Please see question attached
The Attempt at a Solution
Ok so ive been able to do the first few parts and have derived that Wm,n = c pi / L (m^2 + n^2)
I've thus been able to show that the second lowest freq is a factor of root(5/2) times larger than the first and that there are two modes that have this frequency. But how do you do the next two parts?
How do I show that by combining them it is possible to have a node along either diagonal of the square? I guess I have to add the two solutions and subtract, but then how do I show that these nodes are along the diagonal?
Also what about the last part? How am i meant to deduce the fundamental mode for a triangular membrane?