Analytic Solutions of the Helmholtz Equation for Some Polygons with 45 Degree Angles

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This Demonstration considers the Helmholtz equation in two dimensions:


within the interior of a polygonal region , subject to Dirichlet boundary conditions . This equation can be applied to the mechanical vibration of plates with clamped boundaries, in which case . Here, is an allowed eigenfrequency of vibration, with representing the speed of waves in the vibrating medium. In quantum mechanics, the Helmholtz equation in the form

represents a particle in an infinite two-dimensional potential well.

The Helmholtz equation in a unit square has the simple solutions


with .

A linear combination of degenerate solutions

, ,

has the additional property of vanishing along the diagonal . This suggests an additional solution of the Helmholtz equation for isosceles right triangles, and by stacking a set of such triangles, one can build some irregular polygons with the same solutions. We consider here the cases of a parallelogram, a trapezoid and a (nonregular) hexagon. These all contain 45° or 135° vertex angles.

For each of the four irregular polygons, you can choose the quantum numbers and to produce a contour plot of . The corresponding vibration eigenvalue is also shown.


Contributed by: S. M. Blinder (January 2021)
Open content licensed under CC BY-NC-SA




[1] H. P. W. Gottlieb, "Exact Vibration Solutions for Some Irregularly Shaped Membranes and Simply Supported Plates," Journal of Sound and Vibration, 103(3), 1985 pp. 333–339. doi:10.1016/0022-460X(85)90426-2.

[2] W.-K. Li, "A Particle in an Isosceles Right Triangle," Journal of Chemical Education, 61(12), 1984 p. 1034. doi:10.1021/ed061p1034.

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