Rectangular membrane normal mode ($f_{pq}=\frac{c}{2}\sqrt{p^2/a^2+q^2/b^2}$)

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.

For a rectangular membrane mode with peak antinode displacement A, the integrals of each squared sine factor are half the corresponding side length. Integrating local kinetic energy density $\rho ({\partial }_t\psi {)}^2/2$ gives $K=\rho ab{\omega }^2{A}^2{\sin }^2(\omega t+\varphi )/8$. The total energy is $\rho ab{\omega }^2{A}^2/8$, and potential energy has the complementary cosine-squared factor. Its time-averaged kinetic energy is $\rho ab{\omega }^2{A}^2/16$. At equal peak displacement, frequency and density, such mean energies are proportional to area.