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### Sound and Alternating Current

```
Date: 11/17/2000 at 14:04:31
From: Jack Gage
Subject: Sine Waves

I was wondering what characteristics sound and AC electricity have in
common that allow them both to be accurately represented by a sine
wave.
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```
Date: 11/20/2000 at 14:42:06
From: Doctor TWE
Subject: Re: Sine Waves

Hi Jack - thanks for writing to Dr. Math.

Explaining AC electricity's relation to a sine wave is (relatively)
easy. AC electricity is typically generated by rotating a coil through
a magnetic field in a circular path. The current through the coil is
directly proportional to the number of lines of flux the coil cuts per
second. The velocity vector (the speed and direction of the coil) can
be broken down into two component parts: one parallel to the lines of
flux (thus creating no current flow) and one perpendicular to the
lines of flux (creating current flow.) The magnitude ("size") of this
component - and thus the amount of current flow - is based on the sine
function in trigonometry.

Thus, AC electricity is a sine wave because the generator rotates
around in a circle. For more details on how AC generators work, check
out:

AC Theory (Fundamentals of Electricity - Ray Dall)

Chapter 11: Alternating Current (Introduction to Solar Energy,
Lecture notes, John Scofield, Oberlin)
http://www.physics.oberlin.edu/faculty/scofield/p055/Chapt-11/Ch-11.htm

Sound is a bit more complicated. Sound is created by varying pressure
of the air in our ears. Humans can perceive (as sound) these pressure
variations if they vary at a rate between about 20 Hz to 20 kHz.
(That's about 20 cycles per second to about 20,000 cycles per second.)
Most sounds are not true sine waves, but rather other periodic
(repeating) or near-periodic waveforms. Think of the waveform on an
EKG (a heart monitor.) That is not a sine wave, yet we can hear a
heartbeat (especially if magnified, as through a stethoscope.)

Fourier's theorem states that any periodic waveform may be analyzed as
the sum of a series of sine waves with frequencies in a harmonic
series. In other words, all periodic waveforms can be broken down into
a collection of sine waves of specific amplitudes and frequencies. For

Frequency and Spectral Analysis (William Robertson, Middle
Tennessee State University.
http://physics.mtsu.edu/~wmr/fourier_1.htm

Fourier Analysis and Sound (1999 J.D. Edwards Honors Seminar
Presentation, Alan Grow, Univ. of Nebraska-Lincoln)
http://www.cse.unl.edu/~agrow/jdedwards/

Introductory Hearing (Lecture notes, Chris Darwin, Univ. of Sussex)
http://www.biols.susx.ac.uk/Home/Chris_Darwin/Perception/Lecture_Notes/Hearing1/hearing1.html

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
http://mathforum.org/dr.math/
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