wave groups we created using a wrapped normal distribution:

where θ is the angle of propagation, θ 0 is the mean direction and σ θ is the spreading width. For crossing groups (Δ θ  ≠ 0), we superimposed two wrapped normal distributions with mean directions θ 0  = ±Δ θ /2. When Δ θ  = 0, we set the mean direction θ 0  = 0. This means that all the wave groups we created had the same mean direction of propagation, which was along y  = 0 in the positive x direction. Note that the spreading that we implemented in our experiments was independent of frequency.

Extended Data Table 1 details the different directional distributions we examined. The spectral components a n corresponding to each experiment were scaled to give the desired input steepness α 0  =  a 0 k p at the intended point of linear focus at the centre of the wave tank ( x  = 0,  y  = 0), where a 0 is the linearly predicted amplitude at the focus and k p is the peak wavenumber. We performed several experiments for each directional distribution and varied the input steepness with decreasing increments of Δ α 0 reaching a minimum of 0.0125 (Δ a 0  ≈ 2 mm) to find the point at which breaking onset occurs. Breaking onset was identified visually. Note that it is also possible to detect breaking using the surface elevation 51 , 52 and acoustic measurements 40 . Once this threshold was determined, the largest non-breaking wave for each directional distribution was recreated and measured with the high-density gauge array (array A) located in several positions to obtain measurements of surface elevation over the desired area (Extended Data Fig. 1 ). To understand how directionality affects wave evolution beyond breaking-onset steepness, the same measurement process was also carried out for waves at 112.5%, 125% and 150% of the breaking-onset steepness \({\alpha }_{0}^{\star }\) for selected directional spectra (denoted with a dagger symbol in Extended Data Table 1 ).

Definitions of the breaking-onset steepness

The quantities used to parameterize wave-breaking onset can have a significant influence on the perceived results. For example, in 2D, the global and local definitions of steepness (see below) can lead to apparently conflicting parameterizations of breaking-onset steepness (as a function of frequency bandwidth) 19 , 53 , 54 . Additionally, when waves are directionally spread, steepness parameters, which are predominantly 2D, can become ill-defined.

Geometric parameters, such as global steepness or local slope, are generally not considered to be good at distinguishing between breaking and non-breaking waves 18 . Despite this, we investigated geometric and spectral measures of steepness for two main reasons. First, steepness-based parameters are the simplest to measure. Second, recent work by ref. 19 has shown that in 2D, the local slope of waves may function well as a breaking-onset threshold parameter (ref. 19 also showed that the perceived issues associated with geometric parameters are the result of inconsistent definitions). Although slope may be a useful parameter for indicating the onset of wave breaking, it is a local parameter of individual waves so that it cannot be readily used to predict wave-breaking onset in phase-averaged wave models. Instead, the global steepness can be used to predict the breaking onset within a given sea state. Potentially, it has a broader application for predicting wave breaking in wave forecasting. Thus, we sought to parameterize the breaking onset in terms of both the local slope and the global steepness. By performing experiments with focused wave groups, we could gain a stochastic understanding of how the global steepness may relate to the local wave slope and the wave-breaking onset. This approach is based on the theory of quasi-determinism 38 , 39 , which states that extremes within a sea state exist in the form of wave groups. This relies upon the assumption that the largest waves break. In other words, we are concerned with dominant-wave breaking (at length scales that correspond to the spectral peak).

Global steepness S

The global steepness S provides a linear approximation to the maximum local slope \(\max (\partial {\eta }^{(1)}/\partial x)\) that a wave may have for a given amplitude spectrum (distribution). S is calculated discretely as a sum of N wave components:

The global steepness relates only to the spectrum underlying a given set of waves and does not account for nonlinear wave evolution, the phase coherence (focusing) of individual waves or the directions in which the waves are travelling. Thus, the global steepness can be thought of as a measure of spectral steepness (for amplitude spectra). For focused wave groups, in which the phases of wave components are aligned, the value of S may be realized and exceeded for finite-amplitude nonlinear waves. In conditions where the phases of wave components are not predetermined in such a way or where strong nonlinear focusing occurs, the steepness S may not be related well to the actual slope of the waves in question. Additionally, equation ( 4 ) does not take into account the directions in which waves propagate and is a 2D estimate of the slope.

We measured the global steepness of the waves we created in the tank S M using the frequency spectrum of the surface-elevation time series measured at the centre of the tank. If nonlinear focusing were significant, local changes to the spectral shape would cause the measured value of S M to differ from the underlying linear value S . The waves produced in the tank were less steep than the input values S 0 , and this underproduction by the wavemakers was a function of the overall directional spread (consistent with ref. 41 , which studied less steep, non-breaking, wave groups). As a result, the input values of S 0 were not entirely representative of the waves created in the tank, so instead, we report the measured values S M . We believe that this decision is justified, owing to the observations we make in Fig. 1d,e , which suggest that for the more directionally spread waves that we created, nonlinear focusing is not significant.

3D ‘global steepness’

We now introduce a measure akin to the global steepness that accounts for the effects of directional spreading. For directionally spread (3D) linear waves, the surface slope has two orthogonal components:

The directions in which waves propagate affect how they contribute to the total surface slope. When calculating S in equation ( 4 ), which corresponds to the maximum possible 2D linear slope ( M  = 1 and  θ  = 0), the phase argument in equation ( 5 ) is ignored. A similar approach may be used to calculate a 3D equivalent of S :

For broadbanded spectra, the slope maxima of each component are not necessarily colocated in space. As a result, the 3D global slope S 3D can be quite different from the maximum achievable slope for a linear focused wave group \(\max (| \nabla {\eta }^{(1)}| )\) . Thus, we did not use S 3D in ‘Results’. To calculate the linear maximum achievable slope \(\max (| \nabla {\eta }^{(1)}| )({\sigma }_{\theta },\Delta \theta )\) , we searched for the time and position at which the absolute slope was a maximum for a focused wave group based on each directional distribution. Extended Data Fig. 2 demonstrates how two focused wave groups with the same global steepness S can have different waveforms and local slopes (Extended Data Fig. 2b,c ). The wave group in Extended Data Fig. 2a is unidirectional, whereas the wave group in Extended Data Fig. 2d is an axisymmetric standing wave.

Local steepness

Local measures of steepness, unlike the global steepness, implicitly capture the effects that nonlinear evolution and directionality may have on the waveform of steep waves. As a result, such measures may provide better descriptions of the free surface elevation at breaking onset for any given wave. The local steepness has various forms for which either a locally measured wave amplitude a or a height H (Extended Data Fig. 2 ) are non-dimensionalized using a chosen length scale, which is most commonly the wavelength λ ( a k and k H /2, where k  = 2π/ λ ). The choice of length scale and the manner by which it is calculated (in space or time 42 ) have significant implications for the resulting measure of steepness. In highly spread conditions, even if highly resolved spatial measurements of surface elevation are available, the wavelength and other length scales become ill-defined 47 . Even in 2D, where some of the aforementioned issues are resolved, the local steepness is not always a robust indicator of breaking onset 19 . As a result, we have not used local steepness parameters herein.

Local slope ∣ ∇ η ∣

The local surface slope is a form of steepness that is well defined and has been shown to be a robust indicator of breaking and non-breaking behaviour in 2D 19 . In 3D, the local slope has x and y components. We report the magnitude of the gradient vector \(| \nabla \eta | =\sqrt{{\eta }_{x}^{2}+{\eta }_{y}^{2}}\) , where η x  = ∂ η /∂ x and η x  = ∂ η /∂ y . The local slope is akin to \(\max (| \nabla {\eta }^{(1)}| )\) but is locally measured not linearly predicted and is, thus, affected by nonlinear wave evolution as well as by directionality.

Measures of spreading for Ω 0 and Ω 1

The integral measure of spreading ω 0.

The integral measure of spreading,

can be used to parameterize the overall spreading of the waves we created, where θ is the direction in which the waves travelled and Ω ( θ ) is the directional distribution. This parameter is one minus the in-line velocity reduction factor used in ref. 29 and in engineering practice. It is a frequency-independent equivalent of the directional width parameters output by WAVEWATCH III (ref. 43 ) and ECWAM (ref. 44 ). The parameter Ω 0 provides a measure of the degree to which wave components are standing (as opposed to travelling). In the limits σ θ  → ∞ or Δ θ  → π, the value of Ω 0 tends to one. It tends to zero for unidirectional waves, σ θ  → 0 and Δ θ  → 0. The integrand in equation ( 8 ) is a function only of θ , as spreading was independent of frequency in our experiments. In the ocean, spreading can be a function of frequency 32 , 55 . Here, we define frequency-independent spreading to reduce the overall complexity of our experiments. For the directional distributions that we define (equation ( 3 )), equation ( 8 ) can be expressed as

One limitation of Ω 0 is that it has the same value for two unidirectional counter-propagating wave groups ( σ θ  = 0 and Δ θ  = π) as for an axisymmetric wave group ( σ θ  = ∞). These two conditions represent 2D and axisymmetric standing waves, which are known to have different limiting forms 24 , 45 . Figure 3a illustrates how Ω 0 varies as a function of σ θ and Δ θ . The markers show where our experiments are located within this parameter space.

Phase-resolved measure of spreading Ω 1

As mentioned above, the integral measure of spreading Ω 0 does not fully capture the effects of directionality. We, thus, sought to find an alternative parameter. Directionality affects the shape of wave groups, as demonstrated in Extended Data Fig. 2 , which shows the surface elevation (at t  = 0) of two wave groups with the same global steepnesses S but different local slopes and steepnesses. Following arguments made in 2D studies, which demonstrated that certain values of local slope ∂ η /∂ x may trigger breaking 19 , 54 , 56 , alongside our observations in this paper, which suggest that the focusing of directionally spread wave groups is predominantly linear, we introduced a single-parameter measure of spreading that describes how directionality affects linearly predicted wave slope.

For a given directional spectrum, the degree of spreading affects the maximum surface slope in space and time, which can be predicted linearly as \(\max (| \nabla {\eta }^{(1)}| )\) , where η is the free surface surface elevation. Normalizing \(\max (| \nabla {\eta }^{(1)}| )\) by the the 2D global steepness S ( \(\max (| \nabla {\eta }^{(1)}| )/S\) ) gives a measure of directional spreading, which we will call Ω 1 , that reflects how directional spreading affects the potential slope of 3D wave groups (we use the superscript (1) to emphasize that ∇ η (1) is predicted linearly). The phase-resolved spreading measure Ω 1 may appear to be a linear measure of slope. However, because we normalize by the global slope (which is a 2D approximation of the slope), \({\varOmega }_{1}=1-\max (| \nabla {\eta }^{(1)}| )/S\) is a measure of directional spreading, which is purely a function of the directional distribution.

Note that, although Ω 0 may not fully describe the effects of directional spreading, it can be calculated simply using operations (integration) on the directional spectrum and is already an output of phase-averaged wave models. Calculating \(\max (| \nabla {\eta }^{(1)}| )\) involves the use of linear wave theory to search for a maximum slope in space and time. In the narrow-banded limit Ω 1  → 1 −  S 3D / S , where S 3D is a spectral measure of the slope that ignores phase. Like Ω 0 , 1 −  S 3D / S is quick to calculate, but its values are quite different to Ω 1 for the broadband spectra in our experiments.

Parametric fitting coefficients

Extended Data Table 2 details the coefficients for the parametric curves fitted to experimentally measured values of local slope ∣ ∇ η ∣ ⋆ and global steepness \({S}_{{\rm{M}}}^{\star }\) corresponding to breaking onset, which are presented in Fig. 3 .

Error quantification

We identified and quantified three main sources of experimental error: wave gauge calibration error, the error associated with the discrete steps by which we varied the input steepness when identifying the breaking onset, and random error, which we estimated from repeated experiments. All three sources of error affected our estimates of the global steepness S M and the local slope ∣ ∇ η ∣ . The local slope was further affected by an error that results from estimating the slope from discrete points corresponding to gauges in the high-density gauge array.

Error in the global slope S

First, we will discuss how the above sources of error can affect values of the global steepness at breaking onset (Figs. 2 and 3c,f ). The mean calibration error of the wave gauges was ±0.3%, which resulted in an absolute error of Δ a 0  = 0.002 m at worst ( \(\Delta {S}_{{\rm{M}}}^{\star }=0.008\) ). The breaking onset was identified by increasing the input steepness in discrete steps until breaking could be identified visually. We estimated this error as the difference between the threshold values of \({S}_{{\rm{M}}}^{\star }\) ( \({\alpha }_{0}={\alpha }_{0}^{\star }\) , for the steepest non-breaking waves) and values of S M calculated for experiments identified as the least steep breaking waves ( \({\alpha }_{0}={\alpha }_{0}^{\star }+0.0125\) ). The mean value of this error across experiments was \(\Delta {S}_{{\rm{M}}}^{\star }=0.0082\) . To quantify the random error, we used the standard deviation of values of \({S}_{{\rm{M}}}^{\star }\) obtained from repeats of the same experiment to quantify the experimental repeatability. The mean value of this standard deviation across our experiments was \(\Delta {S}_{{\rm{M}}}^{\star }=0.011\) . If these three errors are treated as independent and combined, the resulting error bars are smaller than the markers used to plot \({S}_{{\rm{M}}}^{\star }\) in Figs. 2 and 3 . Therefore, we did not include error bars for measured values of the global steepness at breaking onset \({S}_{{\rm{M}}}^{\star }\) in Figs. 2 and 3c,f .

Error in the post-breaking-onset amplitude a M

For the post-breaking-onset behaviour in Fig. 5 , the error bars correspond to the standard deviation of the maximum amplitudes measured across repeated experiments. For these experiments, the calibration error was negligible in comparison to the random error obtained from repeated experiments (the error associated with identifying the breaking onset in discrete steps was not applicable to these experiments).

Error in the local slope ∣ ∇ η ∣

The high-density gauge array was designed such that the gauges were as closely spaced as possible, while still preventing electric ‘cross-talk’ between the wire gauges so that we could obtain the best possible estimates of local slope. To estimate the local slope ∣ ∇ η ∣ (Fig. 3b,e ), we performed first-order central differencing, which is associated with a truncation error. To obtain an estimate of this error, we performed a second-order bivariate Taylor-series expansion of η ( x ,  y ), from which we obtained an estimate of the error of the gradient vector:

We applied the first-order central differencing twice to obtain estimates of the second derivative and set the error of the magnitude of the local slope to be equal to the magnitude of the error of the gradient vector in equation ( 10 ):

For the measured local slope in Fig. 3b,e , the three aforementioned sources of error were negligible in comparison to the truncation error defined in equation ( 11 ), and the error bars in Fig. 3b,e correspond to ±Δ ∣ ∇ η ∣ .

Data availability

The data generated in this study are available at Zenodo ( https://zenodo.org/records/10818627 ) 57 .

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Acknowledgements

This research was performed with funding from the Engineering and Physical Sciences Research Council [EP/V013114/1] and Science Foundation Ireland [21/EPSRC/3733].

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Contributions

Conceptualization: M.M., S.D., F.D. and T.v.d.B. Methodology: M.M., S.D. and T.v.d.B. Software: M.M. Validation: M.M. and R.C. Formal analysis: M.M. Investigation: M.M., S.D. and R.C. Resources: M.M., R.C. and T.D. Data curation: M.M., R.C. and T.D. Writing—original draft: M.M. Writing—review and editing: M.M., S.D., R.C., T.D., F.D. and T.v.d.B. Visualization: M.M. Supervision: M.M. and T.v.d.B. Project administration: M.M. and T.v.d.B. Funding acquisition M.M., S.D., T.D., F.D.and T.v.d.B.

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Extended data figures and tables

Extended data fig. 1 experimental set-up..

Clockwise from the leftmost image, high-density wave gauge array close-up (left), wave tank (top right), a schematic diagram showing wave tank set-up (bottom right), and a detailed diagram of the high-density gauge array in its 6 locations (bottom middle). Photograph: (top right) © Dave Morris (CC BY).

Extended Data Fig. 2 3D wave steepness and slope.

Diagram showing 3D definitions of zero-crossing wave height H for the mean and normal directions (denoted H x and H y , respectively), wavelength λ ( k  = 2π/ λ ), and local slope \(| \nabla \eta | =\sqrt{{\eta }_{x}^{2}+{\eta }_{y}^{2}}\) (where η x  = ∂ η /∂ x and η y  = ∂ η /∂ y ) for two example wave groups: a unidirectional wave group propagating in the x -direction (panel a) and an axisymmetric wave group (panel d), which are respectively shown by the solid and dashed lines in panels (b) and (c). Both wave groups have the same global steepness S .

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McAllister, M.L., Draycott, S., Calvert, R. et al. Three-dimensional wave breaking. Nature 633 , 601–607 (2024). https://doi.org/10.1038/s41586-024-07886-z

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