for travelling waves

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 16.2 ).

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 wave s are governed by Newton’s laws and require a medium. A medium is the substance mechanical waves propagate 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, v = c = 2.99792458 × 10 8 m/s . v = c = 2.99792458 × 10 8 m/s . 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 .

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 .

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 16.3 . 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 16.3 , 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 f = 1 / T . f = 1 / T . 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 1 Hz = 1 s −1 . 1 Hz = 1 s −1 .

The length of the wave is called the wavelength and is represented by the Greek letter lambda ( λ ) ( λ ) , 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 16.3 , 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 + A + A above the equilibrium position, and the trough is a distance − A − A 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 v → . v → . 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 16.4 (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 16.4 (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 16.4 (b) is shown in Figure 16.5 . Figure 16.5 (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 16.5 (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.

Example 16.1

Wave on a string.

• 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 v = λ / T . v = λ / T .
• The first wave traveled 30.00 m in 6.00 s: v = 30.00 m 6.00 s = 5.00 m s . v = 30.00 m 6.00 s = 5.00 m s .
• The period is equal to the inverse of the frequency: T = 1 f = 1 2.00 s −1 = 0.50 s . T = 1 f = 1 2.00 s −1 = 0.50 s .
• The wavelength is equal to the velocity times the period: λ = v T = 5.00 m s ( 0.50 s ) = 2.50 m . λ = v T = 5.00 m s ( 0.50 s ) = 2.50 m .

Significance

Check your understanding 16.1.

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?

Example 16.2

Characteristics of a 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 v = λ T v = λ T and the frequency from f = 1 T . f = 1 T .
• The distance the wave traveled from time t = 0.00 s t = 0.00 s to time t = 3.00 s t = 3.00 s can be seen in the graph. Consider the red arrow, which shows the distance the crest has moved in 3 s. The distance is 8.00 cm − 2.00 cm = 6.00 cm . 8.00 cm − 2.00 cm = 6.00 cm . The velocity is v = Δ x Δ t = 8.00 cm − 2.00 cm 3.00 s − 0.00 s = 2.00 cm/s . v = Δ x Δ t = 8.00 cm − 2.00 cm 3.00 s − 0.00 s = 2.00 cm/s .
• The period is T = λ v = 8.00 cm 2.00 cm/s = 4.00 s T = λ v = 8.00 cm 2.00 cm/s = 4.00 s and the frequency is f = 1 T = 1 4.00 s = 0.25 Hz . f = 1 T = 1 4.00 s = 0.25 Hz .

Check Your Understanding 16.2

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?

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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) ).

Figure 16.2 From the world of renewable energy sources comes the electric power-generating buoy. Although there are many versions, this one converts the up-and-down motion, as well as side-to-side motion, of the buoy into rotational motion in order to turn an electric generator, which stores the energy in batteries.

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\,×\,{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.

Figure 16.3 An idealized surface water wave passes under a seagull that bobs up and down in simple harmonic motion. The wave has a wavelength [latex] \lambda [/latex], which is the distance between adjacent identical parts of the wave. The amplitude A of the wave is the maximum displacement of the wave from the equilibrium position, which is indicated by the dotted line. In this example, the medium moves up and down, whereas the disturbance of the surface propagates parallel to the surface at a speed v.

The water wave in the figure moves through the medium with a propagation velocity [latex] \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 .

Figure 16.4 (a) In a transverse wave, the medium oscillates perpendicular to the wave velocity. Here, the spring moves vertically up and down, while the wave propagates horizontally to the right. (b) In a longitudinal wave, the medium oscillates parallel to the propagation of the wave. In this case, the spring oscillates back and forth, while the wave propagates to the right.

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.

Figure 16.5 (a) This is a simple, graphical representation of a section of the stretched spring shown in (Figure)(b), representing the spring’s equilibrium position before any waves are induced on the spring. A point on the spring is marked by a blue dot. (b–g) Longitudinal waves are created by oscillating the end of the spring (not shown) back and forth along the x-axis. The longitudinal wave, with a wavelength [latex] \lambda [/latex], moves along the spring in the +x-direction with a wave speed v. For convenience, the wavelength is measured in (d). Note that the point on the spring that was marked with the blue dot moves back and forth a distance A from the equilibrium position, oscillating around the equilibrium position of the point.

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.

Figure 16.6 A transverse wave shown at two instants of time.

• 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]

Figure 16.7 Characteristics of the wave marked on a graph of its displacement.

• 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{\text{Δ}x}{\text{Δ}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⇒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 ⇒\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\,×\,{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] \text{Δ}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)?

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4.2. Travelling Waves

So now we have the definition and key terms out the way we can start looking at wave motion. Moving waves are often called travelling, or progressive waves. Before starting IB you wll have learnt about some of the many different types of waves, such as light, sound, seismic and water waves. You may also have learnt about the 2 main categories we can divide these waves into - longitudinal and transverse. This sections builds upon these key ideas.

Before we start, have a little look at this video by the wonderful Tom Scott. He looks at the sound waves produced by road markings in California, and describes the simple miscommunication that means they sound terrible.

The section has been divided up as follows:

Describing Waves - the wave equation and how we can represent waves graphically.

Transverse and Longitudinal Waves - the 2 categories of waves and their definitions

Electromagnetic Waves - looking at light and the rest of the EM spectrum

Sound Waves - looking at sound and ultrasound

Describing Waves

Some of those ideas we looked at in the previous section on Periodic Motion will be very important here. To begin with, Crash Course have an introductory video about some of the main features of travelling waves.

There are many different examples of wave motion, though they all follow the same idea:

All waves transfer energy without transferring matter

There are a few key terms you should be familiar with. Hopefully you have a good grasp on these from GCSE - if not, refresh your memory on this Isaac Physics lesson.

Representation of Waves - Displacement/ Time

We encountered a displacement/ time representation in the previous section. This graph looks at one particular point and how it moves over time (think a rubber duck floating on top of water waves - bobbing up and down). Here, we can measure the time taken for one oscillation my measuring from one peak to the next.

The purple crosses represent two moments in time. The arrow shows the direction that the duck is moving at each moment in time.

Representation of Waves - Displacement/ Position

This second representation of waves looks very similar to the above. However, this representation is basically a 'snapshot' in time - imagine a photograph taken viewing a wave from the side. Here, we can measure hte wavelength of a wave from one peak to another.

The purple crosses represent two different points on our wave. The arrow shows the direction that these points are moving at that moment in time.

(Note, it is opposite to those shown on the displacement/ time representation)

The Wave Equation

During the time taken for one oscillation, a wave will travel forwards by a distance of λ. As there are f wavesper second, the wave propagates a distance of fλ in one second. This means that the velocity of the wave wave is given by the following equation:

Wave speed (msˉ¹) = Frequency (Hz) x Wavelength (m)

v=fλ

This is called the 'wave equation' and it is important you are familiar with it. Isaac Physics have a set of practice questions you can have a go at to test yourself.

Video Lessons

Transverse and longitudinal waves.

All waves can be characterised as either Transverse or Longitudinal.

Transverse waves include electromagnetic waves, water waves and S-Seismic waves.

For a transverse wave the wave displacement is perpendicular to the direction of wave propagation/ energy transfer. 

In the gif below, the wave is travelling to the right (and transferring energy from left to right), whereas the particles are vibrating up and down. A graphical representation is shown to the right. 

Longitudinal waves include sound waves and P-Seismic waves.

For a longitudinal wave, the wave displacement is parallel to the direction of wave propagation/ energy transfer. 

In the gif below, the wave is travelling to the right (and transferring energy from left to right), while the particles are also vibrating left and right. A graphical representation is shown to the right - it looks similar to the above, but in this case, a positive displacement represents a particle being to the right of its central position. 

Electromagnetic Waves

You will have come across the electromagnetic spectrum in GCSE (and hopefully are already intimately familiar with the Electromagnetic Spectrum song ). You will be familiar with the different parts of the spectrum and perhaps some of their uses, but what exactly do we mean by an 'electromagnetic wave'. The below video by TedEd gives a nice little intro here.

Some of these key points are summarised on the graphic below. It's worth having a general knowledge of some of the rough wavelengths (in particular those for visible light, approximately 400 - 700 nm).

A few pointers that can help remember some of the approximate wavelengths:

Radio waves  ≈ metres or km's  (as radio waves can diffract round tall buildings and hills for communication, so the wavelengths must be a similar order of magnitude as these objects)

Microwaves  ≈ mm's or cm's  (as these are the gaps between local hot and cold bits in food heated in the microwave oven)

Infrared   ≈ 10ˉ⁴ m - 10ˉ⁶ m (wavelengths between red visible and microwaves)

Red visible light ≈ 666 nm   (666 being the number of the red devil) .

Green visible ( ≈ 555 nm ) and blue visible ( ≈ 444 nm )  (easy to remember once you know red's wavelength) UV   ≈ 10ˉ⁷ m - 10ˉ⁹ m  ( wavelengths between violet and X-rays )

X-rays  ≈ 10ˉ⁹ - 10ˉ¹²  m, (X-ray diffraction is used to image atomic structures, because the wavelength is similar to atomic diameters)

Gamma Rays  ≈ <10ˉ¹² m (the most energetic waves have the shortest wavelengths)

All parts of the electromagnetic spectrum travel at 3 x 10⁸ msˉ¹ in a vacuum (though may slow down by different amounts when travelling through different media). At this stage,  make sure you are confident calculating wavelengths and frequencies using your wave equation. 

Electromagnetic waves are transverse waves, in which the displacement is perpendicular to the direction of wave travel, but what exactly is the displacement we are talking about in this case? EM waves are not matter waves (as sound and seismic waves are), so the displacement is not caused by moving particles. Instead, EM waves are made up of fluctuations in the Electric and Magnetic Fields  (which we will look at in subsequent chapters ). These field vectors are oriented perpendicular to one another as shown below, and it is these fluctuations that are what we perceive as light, radio waves or X-rays. As these are fluctuations in the electric and magnetic fields, they do not require a medium to travel through, so our EM waves are able to pass through the vacuum of space.

Sound Waves

Sound waves are a longitudinal wave - and a type of matter wave (therefore they need a medium to travel through). Sound waves are vibrations of matter (e.g. a guitar string, air molecules, an ear drum), that transmit energy. A greater amplitude of vibration is perceived as a louder volume, whereas a higher frequency of vibration is perceived as a higher pitch. Humans are able to hear roughly sounds between 20 Hz and 20 000 Hz (though this upper limit decreases with age, have a go at working out your 'hearing age' here ). 

  PHET have a simulation   that allows you to look at a few properties of sound. Change the pitch and amplitude and look at the effect on the waves. You can also remove the air from the speaker and listen to the effect on the sound. The two source interference is something we will later look at in Section 4.4 .

It's worth making sure you are familiar with a few experiments to measure the speed of sound. THere is the classic GCSE favourite  of banging together two bits of wood and measuring the time taken for sound to get to you with a stopwatch - Note, this is RUBBISH.

A much better way of doing this is digitally with an oscilloscope. Hopefully you've had a go at using one of these in your classroom, otherwise there is a simulation here . This video explains a little bit about how the speed of sound can be measured using two microphones connected to an oscilloscope.

Additional Resources

Ib questions.

A question by question breakdown of the IB papers by year is shown below to allow you to filter questions by topic. Hopefully you have access to many of these papers through your school system. If available, there may be some links to online sources of questions, though please be patient if the links are broken! (DrR: If you do find some broken links, please contact me through the site)

Questions on this topic (Section 4) are shown in light blue.

Use this grid to practice past IB questions topic by topic. You can see from the colours how similar the question topic breakdown is year by year. The more you can familiarise yourself with the IB question style the better - eventually you will come to spot those tricks and types of questions that reappear each year. 

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• Mathematics of Standing Waves

What is a Standing Wave Pattern?

It is however possible to have a wave confined to a given space in a medium and still produce a regular wave pattern that is readily discernible amidst the motion of the medium. For instance, if an elastic rope is held end-to-end and vibrated at just the right frequency , a wave pattern would be produced that assumes the shape of a sine wave and is seen to change over time. The wave pattern is only produced when one end of the rope is vibrated at just the right frequency. When the proper frequency is used, the interference of the incident wave and the reflected wave occur in such a manner that there are specific points along the medium that appear to be standing still. Because the observed wave pattern is characterized by points that appear to be standing still, the pattern is often called a standing wave pattern . There are other points along the medium whose displacement changes over time, but in a regular manner. These points vibrate back and forth from a positive displacement to a negative displacement; the vibrations occur at regular time intervals such that the motion of the medium is regular and repeating. A pattern is readily observable.

We Would Like to Suggest ...

Traveling Waves

• The Sine Wave is the simplest of all possible waves.  A periodic wave is one in which the shape of the wave is repeated "periodically" - at regular fixed intervals.
• Five basic properties which describe periodic waves.
• Wavelength  (lambda)  -  Distance after which the wave begins to repeat (Units: metres).
• Frequency (f)  -  Number of waves passing a fixed point in one second  (Units: Hertz).
• Wave period (T)  -  Time taken for one wave to pass a given point (Units: seconds).
• Wave velocity (v)  -  Distance travelled by the wave per second also called the phase velocity (Units: m/s).
• Amplitude (y m )  -  Maximum displacement of particle which comprise the wave from their equilibrium position  (Units: metres).
• Frequency, wavelength and velocity are related by,

• There are two basic types of traveling waves.
• TRANSVERSE   :  Motion of the constituent particles is at right angles to the wave direction, e.g. waves on a string, "stadium" wave, electromagnetic waves.
• Comparison of SHM and Waves
• The amplitude of the SHM of the particles which comprise a Sine Wave is the same as the amplitude of the wave.
• The frequency of the SHM of the particles which comprise a Sine Wave is the same as the frequency of the wave.
• Particles which comprise the wave typically do not move at the speed of the wave, e.g. molecules of air do not move at the speed of sound  in air.
• Miscellaneous important facts :
• A wave "travels" from A to B.  The particles that comprise the wave do not move from A to B, they oscillate about their fixed (equilibrium) points.
• A wave transmits energy from one point to another.
• Transmission of Electromagnetic waves does not require a medium - there are no "particles".  These waves are comprosed of oscillating electric and magneic fields and are quite happy to propagate in a vacuum.
• Water waves appear to be transverse  -  boats bob up and down due to water waves.  However,  a  detailed study shows that the molecules of water actually perform a circular motion, which can be considered as a combination of transverse and longitudinal wave motion.

• Initial conditions

• Wave Velocity on a Stretched String

Dr. C. L. Davis Physics Department University of Louisville email : [email protected]  

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• Review Article
• Published: 22 March 2018

Cortical travelling waves: mechanisms and computational principles

• Lyle Muller 1 ,
• Frédéric Chavane 2 ,
• John Reynolds 1 &
• Terrence J. Sejnowski 1 , 3  

Nature Reviews Neuroscience volume  19 ,  pages 255–268 ( 2018 ) Cite this article

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• Dynamical systems
• Neural encoding
• Visual system

Multichannel recording technologies have revealed travelling waves of neural activity in multiple sensory, motor and cognitive systems. These waves can be spontaneously generated by recurrent circuits or evoked by external stimuli. They travel along brain networks at multiple scales, transiently modulating spiking and excitability as they pass. Here, we review recent experimental findings that have found evidence for travelling waves at single-area (mesoscopic) and whole-brain (macroscopic) scales. We place these findings in the context of the current theoretical understanding of wave generation and propagation in recurrent networks. During the large low-frequency rhythms of sleep or the relatively desynchronized state of the awake cortex, travelling waves may serve a variety of functions, from long-term memory consolidation to processing of dynamic visual stimuli. We explore new avenues for experimental and computational understanding of the role of spatiotemporal activity patterns in the cortex.

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Acknowledgements

The authors thank Z. Davis, T. Bartol, G. Pao, A. Destexhe, Y. Frégnac and C. F. Stevens for helpful discussions and J. Ogawa for helpful discussions and help with illustrations. L.M. acknowledges support from the US National Institute of Mental Health (5T32MH020002-17). F.C. acknowledges support from Agence Nationale de la Recherche (ANR) projects BalaV1 (ANR-13-BSV4-0014-02) and Trajectory (ANR-15-CE37-0011-01). J.R. acknowledges support from the Fiona and Sanjay Jha Chair in Neuroscience at the Salk Institute. T.J.S. acknowledges support from Howard Hughes Medical Institute, Swartz Foundation and the Office of Naval Research (N000141210299).

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L.M., F.C., J.R. and T.J.S. researched data for the article, made substantial contributions to discussions of the content, wrote the article and reviewed and/or edited the manuscript before submission.

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PowerPoint slides

Powerpoint slide for fig. 1, powerpoint slide for fig. 2, powerpoint slide for fig. 3, powerpoint slide for fig. 4, powerpoint slide for fig. 5, powerpoint slide for fig. 6, powerpoint slide for table 1.

The differences in phase (an amplitude-invariant measure of position in an oscillation cycle) between two (or more) oscillations.

A disturbance that travels through a physical medium that may be water, air or a neural network.

Patterns that can result from the summation of many individual waves. Depending on the properties of the medium, the pattern resulting from these interactions can differ greatly.

A scale between microscopic and macroscopic. In neuroscience, the mesoscopic scale describes single regions (such as cortical areas or subcortical nuclei) spanning millimetres to centimetres. Cortical networks at this scale can be imaged through recently developed recording technologies.

The scale of the whole brain; traditionally recorded with extracranial techniques (electroencephalography and magnetoencephalography) and more recently recorded with intracranial methods (electrocorticography).

(EEG). A neural recording technique in which electrodes are placed on the scalp, outside the skull (extracranial), that is of great use in studying the sensory and cognitive processes of normal human subjects.

(ECoG). A recording technique in which electrodes are placed directly on the cortical surface, offering both high spatial (up to 2 millimetres or greater) and high temporal resolution.

(LFP). The electric potential recorded in the extracellular space of the cortex. The LFP is thought to reflect the synaptic currents from neurons within a few hundred micrometres around the electrode.

(MEAs). One-dimensional or two-dimensional grids of electrodes, which offer the ability to sample local field potential and spiking activity at the mesoscopic scale.

(VSDs). Fluorescent dyes applied directly to the surface of the cortex that allow the subthreshold membrane potential of neural populations to be recorded. The resulting signals are linearly related to the average membrane potential of neurons at each point in the cortex. This technique captures neural activity over a large field of view with very high spatial (up to 20 micrometres) and temporal (up to 1 millisecond) resolution.

The large, 0.1–1.0 Hz rhythm of deep non-rapid-eye-movement sleep.

Passive transmission of an electric field through biological tissue. The fields can be created from a single source of neural activity and will appear as identical, highly synchronous waveforms across electrodes; a cause of spatial smoothing (blurring) in scalp electroencephalography.

A brief ( ∼ 1 second), biphasic waveform composed of a strong negative potential followed by a positive deflection. K-Complexes occur predominantly during stage 2 non-rapid-eye-movement sleep and are driven by transitions from cortical down to up states.

Thalamocortical 11–15 Hz oscillations prevalent in stage 2 non-rapid-eye-movement sleep. These oscillatory periods have long been associated with learning and memory, including sleep-dependent consolidation of long-term memory.

A group of interconnected, repeatedly co-activated neurons whose signature spike pattern is thought to collectively represent a specific sensory stimulus or memory.

Models of emergent collective behaviour in large ensembles. In these networks, individual units are characterized by a state (or phase) between 0 and 2π. Interactions among units are typically modelled as attractive, such that units with different states tend to synchronize depending on the coupling strength of the interaction.

A state of asynchronous, highly irregular firing in spiking network models. This state exhibits the low-correlated firing that is the hallmark of cortical dynamics under general conditions of excitatory and inhibitory balance.

An extension of the neural field model of Wilson and Cowan to include the effects of neural and synaptic noise.

In rodents, the long axis of the hippocampus, running from a dorsal, medial position to a ventral, lateral position; synonymous with septotemporal axis.

Models of chemical dynamics that take into account local reactions and diffusion across space. These reactions exhibit complex dynamics, including travelling waves and emergent patterns.

A property of a system whose dynamics remain the same when time is reversed. This feature implies important mathematical properties for the system under study.

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Muller, L., Chavane, F., Reynolds, J. et al. Cortical travelling waves: mechanisms and computational principles. Nat Rev Neurosci 19 , 255–268 (2018). https://doi.org/10.1038/nrn.2018.20

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DOI : https://doi.org/10.1038/nrn.2018.20

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Travelling waves

Shifting the position, turning standing waves into travelling waves, another way to make waves travel, the speed of a wave on a string, for more information.

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Chapter 16: Waves

Back to chapter, travelling waves, next video 16.2: wave parameters.

When a stone is thrown on the surface of a calm lake, we see circular rings form and spread out from the drop point. These circular rings observed on the water surface are an example of wave motion.

Wave motion is a disturbance that propagates from a state of rest or equilibrium without bulk motion of matter.

Traveling waves can be divided into transverse waves and longitudinal waves.

In a longitudinal wave, the particles of the propagating medium displace parallel to the wave propagation in back and forth motion.

Sound waves traveling in air or the sound waves traveling in water are examples of longitudinal waves.

Whereas, in a transverse wave, the particle displacement is perpendicular to the direction of wave propagation.

Ripples on the surface of the water and the waves produced from the stretched string of a guitar are examples of transverse waves.

A wave is a disturbance that propagates from its source, repeating itself periodically, and is typically associated with simple harmonic motion. Mechanical waves are governed by Newton's laws and require a medium to travel. A medium is a substance in which a mechanical wave propagates, and the medium produces an elastic restoring force when it is deformed.

Water waves, sound waves, and seismic waves are some examples of mechanical waves. For water waves, the wave propagation medium is water; for sound waves, the medium is usually air, but could also be water. The wave disturbance in sound waves is generally a change in air pressure. In the case of mechanical waves, both energy and momentum are transferred with the wave's motion, whereas the mass oscillates around an equilibrium point.

Waves can be transverse, longitudinal, or a combination of the two. For example, the waves on the strings of musical instruments are transverse. Meanwhile, sound waves in air and water are longitudinal; their disturbances are periodic variations in pressure that are transmitted in fluids. Fluids do not have appreciable shear strength, and thus, the sound waves in them must be longitudinal or compressional. However, sound waves that travel in solids can be both longitudinal and transverse.

Earthquake waves under the Earth's surface also have both longitudinal and transverse components. These components have important individual characteristics—they propagate at different speeds. Earthquakes also have surface waves that are similar to surface waves on water.

This text is adapted from Openstax, University Physics Volume 1, Section 16.1: Traveling Waves.

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• Travelling Wave

Have you ever sat by a lake and observed the waves created on the surface of the water when you throw a stone into it? This is a good visual example of the propagation of waves and makes it simpler for you to understand travelling of waves and all other concepts related to it. Our universe has an amazing way of informing us about any changes in the physical world. When there are changes the information about that disturbance moves gradually outwards. It moves far from the source of disturbance in all the directions. When the said information travels, it travels in the form of a wave, just like the way waves are created when you throw a stone in the still water. This is known as the travelling wave.

Define Travelling Wave

(image will be uploaded soon)

Before understanding what is travelling wave, let’s understand waves. Wave can be defined as a disturbance in a medium that travels transferring momentum and energy without any actual movement of the medium. However, the medium must have elastic properties. In our everyday life, there are many examples of waves, for example, ocean waves, strings of musical instruments, etc. On the other hand, a travelling wave is a wave in which the positions of minimum and maximum amplitude travel through the medium.

Points To Remember

Here are some of the points that are necessary to keep in mind about the wave:

Every wave has a high point and a low point. The high points are known by the name of crests. On the other hand, the low points are named by troughs.

Amplitude is the maximum distance of the disturbance from the midpoint of the wave to either the top of the crest or the bottom of a trough.

The maximum distance between the two adjacent troughs or the two adjacent crests is known as a wavelength.

Now, the time period is actually the time taken to complete one vibration.

Frequency is the number of vibrations the wave undergoes in one second.

You can witness an inverse relationship between both frequency and time period. The relationship is given below,

The speed of a wave is given by the travelling wave equation,

Where 𝛌 is the wavelength.

What are the Various Types of Travelling Waves?

Each type of wave contains different characteristics. And with these characteristics, we can easily distinguish between them. Here is a list of different types of waves that have been categorized based upon their particle motion.

Pulse Waves - the sudden disturbance that travels through a medium is known as a pulse wave. The disturbance can be caused by a chain reaction or sudden compression of air caused by an explosion. One example of a pulse wave is thunder. It comprises only one crest that travels through the transmission medium.

Continuous Waves - it is an electromagnetic wave that has constant amplitude and frequency. It is a typical sine wave and is considered to be of infinite duration. It was used in the earlier days of radio transmission.

Transverse Waves - in the transverse wave, the movement of the particles is at right angles to the motion of the energy. It is generated through a solid object like a stretched rope. Trampoline is the best example to understand this wave.

Longitudinal Waves - in this type of travelling wave the motion of the wave-particle is in the same direction as the propagation of the wave. In simple words, the movement of the particles is parallel to the motion of the energy. The best example for longitudinal waves is sound waves moving through the air when you hear a loudspeaker playing in the distance.

There is a second way to characterize the waves by types of matter they are able to move or travel through.

Electromagnetic Waves - this type of wave can travel easily through a vacuum. It does not need any medium, soft or hard to travel. An example of an electromagnetic wave is mobile phone waves or sound waves. They don't need any vacuum to travel.

Physical waves - Unlike electromagnetic waves, they require a medium to travel. They are further distinguished on the basis of phases of matter through which they can move.

Longitudinal Waves - these waves can easily pass through liquids and games.

Transverse Waves - they require a solid material or medium to propagate.

Problems based on travelling wave equation

Solved Examples

1: A wave on a rope is shown on the right at some time t.  What is the wavelength of this wave?  If the said frequency is about 4 Hz, what will be the wave speed?

Now, for all the periodic waves, you will find v = λ/T = λf.

Details of the calculation:

The wavelength λ is 3 m. The speed is v = λf = (3 m)(4/s) = 12 m/s.

FAQs on Travelling Wave

1. Do all the Waves have a Defined Wavelength?

Yes, every wave has a specific wavelength, which is defined as the length from one wave crest to the next.

Different kinds of waves have different wavelengths. In water, the surf waves produced by a surfer, have wavelengths of 30–50 m, and the large tsunamis have much longer wavelengths (about 100km). The Sound waves vary in wavelength according to the pitch of it.

2. At What Wavelengths can a Human Hear a Sound?

As per experts and facts, the humans can very well hear sounds possessing wavelengths between 70 mm and 70 m. Any sound above or below this level can not be heard by a human. Example- our universe is filled with cosmic noises. The wavelength of the noise is so high that we cannot hear it on earth.

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Funds are cutting aid for women seeking abortions as costs rise

Organizations that help pay abortion costs are capping how much they can help as travel costs rise and the wave of “rage giving” that fueled them two years ago has subsided.

Abortion funds, which have operated across the U.S. for decades, in many cases as volunteer groups, ramped up their capacity fast after the Supreme Court overturned Roe v. Wade in 2022, ending a national right to abortion. Donations rolled in from supporters who saw the groups as key to maintaining abortion access as most Republican-controlled states implemented bans.

The expansion of the funds and increasing access to abortion pills are major reasons the number of abortions has risen slightly despite bans on abortion at all stages of pregnancy in 14 states and after about six weeks of pregnancy, before many women know they are pregnant, in another four.

But the funds have found that even with record budgets, it’s not enough to fill all the gaps between the cost of obtaining abortions and what women seeking them can afford as they have to travel farther for legal procedures.

The National Abortion Federation, which helps people seeking abortions across the country, used to cover half the cost of the abortion for callers who couldn’t afford it. Since July, it’s pulled back to 30%. Brittany Fonteno, the organization’s president and CEO, said the allocations had to be cut because of the rising demand and costs — even though the fund has a record $55 million budget this year.

“We’re at the point now where we know that people who are most impacted by funding shifts — and by abortion bans which have caused the funding shifts — are the people who can least afford to be kept away from care,” Fonteno said. “And that includes people of color, younger people, immigrants and people with lower incomes.”

Other groups have also imposed limits on aid to keep from exhausting their funds.

The Blue Ridge Abortion Fund, based in Virginia, hits its budget limit nearly every week and has to put requests on hold until the next week.

The Cobalt Abortion Fund in Colorado has had to cap how much it can spend. Its president, Karen Middleton, said groups like hers are used to being scrappy.

“It’s the bake sale of the abortion rights movement,” she said.

Abortion funds have existed for decades out of the spotlight. Many were — and some remain — volunteer-run. Nearly all of them ramped up as the abortion landscape shifted.

Cobalt, for instance, spent $206,000 in 2021. Of that, only about $6,000 was for for travel costs — and much of that came in the form of gas cards to help people in outlying parts of Colorado get to clinics.

This year, the group expects to spend $2.2 million — 10 times as much as in 2021. In the first six months of this year, it spent more than $600,000 on travel and other logistical costs. Now they’re booking hotel rooms and flights — mostly on short notice.

“We’re a travel agency as much as we’re an abortion fund,” Middleton said.

In Colorado, like other states between the coasts, the influx of patients began late in 2021 when a ban on abortion after the first six weeks of pregnancy took effect in Texas. That’s since been replaced by a ban on abortion at all stages of pregnancy.

For the Blue Ridge Abortion Fund, a big change arrived after May 1, when Florida’s ban on abortion after the first six weeks of pregnancy took effect. Before that, Florida, the nation’s third most populous state, was a destination for people traveling from other Southern states that had stricter limits.

Greene said her fund helped 20 people from Florida from January through April. When the ban took effect, it left Virginia as the nearest state where abortion was available past 12 weeks and without a 72-hour waiting period. Greene said the fund helped about 40 Florida residents from May through August.

Their average travel cost per Floridian has been about $3,000, she said — more than any other state.

Fonteno said the spike in requests from Florida — six times as many each month since the ban began — was an impetus for its abrupt policy change earlier this year. Blue Ridge and other funds have been trying to make up the difference.

“We’re seeing more and more patients with more funding gaps,” Greene said.

To try to stick to its weekly aid budget, the fund has cut back when it accepts calls requesting help to two mornings a week instead of two full work days. Greene said her fund collaborates with others to try to cover costs.

The New Jersey Abortion Access Fund responded to the National Abortion Federation’s cuts by increasing what it sends every week to a solidarity fund to help people seeking abortions from other states to $5,000 from $3,000, said Quadira Coles, the group’s president.

The organization also sends block grants to New Jersey abortion clinics to use to help pay for patients who cannot afford their fees. Coles said the group has increased that funding, too, after hearing from clinics that it had been running out halfway through the month.

The groups could see some of their financial pressure eased depending on the outcomes of measures on the November ballots that would add a state constitutional right to abortion in nine states.

In four of the states — Florida, Missouri, Nebraska and South Dakota — passage would overturn current bans and potentially mean that many people could access abortion without traveling.

In other states, the change would be more subtle. For instance, part of the Colorado amendment would allow state government employee health plans to cover abortion — possibly reducing the number of people who would seek help paying.

In the meantime, organizers of some of the funds say that some supporters have been contributing to the ballot measure campaigns, at the expense of the funds.

Joan Lamunyon Sanford, executive director of Faith Roots, which helps pay the costs of people traveling to New Mexico for abortion, said many donors who started around the time of Texas’ restrictions, often called Senate Bill 8, or the Supreme Court’s 2022 ruling make recurring gifts — though others gave just once.

“For those who felt that, whether it was the righteous anger or compassion, that led them to donate after S.B. 8 and after Dobbs, we’re still here, and the need is still here,” she said. “We still need them.”

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Mathematics > Analysis of PDEs

Title: modulational instability of small amplitude periodic traveling waves in the novikov equation.

Abstract: We study the spectral stability of smooth, small-amplitude periodic traveling wave solutions of the Novikov equation, which is a Camassa-Holm type equation with cubic nonlinearities. Specifically, we investigate the $L^2(\mathbb{R})$-spectrum of the associated linearized operator, which in this case is an integro-differential operator with periodic coefficients, in a neighborhood of the origin in the spectral plane. Our analysis shows that such small-amplitude periodic solutions are spectrally unstable to long-wavelength perturbations if the wave number if greater than a critical value, bearing out the famous Benmajin-Feir instability for the Novikov equation. On the other hand, such waves with wave number less than the critical value are shown to be spectrally stable. Our methods are based on applying spectral perturbation theory to the associated linearization.

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The Wave and Box twist cowl is knit sideways in a tube to give a super soft and squishy finish that will keep you cosy on chilly autumn days. It is a quick knit that is perfect for tv or travel knitting and the pattern allows for some really fun colour experimentation!

This pattern has been Tech Edited by Hannah Middleton

Skills and Techniques used

Provisional cast on Kitchener stitch OR Cast on & bind off (see notes on page 2) Knit Slipping stitches Working in the round on circular needles Horizontal mattress stitch

Gauge 20 sts/30 rounds = 4 inches square in slipped stitch pattern using 4.5mm needles after blocking

Size 23.4” wide by 8”tall when lay flat before seaming

10” wide by 8” tall when seamed - the twist takes away some of the width.

You can make the cowl shorter or longer by adding or subtracting a pattern repeat. One full pattern repeat will add 2.1”. All measurements are in inches.

Yarn: Drops Daisy DK in Cobalt Blue (MC) and Off White (CC) (100% Wool, 110m/50g) - 2 balls of MC and 1 ball of CC

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