Is there a mathematical equation to decompose complex sound wave into it's simple parts?

Question

Ie., to decompose a sound wave of multiple instruments, notes, and vibrations into it's component parts, the individual tones.

Answer 1

This question is one that falls into what is called Digital Signal Processing. I am taking a graduate course in it now. What you need is either the Fourier Transform, if you have a continuous function for the wave, or much more likely the Z-Transform, which is basically the Fourier Transform for when you have, instead of a continuous function, a whole bunch of discrete data points, like in a computer sound file. As I said, this is part of a whole subject of mathematics and engineering. If you want more information, google and wikipedia searches should give some more introductory information.

Answer 2

Not really.
The complex sound wave is not a mathematical REGULAR function: it is changing over time (as notes are played one after the other). You cannot get a function to represent a whole symphony!
However, if you cut out bits of a sound (a short period of time), you will find that during that period, the wave is relatively constant, and you could derive a function from it.
Then, to decompose that part, use the Fourrier and Laplace analysis to decompose the function into several sine functions.
Each of them will have a given frequency and amplitude.
You will find, for example, a root of 440Hz, say 10units amplitude, followed by a 880Hz, 5units (harmonic 2), another of 1760Hz, 3units etc. The combination of harmonic frequencies gives the "tone" of an instrument.
Hard work, but feasible...
You could also, and it is much easier, use a spectrum analyser scope. This apparatus decomposes the signal into all its unique phases, frequencies and amplitudes. The output could be read and stored in a PC and manipulated afterward.
If you are very smart, you could even use that data to write the score of the music!
(I believe there are some PC programs that do something similar: look for synthetizers, music writing programs and sound mixers).
Good luck!

Answer 3

For the simpler math student, manipulating the function sin(x) to change amplitude and wavelength can be a good way to picture waves. In higher level courses like differential equations, there are partial differential equations to represent complex waveforms.

Answer 4

yeah there is a math eqation for everything