Does anyone know where I need to start on this particular question?

Question

Assume that a violin string of length 10 is held so that at time zero its shape is given by the function:


f(x) = (x/4) at (0< x < 4), (-x/6 +10/6) at (4< x <10)

and released, so that its initial velocity is zero. Find the function u(x,t) that describes the height of the point x of the string at time t. Use 5 terms of your series to determine the position of the midpoint of the string at time t =5.

We only barely covered partial differentials in class so any help would be useful.

Answer 1

All shapes on a vibrating string are simply linear additions of the sinusoidal basis functions. If the fundamental frequency is 100 Hz, the other frequencies are all integer multiples of this (200 Hz, 300 Hz, 400 Hz, etc.). Each basis function is described by a function of the form

f_n = sin(n*pi*x/L), where n = 1, 2, 3...

You need to calculate the Fourier series of this initial shape. If the starting shape of the string is f(x), the Fourier coefficients are...

a_n = 1/2 * integral[ sin(n*pi*x/L)*f(x), x, 0, L ]

where n = 1, 2, 3... and L = the length of the string. Doing each integral will actually take two integrations, since your equation is defined piece-wise (first from 0 to 4, then from 4 to 10).

The Fourier coefficients tell you how much of each frequency is present in the initial shape. I get this result:

a_n = -25/3 * 1/(n^2*pi^2) * (2*n*pi*cos( 2*n*pi/5 ) - 2*n*pi*cos( n*pi ) - 5*sin( 2*n*pi/5 ) + 2*sin( n*pi ))
where n = 1, 2, 3...

The frequency with which each function oscillates is simply
w_n = w0 * n
where w0 is the fundamental frequency, and n = 1, 2, 3...

Now you simply multiply each basis function by cos(w_n * t) to yield a series of the following form:

F[x,t] = a_n * sin(n*pi*x/L) * cos(w_n * t)

This describes the shape of the string at ALL subsequent points in time. For a rough approximation, you can only use the terms n=1 through n=5, as your problem suggested. Theoretically, you have to sum from n=1 to n=infinity to yield the exact solution.