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### Simulating Sound Waves

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Date: 10/27/2000 at 06:21:36
From: Geoff Kay
Subject: Graphing the sound wave

Dr. Math,

I cannot find any way of creating an equation that would (using a
graphing program) simulate the basic structure of the sound wave. By
this I mean a wave with a host of different amplitudes at different
locations. In particular, I am after a wave that will reach a maximum
(identical) height every seventh crest. Can this be done? Can you
help?
```

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Date: 10/27/2000 at 12:38:45
From: Doctor Rick
Subject: Re: Graphing the sound wave

Hi, Geoff, I'll be glad to help.

Sound waves are a sum (or "superposition") of sinusoidal waves of
different frequencies as well as amplitudes. The waveform you are
seeking can be formed in two ways.

One, the easiest to comprehend, is a product of sinusoids, one varying
more slowly and constituting, in effect, the varying amplitude of the
other. Thus,

x(t) = sin(2*pi*t/T)*cos(2*pi*t/(14T))

is a sinusoid of period T (the sine function), modulated by a sinusoid
with a period of 14T (the cosine function). Twice every 14 periods the
amplitude goes to zero. When the cosine rises to 1, the function x(t)
has maximum amplitude. When the cosine goes to -1, x(t) again has
maximum amplitude, but its phase is reversed. Thus, the peak of the
function occurs twice every 14 periods, or once every 7 periods, which
is what you were looking for.

The same signal can be obtained as a sum of sinusoids with similar
frequencies. Mathematically, this follows from the trigonometric
identity

sin(a) + sin(b) = 2*sin((a+b)/2)*cos((a-b)/2)

If we let

(a+b)/2 = 2*pi*t/T
(a-b)/2 = 2*pi*t/(14T)

and solve for a and b, we get

a = 2*pi*t/T*(1 + 1/14)
b = 2*pi*t/T*(1 - 1/14)

x(t) = (1/2)(sin(2*pi*t/(14T/15)) + cos(2*pi*t/(14T/13)))

This signal is half the sum of a sine and a cosine, whose frequencies
are in the ratio 15:13.

The second approach models the nature of sound waves as a
superposition, or sum, of sinusoids. In sound physics, the phenomenon
is called a "beat frequency": the sum of the two signals has an
amplitude that varies with a frequency equal to the difference in the
frequencies of the two signals. The frequencies of our signals are
13/(14T) and 15/(14T); the difference between these frequencies is
2/(14T) = 1/(7T), which is 1/7 the base ("carrier") frequency.

I hope this meets your needs.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Physics/Chemistry
High School Trigonometry

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