On Nov 23, 2007 7:15 AM, Mathieu Bouchard <<a href="mailto:matju@artengine.ca">matju@artengine.ca</a>> wrote:<br><div class="gmail_quote"><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
<div class="Ih2E3d">On Sat, 17 Nov 2007, Uur Güney wrote:<br>> An example of sound producing mechanism is<br>> plucked and vibrating string (or vibrating membrane) It is a continuum<br>> and so has infinite dimensions.
<br><br></div>It's not because it's a continuum, that it has infinite dimensions. Real<br>numbers form a continuum, but have only 1 dimension.</blockquote><div> <br></div><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
The set of all possible continuous functions over a given finite interval,<br>forms a continuum that has infinitely (countably) many dimensions. This<br>continuum also happens to include some simple (Fourier-compatible)<br>
discontinuities as well. (Including all possible discontinuities is<br>another story.) Physical sounds can be understood to have no<br>discontinuities, as several factors tend to "low-pass" the sound enough to<br>
remove discontinuities.<br></blockquote><div><br># Ok. I got it. Thanks for clarification.<br># 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." <br># 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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