Chapter 13.6, Problem 3P

Separate the wave equation in two-dimensional rectangular coordinates x, y. Consider a rectangular membrane as shown, rigidly attached to supports along its sides. Show that its characteristic frequencies are ${\nu }_{nm}=(v/2)\sqrt{{(n/a)}^2+{(m/b)}^2}$, where $n$ and $m$ are positive integers, and sketch the normal modes of vibration corresponding to the first few frequencies. That is, indicate the nodal lines as we did for the circular membrane in Figure 6.1 and Problem 1.

Next suppose the membrane is square. Show that in this case there may be I more normal modes of vibration corresponding to a single frequency. (Hint Eror one example: $7^2+1^1=1^2+7^2=5^2+5^2.$ ) This is an example of what iscalled degeneracy, we say that there is degeneracy when several different solutions of the wave equation (eigenfunctions) correspond to the same frequency (eigenvalue). Sketch several normal modes giving rise to the same frequency. Comment: Compare Chapter 3, Section 11, where an eigenvalue of a matrix is called degenerate if several eigenvectors correspond to it.