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.

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)
  http://www.vvm.com/~radioray/html/e101-23.htm   
   
  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 
more information on sound waves and Fourier's theorem, see:

  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/