W a v e E q u a t i o n
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Suppose you pluck a string at a point, say, one-quarter way from its left end, so that it looks something like nrows=33&ncols=65&ncoefs=99& fgblue=135&f=1,.75,.5,.25>imestep=1&bigf=1 just before you release it. Exactly how does the string behave after you release it? The one-dimensional wave equation is essentially the plucked string problem. Our plucked string behaves like nrows=33&ncols=65&ncoefs=99& fgblue=135&f=1,.75,.5,.25>imestep=0&bigf=1&dt=.05 , and this page discusses how we arrive at that solution.
All problems in classical mechanics boil down to (plus conservation). Point masses subject to constant forces are typically trivial. Extended masses subject to forces that vary over space and time can be more complicated to treat. Reasonable simplifying assumptions often make such complicated problems more tractable. We make two assumptions:
Keeping these assumptions in mind, the forces acting on an element of the stretched string are illustrated in more detail by
Due to the
so there's no
where you'll notice just defines the derivative.
acts on the
whose mass is
string's linear density. And
For comp.text.tex ng readers:
Cropped test image (dvips bounding box determined by A and B)...
Uncropped test image (note the two dots in upper left- and right-corners)...