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Interference and Sound Waves

We will not cover interference and diffraction of harmonic sound waves in this course. Beats are a common experience in sound as is the doppler shift, but sound wave interference is not so common an experience (although it can definitely and annoyingly occur if you hook up speakers in your stereo out of phase). Interference will be treated next semester in the context of coherent light waves. Just to give you a head start on that, we'll indicate the basic ideas underlying interference here.

Suppose you have two sources that are at the same frequency and have the same amplitude and phase but are at different locations. One source might be a distance $x$ away from you and the other a distance $x+\Delta x$ away from you. The waves from these two sources add like:

$\displaystyle s(x,t)$ $\textstyle =$ $\displaystyle s_0 \sin(k x - \omega t) + s_0 \sin(k (x + \Delta x) -
\omega t)$ (196)
  $\textstyle =$ $\displaystyle 2 s_0 \sin(k(x + \frac{\Delta x}{2} - \omega t)
\cos(k\frac{\Delta x}{2})$ (197)

The sine part describes a wave with twice the amplitude, the same frequency, but shifted slightly in phase by $k \Delta x/2$. The cosine part is time independent and modulates the first part. For some values of $\Delta x$ it can vanish. For others it can have magnitude one.

The intensity of the wave is what our ears hear - they are insensitive to the phase (although certain echolocating species such as bats may be sensitive to phase information as well as frequency). The average intensity is proportional to the wave amplitude squared:

\begin{displaymath}
I_0 = \frac{1}{2}\rho \omega^2 s_0^2 v
\end{displaymath} (198)

With two sources (and a maximum amplitude of two) we get:

$\displaystyle I$ $\textstyle =$ $\displaystyle \frac{1}{2}\rho \omega^2 (2^2 s_0^2 \cos^2(k\frac{\Delta x}{2})
v$ (199)
  $\textstyle =$ $\displaystyle 4 I_0 \cos^2(k\frac{\Delta x}{2})$ (200)

There are two cases of particular interest in this expression. When

\begin{displaymath}
\cos^2(k\frac{\Delta x}{2}) = 1
\end{displaymath} (201)

one has four times the intensity of one source at peak. This occurs when:
\begin{displaymath}
k\frac{\Delta x}{2} = n \pi
\end{displaymath} (202)

(for $n = 0, 1, 2...$) or
\begin{displaymath}
\Delta x = n \lambda
\end{displaymath} (203)

If the path difference contains an integral number of wavelengths the waves from the two sources arrive in phase, add, and produce sound that has twice the amplitude and four times the intensity. This is called complete constructive interference.

On the other hand, when

\begin{displaymath}
\cos^2(k\frac{\Delta x}{2}) = 0
\end{displaymath} (204)

the sound intensity vanishes. This is called destructive interference. This occurs when
\begin{displaymath}
k\frac{\Delta x}{2} = \frac{2n+1}{2} \pi
\end{displaymath} (205)

(for $n = 0, 1, 2...$) or
\begin{displaymath}
\Delta x = \frac{2n+1}{2} \lambda
\end{displaymath} (206)

If the path difference contains a half integral number of wavelengths, the waves from two sources arrive exactly out of phase, and cancel. The sound intensity vanishes.

You can see why this would make hooking your speakers up out of phase a bad idea. If you hook them up out of phase the waves start with a phase difference of $\pi$ - one speaker is pushing out while the other is pulling in. If you sit equidistant from the two speakers and then harmonic waves with the same frequency from a single source coming from the two speakers cancel as they reach you (usually not perfectly) and the music sounds very odd indeed, because other parts of the music are not being played equally from the two speakers and don't cancel.

You can also see that there are many other situations where constructive or destructive interference can occur, both for sound waves and for other waves including water waves, light waves, even waves on strings. Our ``standing wave solution'' can be rederived as the superposition of a left- and right-travelling harmonic wave, for example. You can have interference from more than one source, it doesn't have to be just two.

This leads to some really excellent engineering. Ultrasonic probe arrays, radiotelescope arrays, sonar arrays, diffraction gratings, holograms, are all examples of interference being put to work. So it is worth it to learn the general idea as early as possible, even if it isn't assigned.


next up previous contents
Next: Fluids Up: Sound Previous: Beats   Contents
Robert G. Brown 2004-04-12