16.3.1 The Fundamental Relationship for Sinusoidal Waves

There is an important relationship between the wavelength and the period of a wave. FIGURE 16.12 shows this relationship through five snapshot graphs of a sinusoidal wave at time increments of one-quarter of the period T. One full period has elapsed between the first graph and the last, which you can see by observing the motion at a fixed point on the x-axis. Each point in the medium has undergone exactly one complete oscillation.

The figure consists of five parts that illustrate a series of snapshot graphs at time increments of one-quarter of the period T.

FIGURE 16.12 Full Alternative Text

The critical observation is that the wave crest marked by an arrow has moved one full wavelength between the first graph and the last. That is, each crest of a sinusoidal wave travels forward a distance of exactly one wavelength $λ$ during a time interval of exactly one period T. Because speed is distance divided by time, the wave speed must be

v = \frac{distance}{time} = \frac{λ}{T}

(16.4)

Because $f = 1 / T$ , it is customary to write Equation 16.4 in the form

v = λ f

(16.5)

Although Equation 16.5 has no special name, it is the fundamental relationship for periodic waves. When using it, keep in mind the physical meaning that a wave moves forward a distance of one wavelength during a time interval of one period.

NOTE Wavelength and period are defined only for periodic waves, so Equations 16.4 and 16.5 apply only to periodic waves. A wave pulse has a wave speed, but it doesn’t have a wavelength or a period. Hence Equations 16.4 and 16.5 cannot be applied to wave pulses.

Because the wave speed is a property of the medium while the wave frequency is a property of the oscillating source, it is often useful to write Equation 16.5 as

λ = \frac{v}{f} = \frac{property of the medium}{property of the source}

(16.6)

The wavelength is a consequence of a wave of frequency $f$ traveling through a medium in which the wave speed is v.