Sound and Alternating CurrentDate: 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/ |
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