= Energy in a rectangular membrane mode
{title2=$\overline K=\rho ab\omega^2A^2/16$}
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 $K=\rho ab\omega^2A^2\sin^2(\omega t+\phi)/8$. The total <energy> is $\rho ab\omega^2A^2/8$, and <potential energy> has the complementary cosine-squared factor. Its time-averaged <kinetic energy> is $\rho ab\omega^2A^2/16$. At equal peak displacement, <frequency> and <density>, such mean energies are proportional to area.
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