Simulating Sound Waves

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?

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/