ugurguney at gmail.com
Fri Nov 23 12:23:13 CET 2007
On Nov 23, 2007 7:15 AM, Mathieu Bouchard <matju at 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".
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