strings and things

Continuing the mathematics portion of our scheduled programming, it might be handy to look an example from physics at this point.

One common example that comes along during an undergraduate degree in physics, math or engineering (if I left our your major, settle down egghead) is that of a vibrating string. It is an application of the wave equation and is often the first place that students sill see eigenvalues in the way described last time.

The position of the string is determined by its displacement at a particular spot from its rest position at a given time. So the state u of the string is described by two variables, x for position on the string and t for time, and has values that are the displacement of the string from rest. We may take a derivative with respect to either of those two variables (a partial derivative, this is called), which will use the following notation: u_t is the partial derivative with respect to t, which u_tt is the second partial derivate with respect to t. It is possible to take a derivative with respect to one variable, then the other (producing "mixed partials"), but we won't have to worry about those here. The wave equation, which is derived from some physical assumptions about the string, is then:

u_tt = c*c*u_xx

where c is the (constant here) speed of propagation of the wave (if you are talking about a sound wave, it is the speed of sound, for example).

It is well known (you can learn why in a partial differential equations course) that solutions to this equation have the form u(x,t) = y(x) z(t), meaning that u is a function of position times a function of time. This means that u_tt = y(x) z''(t) and u_xx = y''(x) z(t). Rearranging the above equation (I won't do all the calculations, because that is not really the point here), we find that

c*c*z''/z = -y''/y = 1/(k*k).

The first term is a function of t, and it is equal to a function of x (the middle term), so they both must be equal to a constant. We may rewrite the second equation as

y'' = -k*k*y,

which is the same differential equation encountered in the previous math post. We now just apply boundary conditions and we are away. The solutions look like standing waves, and represent the different modes you can have in a vibrating string, on a violin for example. Higher modes (larger eigenvalues) represent faster vibration and higher pitches.

Posted by warcode at December 02, 2002 09:40 PM