For a string with fixed ends, which normal modes are missing?

1. Nov 22, 2015

snickersnee

1. The problem statement, all variables and given/known data

Here's the problem. I was able to find the a_n and b_n values, my question is mainly on part (c), how do I find which modes are missing? The function is odd, so even modes should disappear, but cos(n*pi) doesn't disappear, it's either +1 or -1. I'd greatly appreciate any help.

2. Relevant equations

b_n = 0 because released from rest

3. The attempt at a solution

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2. Nov 23, 2015

RUber

When it says fixed ends, are we to assume that at x = 0 and x = L, the ends are fixed at $\xi(0,t) = \xi(L,t) = 0$?
If so, then you should be able to say something about $\xi( L/2,t)$.

Using your equation for a_n, break that into two integrals, one from 0 to L/2 and one from L/2 to L. Then enforce an appropriate matching condition for $\xi(L/2,t)$ and $\xi_t(L/2,t)$