 Rectangular membrane normal mode

For a rectangle of sides a and b with fixed edges, separation of variables gives $\psi =A\sin (p\pi x/a)\sin (q\pi y/b)\cos (\omega t+\varphi )$ with positive integers p and q. Its angular frequency is $\omega =c\pi \sqrt{p^2/a^2+q^2/b^2}$. Zero integers produce the trivial displacement, not a lower mode. The p−1 and q−1 interior nodal lines divide alternating-sign lobes.