 Energy in a rectangular membrane mode

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.