Suppose you were to take a photograph of ripples spreading on a pond. If you mark the location of the crests on the photo, your picture would look like FIGURE 16.25a. The lines that locate the crests are called wave fronts, and they are spaced precisely one wavelength apart. The diagram shows only a single instant of time, but you can imagine a movie in which you would see the wave fronts moving outward from the source at speed v. A wave like this is called a circular wave. It is a two-dimensional wave that spreads across a surface.

FIGURE 16.25 The wave fronts of a circular or spherical wave.

Although the wave fronts are circles, you would hardly notice the curvature if you observed a small section of the wave front very, very far away from the source. The wave fronts would appear to be parallel lines, still spaced one wavelength apart and traveling at speed v. A good example is an ocean wave reaching a beach. Ocean waves are generated by storms and wind far out at sea, hundreds or thousands of miles away. By the time they reach the beach where you are working on your tan, the crests appear to be straight lines. An aerial view of the ocean would show a wave diagram like FIGURE 16.25b.

Many waves of interest, such as sound waves or light waves, move in three dimensions. For example, loudspeakers and lightbulbs emit spherical waves. That is, the crests of the wave form a series of concentric spherical shells separated by the wavelength $\text{λ}$. In essence, the waves are three-dimensional ripples. It will still be useful to draw wave-front diagrams such as Figure 16.25, but now the circles are slices through the spherical shells locating the wave crests.

If you observe a spherical wave very, very far from its source, the small piece of the wave front that you can see is a little patch on the surface of a very large sphere. If the radius of the sphere is sufficiently large, you will not notice the curvature and this little patch of the wave front appears to be a plane. FIGURE 16.26 illustrates the idea of a plane wave.

FIGURE 16.26 A plane wave.

To visualize a plane wave, imagine standing on the x-axis facing a sound wave as it comes toward you from a very distant loudspeaker. Sound is a longitudinal wave, so the particles of medium oscillate toward you and away from you. If you were to locate all of the particles that, at one instant of time, were at their maximum displacement toward you, they would all be located in a plane perpendicular to the travel direction. This is one of the wave fronts in Figure 16.26, and all the particles in this plane are doing exactly the same thing at that instant of time. This plane is moving toward you at speed v. There is another plane one wavelength behind it where the molecules are also at maximum displacement, yet another two wavelengths behind the first, and so on.

Because a plane wave’s displacement depends on x but not on y or z, the displacement function $D\left(x\text{,\hspace{0.17em}}t\right)$ describes a plane wave just as readily as it does a one-dimensional wave. Once you specify a value for x, the displacement is the same at every point in the yz-plane that slices the x-axis at that value (i.e., one of the planes shown in Figure 16.26).

We can describe a circular wave or a spherical wave by changing the mathematical description from $D\left(x\text{,\hspace{0.17em}}t\right)$ to $D\left(r\text{,}t\right)$, where r is the radial distance measured outward from the source. Then the displacement of the medium will be the same at every point on a spherical surface. In particular, a sinusoidal spherical wave with wave number k and angular frequency $\omega$ is written

Other than the change of x to r, the only difference is that the amplitude is now a function of r. A one-dimensional wave propagates with no change in the wave amplitude. But circular and spherical waves spread out to fill larger and larger volumes of space. To conserve energy, an issue we’ll look at later in the chapter, the wave’s amplitude has to decrease with increasing distance r. This is why sound and light decrease in intensity as you get farther from the source. We don’t need to specify exactly how the amplitude decreases with distance, but you should be aware that it does.