5.5: Waves

Learning Objectives

• State the characteristics of a wave.
• Calculate the velocity of wave propagation, given frequency and wavelength

Figure \(\PageIndex{1}\): Waves in the ocean behave similarly to all other types of waves. (credit: Steve Jurveston, Flickr)

What do we mean when we say something is a wave? The most intuitive and easiest wave to imagine is the familiar water wave. More precisely, a wave is a disturbance that propagates, or moves from the place it was created. For water waves, the disturbance is in 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. For earthquakes, there are several types of disturbances, including disturbance of Earth’s surface and pressure disturbances under the surface. Even radio waves are most easily understood using an analogy with water waves. Visualizing water waves is useful because there is more to it than just a mental image. Water waves exhibit characteristics common to all waves, such as amplitude, period, frequency and energy. All wave characteristics can be described by a small set of underlying principles.

A wave is a disturbance that propagates, or moves from the place it was created. The simplest waves repeat themselves for several cycles and are associated with simple harmonic motion. Let us start by considering the simplified water wave in Figure \(\PageIndex{2}\). The wave is an up and down disturbance of the water surface. It causes a sea gull to move up and down in simple harmonic motion as the wave crests and troughs (peaks and valleys) pass under the bird. The time for one complete up and down motion is the wave’s period \(T\). The wave’s frequency is \(f=1 / T\), as usual. The wave itself moves to the right in the figure. This movement of the wave is actually the disturbance moving to the right, not the water itself (or the bird would move to the right). We define wave velocity \(v_{\text {w }}\) to be the speed at which the disturbance moves. Wave velocity is sometimes also called the propagation velocity or propagation speed, because the disturbance propagates from one location to another.

MISCONCEPTION ALERT

Many people think that water waves push water from one direction to another. In fact, the particles of water tend to stay in one location, save for moving up and down due to the energy in the wave. The energy moves forward through the water, but the water stays in one place. If you feel yourself pushed in an ocean, what you feel is the energy of the wave, not a rush of water.

Figure \(\PageIndex{2}\): An idealized ocean wave passes under a sea gull that bobs up and down in simple harmonic motion. The wave has a wavelength \(\lambda\), which is the distance between adjacent identical parts of the wave. The up and down disturbance of the surface propagates parallel to the surface at a speed \(v_{\mathrm{w}}\).

he water wave in the figure also has a length associated with it, called its wavelength \(\lambda\), the distance between adjacent identical parts of a wave. (\(\lambda\) is the distance parallel to the direction of propagation.) The speed of propagation \(v_{\mathrm{w}}\) is the distance the wave travels in a given time, which is one wavelength in the time of one period. In equation form, that is

\[v_{\mathrm{w}}=\frac{\lambda}{T} \nonumber \]

or

\[v_{\mathrm{w}}=f \lambda. \label{1} \]

This fundamental relationship holds for all types of waves. For water waves, \(v_{\mathrm{w}}\) is the speed of a surface wave; for sound, \(v_{\mathrm{w}}\) is the speed of sound; and for visible light, \(v_{\mathrm{w}}\) is the speed of light, for example.

One note of caution with Equation \(\eqref{1}\): despite the appearance, wave speed \(v_{\text {w }}\) is a not a function of frequency ff and wavelength \(\lambda\). When ff changes (or \(\lambda\) changes), \(v_{\text {w }}\) does not change (you will see this explicitly later with sound and light waves). Wave speed is a property of the medium the wave travels in. Unless the medium itself changes, the wave speed remains constant. This is true of sound and light waves, as well as almost all other types of waves you will see in this textbook (water waves, waves on a string, etc.). So, what happens when frequency or wavelength does change, then? The wavelength of frequency changes to compensate. For example, if the frequency doubles for a given wave, instead of the wave speed doubling, the wavelength will decrease to half, so that the product \(f \lambda\) remains constant.

TAKE-HOME EXPERIMENT: WAVES IN A BOWL

Fill a large bowl or basin with water and wait for the water to settle so there are no ripples. Gently drop a cork into the middle of the bowl. Estimate the wavelength and period of oscillation of the water wave that propagates away from the cork. Remove the cork from the bowl and wait for the water to settle again. Gently drop the cork at a height that is different from the first drop. Does the wavelength depend upon how high above the water the cork is dropped?

Calculate the Velocity of Wave Propagation: Gull in the Ocean

Calculate the wave velocity of the ocean wave in Figure \(\PageIndex{2}\) if the distance between wave crests is 10.0 m and the time for a sea gull to bob up and down is 5.00 s.

Strategy

We are asked to find \(v_{\mathrm{w}}\). The given information tells us that \(\lambda=10.0 \mathrm{~m}\) and \(T=5.00 \mathrm{~s}\). Therefore, we can use \(v_{\mathrm{w}}=\frac{\lambda}{T}\) to find the wave velocity.

Solution

1. Enter the known values into \(v_{\mathrm{w}}=\frac{\lambda}{T}\):

   \[v_{\mathrm{w}}=\frac{10.0 \mathrm{~m}}{5.00 \mathrm{~s}}. \nonumber\]

2. Solve for \(v_{\mathrm{w}}\) to find \(v_{\mathrm{w}}=2.00 \mathrm{~m} / \mathrm{s}\).

Discussion

This slow speed seems reasonable for an ocean wave. Note that the wave moves to the right in the figure at this speed, not the varying speed at which the sea gull moves up and down.

Transverse and Longitudinal Waves

A simple wave consists of a periodic disturbance that propagates from one place to another. The wave in Figure \(\PageIndex{3}\) propagates in the horizontal direction while the surface is disturbed in the vertical direction. Such a wave is called a transverse wave or shear wave; in such a wave, the disturbance 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 \(\PageIndex{4}\) shows an example of a longitudinal wave. The size of the disturbance is its amplitude \(X\) and is completely independent of the speed of propagation \(v_{\mathrm{w}}\).

Figure \(\PageIndex{3}\): In this example of a transverse wave, the wave propagates horizontally, and the disturbance in the cord is in the vertical direction.

Figure \(\PageIndex{4}\): In this example of a longitudinal wave, the wave propagates horizontally, and the disturbance in the cord is also in the horizontal direction.

Waves may be transverse, longitudinal, or a combination of the two. (Water waves are actually a combination of transverse and longitudinal. The simplified water wave illustrated in Figure \(\PageIndex{2}\) shows no longitudinal motion of the bird.) The waves on the strings of musical instruments are transverse—so are electromagnetic waves, such as visible light.

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. Sound in solids can be both longitudinal and transverse.

Figure \(\PageIndex{5}\): The wave on a guitar string is transverse. The sound wave rattles a sheet of paper in a direction that shows the sound wave is longitudinal.

Earthquake waves under Earth’s surface also have both longitudinal and transverse components (called compressional or P-waves and shear or S-waves, respectively). These components have important individual characteristics—they propagate at different speeds, for example. Earthquakes also have surface waves that are similar to surface waves on water.

Exercise \(\PageIndex{1}\)

Why is it important to differentiate between longitudinal and transverse waves?

Answer

In the different types of waves, energy can propagate in a different direction relative to the motion of the wave. This is important to understand how different types of waves affect the materials around them.

Section Summary

• A wave is a disturbance that moves from the point of creation with a wave velocity \(v_{\mathrm{w}}\).
• A wave has a wavelength \(\lambda\), 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 \(v_{\mathrm{w}}=\frac{\lambda}{T}\) or \( v_{\mathrm{w}}=f \lambda \). However, wave velocity is a property of the medium and remains constant as wavelength or frequency changes.
• A transverse wave has a disturbance perpendicular to its direction of propagation, whereas a longitudinal wave has a disturbance parallel to its direction of propagation.

Glossary

longitudinal wave

a wave in which the disturbance is parallel to the direction of propagation

transverse wave

a wave in which the disturbance is perpendicular to the direction of propagation

wave velocity

the speed at which the disturbance moves; also called wave speed, propagation velocity, or propagation speed

wavelength

the distance between adjacent identical parts of a wave