Lesson Explainer: Comparing Transverse and Longitudinal Waves Physics • 9th Grade

In this explainer, we will learn how to identify the differences and similarities between transverse and longitudinal waves in terms of amplitude, zzwavelength, and period.

To make this comparison, we first need to define what a wave is.

Definition: A Wave

In the simplest terms, a wave is a method of transferring energy. This transfer occurs thanks to some kind of disturbance (or oscillation) that travels from the source to the destination without net transfer of matter.

To make this definition clearer, we can use an example.

Shown in the figure is a string. This string is made up of a series of balls. Person A moves the string up and then down. This creates a wave in the string, which consists of an upward disturbance and a downward disturbance. This wave travels the length of the string and transfers the energy (the kinetic energy of movement) to person B at the other end.

As well as what is happening to the wave, it is important to understand what is happening to the balls. If we look at one specific ball, say, the one labeled 6, and carefully think about what is happening, we notice that it does not move left or right, even when the wave passes. In fact, all it does is oscillate vertically up and down. This is what we mean when we say without net transfer of matter; the balls (i.e., matter) are disturbed upward and then disturbed downward by an equal amount, returning to their original positions. As the wave passes, ball 6 will start without any disturbance on the gray line. It will then oscillate downward, reaching the same depth as point T, and then oscillate upward, reaching the same height as point P. It will then return to its original position. This means there is no net motion (transfer) of matter, as required by our definition of a wave.

A second example may help with this definition. Consider the phenomenon known as a stadium wave. This is where successive groups in a large stadium will raise their arms then lower them again. The effect is a wave of raised arms traveling across the stadium. Just like our string example, the people do not move and their net vertical disturbance is zero and yet a wave, a traveling disturbance, moves horizontally across the stadium.

Now that we have defined, in general terms, what a wave is, we are now in a position to talk about and, more importantly, compare the two types of waves. Central to both definitions is the direction of the wave compared to the direction of oscillation.

To explain what a transverse wave is, we will return to our first example of a string of balls. The wave made by this up-and-down motion is a transverse wave. In this example of a transverse wave, the oscillating motion of the balls is vertical, whereas the direction the wave is traveling in is horizontal. The key feature here is that the two are perpendicular (at right angles) to each other.

Definition: Transverse Waves

Transverse waves are defined as having oscillating motion perpendicular to the direction of wave propagation.

Transverse waves have two unique and important features. The first of these is called the crest or peak of the wave. This is the maximum displacement up (in the positive direction) from the original position of the balls as they oscillate. It is shown by P in the figure.

Definition: Crests (Peaks)

Maximum oscillation up from the original (equilibrium) position

The second feature is the trough of the wave. It is shown by T in the figure.

Definition: Troughs

Maximum oscillation down from the original (equilibrium) position

To explain longitudinal waves, we again need to talk about the direction of oscillation and the direction of wave motion. Consider the figure below. This is a representation of a sound wave. The individual dots represent particles of air. These particles are the matter that is oscillating, creating the wave, and are equivalent to the balls on our string. Just like before, if we focus on the motion of one particular particle, we can get an overall picture of what is happening as the wave travels from left to right.

An individual air particle will not move vertically as the sound wave travels. It will instead oscillate about its original point horizontally. Its net motion will still be zero, as it starts and finishes at its original position. The oscillation of all the air particles together makes the sound wave propagate. To picture this motion, just think of the areas of high particle concentration moving left to right across the screen, keeping in mind that the particles themselves, just like in transverse waves, do not travel with the wave as it moves, as they can only oscillate about their original point. This is demonstrated by the blue ball and the orange ball. The blue ball oscillates only within its red arrow and the orange ball oscillates within its own red arrow as the wave propagates.

Definition: Longitudinal Waves

Longitudinal waves are defined as having oscillating motion parallel to the direction of wave propagation.

Longitudinal waves have two unique important features that are similar to those seen in a transverse wave. To explain what these are, we again need to look at our figure.

Definition: Compressions

Compression regions are the areas of high particle concentration.

Definition: Rarefactions

Rarefactions are the areas of low particle concentration.

These two are roughly equivalent to the peaks and troughs of a transverse wave. It is these areas of compression and rarefaction that travel left to right across the screen. These regions are shown by the C and R labels on the diagram.

Example 1: Transverse and Longitudinal Waves

A rectangular-prism-shaped tile is oscillated vertically along the axis shown in the diagram. The tile moves in a medium that cannot support transverse waves. Along which wave direction axis will waves travel away from the tile surface?

Answer

Key to understanding this question is knowing that longitudinal waves travel parallel to the direction of oscillation. Therefore, the waves will radiate from the tile along wave direction axis I, as this axis is parallel to the oscillation axis.

Next, we will use the idea of compressions, rarefactions, peaks, and troughs to set out four new terms. These terms are not unique to just one wave type, and they can be used to describe both types of waves.

The first term that applies to both types of waves is the medium.

Definition: The Medium

This is simply defined as what the wave is traveling through.

Specifically, thinking back to the first example, the oscillating balls on our string represent the medium. This is perhaps more clearly demonstrated in our example in which we talked about the longitudinal wave, sound. There, the medium sound travels through is the air. The air particles are the ones that are doing the oscillating; they are the matter with no net motion required for the wave definition. This means that they are the medium the sound wave travels through.

A good real-world example of transverse waves would be water waves, seen when you drop a rock into water. When you do this, transverse waves spread in the form of ripples in the surface. Here, the water is the medium, as it is what the wave is traveling through.

The next term common to both wave types is the wavelength.

Definition: Wavelength

Wavelength is defined as the distance covered by one full cycle of the wave. It is given the symbol 𝜆 and, as a distance, its unit is the metre.

Using the figure below will allow us to expand upon this general definition and apply it to our two wave types. Looking at it, we can see the similarity in the shape of the two wave types, as the peaks line up with the compressions and the troughs line up with the rarefactions.

First, consider a transverse wave. A full cycle can be thought of as the distance it takes for the wave to return to its original position. This does not mean the next point it crosses the dotted line (the equilibrium position), as at this point the wave is still decreasing. We must go all the way to the start of the second red region to complete a full cycle, as the wave is increasing there just as it is at the start of the first red area. An important point is that we can choose any point on the wave to be our starting point. Therefore, the wavelength of a transverse wave can be also thought of as the peak-to-peak distance or even the trough-to-trough distance.

The idea of wavelength may be most familiar when we talk about light. Light is a transverse wave. It is created by the oscillation of electric and magnetic fields perpendicular to the direction in which the light travels. For example, visible light has a wavelength range of 350 nm to 750 nm.

We next consider the wavelength of a longitudinal wave. Since the shape of the two waves are similar, we can simply say that the wavelength of a longitudinal wave is the distance between the centers of compression regions (peak-to-peak distance). We could also define it as the distance between the centers of rarefaction regions (trough-to-trough distance). Again, we can pick any starting point in our longitudinal wave to find one full cycle. We could instead choose the distance between the start of one compression region and the start of the next or the distance between the start of one rarefaction region and the start of the next.

Next is an example testing your knowledge of both wavelength and graph interpretation.

Example 2: Transverse and Longitudinal Waves

A transverse wave is shown in the diagram. What is the wavelength of the wave?

Answer

This question relies on knowledge of two things: firstly, the definition of what a wavelength is for a transverse wave and secondly, knowledge of interpreting graphs.

To address that first point, the wavelength is the distance covered by one full cycle. This means it can be thought of as the distance between any point on the wave and the location of that point in the next cycle. For example the trough-to-trough distance is equal to the wavelength and so is the peak-to-peak distance.

Now, we rely on the second part of our knowledge to read the peak-to-peak distance off the graph. The first peak is at a distance of 1 m and the second is at a distance of 5 m. This gives us a peak-to-peak distance of 4 m or 𝜆=4.m

Remember to include the unit, metres.

The next term that can be defined by both types of waves is the amplitude.

Definition: Amplitude

In general terms, the amplitude is defined as the maximum possible displacement of the oscillating medium. It has the label 𝐴 and, as a distance, its unit is the metre.

Again, it is easier to first define this for a transverse wave, where we see the amplitude is the distance between one of the peaks and the equilibrium position (the dotted line). It can also be defined as the distance between the equilibrium position and one of the troughs. Remember, the equilibrium position can be thought of as the position of the medium before the wave passes that position (i.e., no displacement). Notice that for a transverse wave, its amplitude (a measure of its oscillation) is perpendicular to its wavelength (a measure of the distance covered by a wave cycle). This makes sense, as our definition of a transverse wave has the direction of oscillation perpendicular to the direction of the wave.

Moving onto the amplitude of longitudinal waves, we should therefore expect the amplitude to be parallel to the wavelength. Remembering that amplitude is the maximum oscillation of the medium, if we consider a single particle at the center of an area of rarefaction, we can reach a clearer definition. As the wave passes our particle, it will oscillate an equal distance on either side of its original position. So, the maximum displacement, the amplitude, is half the width of the region of rarefaction. Equally, it can be thought of as half the width of the region of compression. It can be made clearer why we take half the region distance if we look at a transverse wave. To define its amplitude, we do not measure the whole peak-to-trough distance; we only measure half of it or the peak-to-equilibrium-position distance.

Since longitudinal waves have a similar shape to transverse waves, you can represent one type of wave with another. More specifically, they are both just a measure of the amount of oscillation. Regions of compression/peaks represent either oscillations up from the equilibrium (peaks) or oscillations horizontally creating a compression region.

Remember, from our air particles example, no oscillation means we are at the equilibrium position. Therefore, there will be an equal distribution of air particles. This will be at a lower density than in compression regions but at a higher density than in rarefaction regions. So, a rarefaction is also a deviation from this equilibrium, also caused by oscillations, just the other way. For transverse waves, displacement (the other way) is also due to oscillation and is represented by the troughs. Looking at the figure above with both types of waves makes this clearer.

Below is an example that explores your understanding of this concept.

Example 3: Transverse and Longitudinal Waves

The diagram shows a longitudinal wave traveling along a spring and four displacement–time graphs. If positive displacement corresponds to compression of the spring, which of the graphs correctly shows the change in displacement with time of the wave on the spring?

Answer

This question relies on you being able to understand the idea of representing a longitudinal wave as a transverse one.

We need to know to associate the peaks of a transverse wave with the areas of compression of a longitudinal wave. Equally true is that you associate the troughs of a transverse wave with the areas of rarefaction of a longitudinal wave.

Hence, the correct answer will be D, as the peaks line up with the centers of the compression regions and the troughs line up with the centers of the rarefaction regions.

Next is an example testing your knowledge of both amplitude and graph interpretation. The graph in this next question is slightly more difficult than that in example 1.

Example 4: Transverse and Longitudinal Waves

A transverse wave is shown in the diagram. What is the amplitude of the wave?

Answer

This question relies on knowledge of two things: firstly, the definition of what an amplitude is for a transverse wave and secondly, knowledge of interpreting graphs.

To address that first point, the amplitude is the maximum possible displacement of the oscillating medium from the equilibrium point. Remember, the medium is whatever our wave is propagating through, in this case, whatever the blue line is made of.

Be careful with this definition. You may want to find the peak-to-trough distance thinking that this is the maximum displacement, but this would be wrong. The key part of this definition is that we measure from the equilibrium point (i.e., the point where the medium would be without any disturbance, in our case, the 𝑥-axis). This means the amplitude is the distance between the peak and the 𝑥-axis. An equally valid option is the distance between the trough and the 𝑥-axis.

Now, we rely on the second part of our knowledge to read the peak-to-axis distance off the graph. The first peak is at a displacement somewhere between 1.5 m and 2 m, in the large box. To find out where our peak lies in this range, we need to work out the value of the smaller boxes. A large box has a size of 21.5=0.5.

Since this 0.5 is shared between the 5 smaller boxes, the value of one smaller box is then 0.55=0.1.

Therefore, since our peak is two little boxes above 1.5, its value is peak=(0.1×2)+1.5=1.7.

Since, by definition, our 𝑥-axis is at a displacement = 0, we have our final answer: 𝐴=1.7.m

Remember to include the unit, metres.

Next is a similar question but with a different transverse wave.

Example 5: Transverse and Longitudinal Waves

A transverse wave is shown in the diagram. What is the amplitude of the wave?

Answer

The amplitude is the distance between the peak and the 𝑥-axis. An equally valid option is the distance between the trough and the 𝑥-axis.

A mistake often made is to assume the amplitude is the displacement of the curve at time 0. This is not the case. A wave can start with any value of displacement between zero and maximum displacement. In our case, it starts just short of the maximum displacement.

The first peak can be read straight from the graph and is at a displacement of 1.5 m. Since, by definition, our 𝑥-axis is at a displacement = 0, we have our final answer: 𝐴=1.5.m

Remember to include the unit, metres.

The next term that describes both waves is called the frequency.

Definition: Frequency

Frequency is defined as the number of full wave cycles that pass a particular point per second (per unit time).

You can think of this as a measure of how fast the wave is traveling from left to right. We need to remember that what is being shown in these figures is a snapshot of the wave and that it is propagating as time progresses. To demonstrate this, a new figure is shown below.

For a transverse wave, the frequency will tell you the number of peaks passing a given point (shown by the green line) per second. It will also tell you the number of troughs passing per second.

Equivalently, for a longitudinal wave, the frequency will tell you the number of compressions passing a given point per second as well as the number of rarefactions.

There is another property of waves that is closely linked to frequency; it is the period of the wave. It also describes both waves. It can be thought of as the inverse of frequency.

Where frequency is the number of wave cycles in a given time (one second), a period is the amount of time that it takes the wave to go through one cycle.

Definition: Period

It is the time taken for one complete oscillation. It is given by the equation 𝑇=1𝑓.

We have now summed up the ways to define longitudinal and transverse waves. There are four terms that apply to both types: medium, wavelength, amplitude, and frequency. There are four that apply to specific types: peaks and troughs (transverse waves) and compressions and rarefactions (longitudinal waves).

One final thing covered by this explainer is the idea that often waves in real life are more complicated. Parts of a wave can contain longitudinal wave motion and other parts can contain transverse wave motion. This is demonstrated in the figure below. The spring in the diagram has been disturbed in such a way as to create the wave motion seen. If we look at the spring’s motion as a whole, we can clearly see there is a transverse wave.

However, if we look closer at parts of the spring, we see a different kind of wave motion. The blue areas are parts of the spring being extended or stretched out, and the red areas are parts of the spring being compressed. If we consider these areas only as the wave propagates, the spring in these regions will have an oscillation parallel to the wave motion. If we picture the spring moving, the areas of compression and extension will move along with the wave as a whole. A longitudinal wave is defined as areas of compression and extension (or rarefaction) moving parallel to oscillation. Remember, it is not the spring itself moving with these areas of compression/extension; the spring is the medium and as such only oscillates about its equilibrium point. So, we have one spring in wave motion that contains both transverse and longitudinal components.

Key Points

  • Waves transfer energy from one point to another without a net transfer of matter.
  • Transverse waves have an oscillating medium perpendicular to the direction of wave propagation, whereas longitudinal waves have an oscillating medium parallel to the direction of wave propagation.
  • There are four properties of waves that are specific to certain types of waves. Peaks and troughs describe transverse waves. Compressions and rarefactions describe longitudinal waves.
  • There are four properties that apply to both types of waves: medium, wavelength, amplitude, and frequency. Each of these has its own definition.
  • Wave motion can consist of both transverse and longitudinal components.

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