16.3 Sinusoidal Waves

A wave source that oscillates with simple harmonic motion (SHM) generates a sinusoidal wave. For example, a loudspeaker cone that oscillates in SHM radiates a sinusoidal sound wave. The sinusoidal electromagnetic waves broadcast by cell towers and FM radio stations are generated by electrons oscillating back and forth in the antenna wire with SHM. The frequency $f$ of the wave is the frequency of the oscillating source.

FIGURE 16.10 shows a sinusoidal wave moving through a medium. To understand how this wave propagates, let’s look at history and snapshot graphs. FIGURE 16.11a is a history graph, showing the displacement of the medium at one point in space. Each particle in the medium undergoes simple harmonic motion with frequency f, so this graph of SHM is identical to the graphs you learned to work with in Chapter 15. The period of the wave, shown on the graph, is the time interval for one cycle of the motion. The period is related to the wave frequency f by

T = \frac{1}{f}

(16.3)

exactly as in simple harmonic motion. The amplitude A of the wave is the maximum value of the displacement. The crests of the wave have displacement $D_{crest} = A$ and the troughs have displacement $D_{trough} = - A$ .

The figure illustrates a sinusoid wave moving along the x axis.

FIGURE 16.10 Full Alternative Text

The figure consists of two parts labeled ("a") and (b) that illustrate history and snapshot curves for a sinusoidal wave.

FIGURE 16.11 Full Alternative Text

Displacement versus time is only half the story. FIGURE 16.11b shows a snapshot graph for the same wave at one instant in time. Here we see the wave stretched out in space, moving to the right with speed v. An important characteristic of a sinusoidal wave is that it is periodic in space as well as in time. As you move from left to right along the “frozen” wave in the snapshot graph, the disturbance repeats itself over and over. The distance spanned by one cycle of the motion is called the wavelength of the wave. Wavelength is symbolized by $λ$ (lowercase Greek lambda) and, because it is a length, it is measured in units of meters. The wavelength is shown in Figure 16.11b as the distance between two crests, but it could equally well be the distance between two troughs.

NOTE Wavelength is the spatial analog of period. The period T is the time in which the disturbance at a single point in space repeats itself. The wavelength $λ$ is the distance in which the disturbance at one instant of time repeats itself.

16.3 Sinusoidal Waves

FIGURE 16.10 A sinusoidal wave moving along the x-axis.

FIGURE 16.11 History and snapshot graphs for a sinusoidal wave.