16.4. The Mathematical Description of a Wave

When a wave travels through a medium, it displaces the particles of the medium from their undisturbed positions. Suppose a particle is located at a distance x from a coordinate origin. We would like to know the displacement y of this particle from its undisturbed position at any time t as the wave passes. For periodic waves that result from simple harmonic motion of the source, the expression for the displacement involves a sine or cosine, a fact that is not surprising. After all, in Chapter 10 simple harmonic motion is described using sinusoidal equations, and the graphs for a wave in Figure 16.6 look like a plot of displacement versus time for an object oscillating on a spring (see Figure 10.6).

Our tack will be to present the expression for the displacement and then show graphically that it gives a correct description. Equation 16.3 represents the displacement of a particle caused by a wave traveling in the +x direction (to the right), with an amplitude A, frequency f, and wavelength l. Equation 16.4 applies to a wave moving in the –x direction (to the left).

(16.3)

(16.4)

These equations apply to transverse or longitudinal waves and assume that y=0 m when x=0 m and t=0 s.

Consider a transverse wave moving in the +x direction along a string. The term (2pft–2px/l) in Equation 16.3 is called the phase angle of the wave. A string particle located at the origin (x=0 m) exhibits simple harmonic motion with a phase angle of 2pft; that is, its displacement as a function of time is y=A sin (2pft). A particle located at a distance x also exhibits simple harmonic motion, but its phase angle is

The quantity x/v is the time needed for the wave to travel the distance x. In other words, the simple harmonic motion that occurs at x is delayed by the time interval x/v compared to the motion at the origin.

Figure 16.11 shows the displacement y plotted as a function of position x along the string at a series of time intervals separated by one-fourth of the period. These graphs are constructed by substituting the corresponding value for t into Equation 16.3, remembering that f=1/T, and then calculating y at a series of values for x. The graphs are like photographs taken at various times as the wave moves to the right. For reference, the colored square on each graph marks the place on the wave that is located at x=0 m when t=0 s. As time passes, the colored square moves to the right, along with the wave. In a similar manner, it can be shown that Equation 16.4 represents a wave moving in the –x direction. Note that the phase angles (2pft–2px/l) in Equation 16.3 and (2pft+2px/l) in Equation 16.4 are measured in radians, not degrees. When a calculator is used to evaluate the functions sin (2pft–2px/l) or sin (2pft+2px/l), it must be set to its radian mode.

Equation 16.3 is plotted here at a series of times separated by one-fourth of the period T. The colored square in each graph marks the place on the wave that is located at x

0 m when t

0 s. As time passes, the wave moves to the right.

Figure 16.11 Equation 16.3 is plotted here at a series of times separated by one-fourth of the period T. The colored square in each graph marks the place on the wave that is located at x

0 m when t

0 s. As time passes, the wave moves to the right.

Self-Assessment Test 16.1

Check your understanding of the material in Sections 16.1, 16.2, 16.3, and 16.4:

·Periodic Waves ·The Speed of a Wave on a String ·The Mathematical Description of a Wave