Mathematical Methods in the Physical Sciences · Problem 9.27

This problem asks you to find the natural frequencies and mode shapes of a vibrating rectangular membrane—a 2D extension of the 1D string problem. The key insight is that the wave equation separates into independent standing-wave patterns in each direction, so you can build the full solution by combining $\sin$ modes in $x$ and $y$ separately, each with its own integer quantum number. The frequency depends on how many half-wavelengths fit along each dimension.

This problem asks you to find the natural frequencies and mode shapes of a vibrating rectangular membrane—a 2D extension of the 1D string problem. The key insight is that the wave equation separates into independent standing-wave patterns in each direction, so you can build the full solution by combining $\sin$ modes in $x$ and $y$ separately, each with its own integer quantum number. The frequency depends on how many half-wavelengths fit along each dimension.