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/
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