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Interference in Time

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

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

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


Figure:   Beats.
\begin{figure}\begin{center}\mbox{
\epsfig{file=PS/beats.ps,height=3.85in}}\end{center}\end{figure}

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

next up previous
Next: Interference in Space Up: Interference Previous: Interference
Jess H. Brewer - Last modified: Sun Nov 15 21:36:45 PST 2015