Further properties of the nonseparable solutions of the Helmholtz wave equation

Nonseparable solutions ${W^{\left ( n \right )}}$ of $\left ( {{\nabla ^2} + {k^2}} \right ){W^{\left ( n \right )}} = 0$ are linearly independent, but inter-related through a generative differential operator. The nonseparable of order $n = 0$ is the familiar separable solution. In two cartesian coordinates, a sum of zero and second order solutions can describe transverse motion of a membrane of unique boundary contour. In three coordinates the same sum can describe acoustic pressure in a uniquely shaped cavity with pressure-release walls.