Wave Equation--Triangle

The equation of motion for a membrane shaped as a right isosceles triangle of length  on a side and with the sides oriented along the positive  and  axes is given by

where

and ,  integers with . This solution can be obtained by subtracting two wave solutions for a square membrane with the indices reversed. Since points on the diagonal which are equidistant from the center must have the same wave equation solution (by symmetry), this procedure gives a wavefunction which will vanish along the diagonal as long as  and  are both even or odd. We must further restrict the modes since those with  give wavefunctions which are just the negative of  and  give an identically zero wavefunction.

The plots above show the lowest order spatial modes.