## Acoustic Waves

We noted that Eq. (3.52) is the governing equation for acoustic waves. Waves on a string can be looked upon as one-dimensional acoustic waves in that they are governed by this equation in one dimension. And waves on the strings of musical instruments excite three-dimensional acoustic waves in the air surrounding them.

Although our primary interest is (vector) electromagnetic waves, acoustic waves in fluids are scalar waves and hence simpler. For this reason we often draw analogies between acoustic and electromagnetic waves. Although the two are similar, they are also different, most notably in the way they usually are detected, including by humans. Detectors of light, such as our eyes and photomultiplier tubes, are power detectors: the detected signal is the time-averaged power because of the very short period (inverse frequency) of light waves relative to the detector response. But the instantaneous amplitude of sound waves is detected by the human ear.

Suppose that two time-harmonic waves of different frequency but equal amplitude are superposed:

By using the identities

2 . , . 1 + cos x 2, , . 1 — cos x cos (x/2) = ——-, sin (x/2) =---, (3.61)

fx ± y \ x y x y cos - = cos — cos — =F sin — sin —, (3.62)

If the two frequencies are relatively close to each other we may interpret this equation as describing a wave of frequency equal to the average of the two frequencies, called the carrier frequency, with amplitude varying (modulated) at a frequency equal to half the difference of the two frequencies. This frequency difference is called the beat frequency. Two harmonic sound waves of different frequency when superposed give rise to the phenomenon of beats. And the human ear often can hear these beats.