On Nov 23, 2007 7:15 AM, Mathieu Bouchard matju@artengine.ca wrote:
On Sat, 17 Nov 2007, Uur Güney wrote:
An example of sound producing mechanism is plucked and vibrating string (or vibrating membrane) It is a continuum and so has infinite dimensions.
It's not because it's a continuum, that it has infinite dimensions. Real numbers form a continuum, but have only 1 dimension.
The set of all possible continuous functions over a given finite interval, forms a continuum that has infinitely (countably) many dimensions. This continuum also happens to include some simple (Fourier-compatible) discontinuities as well. (Including all possible discontinuities is another story.) Physical sounds can be understood to have no discontinuities, as several factors tend to "low-pass" the sound enough to remove discontinuities.
# Ok. I got it. Thanks for clarification. # Once I asked to my Non-linear Dynamics teacher. "Isn't the shape of a string a 1D function of its length? Why we are calling it as continuum?" And she said that: "A simple harmonic oscillator makes a 1D motion (in time). It goes back and forth. You can approximate a string as N connected harmonic oscillator lying along a line. if N goes to infinity we'll have a SHO at every point in space, which makes a 1D motion in time. And this is a field, and hence it is a continuum." # This is in accordance with your definition, an ideal string can have any shape, so its possible shapes form "the set of all possible continous functions over its length".