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Interference in Time

Suppose we add together two equal amplitude waves with slightly different frequencies

$$\omega_1 = \bar{\omega} + \delta/2 \quad \hbox{\rm and} \quad \omega_2 = \bar{\omega} - \delta/2$$ (14.44)

where $\bar{\omega}$ is the average frequency and $\delta$ is the difference between the two frequencies.

Figure:   Beats.

If we measure the combined amplitude at a fixed point in space, a little algebra reveals the phenomenon of BEATS. This is usually done with $\sin$ or $\cos$ functions and a lot of trigonometric identities; let's use the complex notation instead - I find it more self-evident, at least algebraically:

$$\psi(z,t) = \psi_{_0} \; \left[ e^{i \omega_1 t} + e^{i \omega_2 t} \right] \cr$$ (14.45)

That is, the combined signal consists of an oscillation at the average frequency, modulated by an oscillation at one-half the difference frequency. This phenomenon of "BEATS" is familiar to any musician, automotive mechanic or pilot of a twin engine aircraft.

One seemingly counterintuitive feature of BEATS is that the "envelope function" $\cos [(\delta/2) t]$ has only half the angular frequency of the difference between the two original frequencies. What we hear when two frequencies interfere is the variation of the sound INTENSITY with time; and the intensity is proportional to the square of the displacement.^14.23Squaring the envelope effectively doubles its frequency (see Fig. 14.12) and so the detected BEAT FREQUENCY is the full frequency difference $\delta = \omega_1 - \omega_2$.

This is a universal feature of waves and interference: the detected signal is the average intensity, which is proportional to the square of the amplitude of the displacement oscillations; and it is the displacements themselves that add linearly to form the interference pattern. Be sure to keep this straight.

Footnotes

. . . displacement.^14.23

Actually the INTENSITY is defined in terms of the average of the square of the displacement over times long compared with the average frequency $\bar{\omega}$. This makes sense as long as the beat frequency $\delta \ll \bar{\omega}$; but if $\omega_1$ and $\omega_2$ differ by an amount $\delta \sim \bar{\omega}$ then it is hard to define what is meant by a "time average". We will just duck this issue.