Vibrational modes of membranes and plates of a variety of shapes

Abstract

The linearity of the equations of motion for small amplitude vibrations of plates and membranes allows new solutions of the equations to be constructed from known solutions by superposition. In particular, for an infinite membrane or plate, a new solution can be formed by taking a known solution, displacing it, and then adding it to the undisplaced solution. For example, since y1 = AJ0(k|r⃗|)e−iωt is a solution for an infinite membrane, so is y2 = A[J0(k|r⃗|) + J0(k|r⃗ + a⃗|)]e−iωt for any a⃗ in the plane of equilibrium. The membrane can then be considered clamped along any closed nodal line and the portion outside the line disregarded. The solutions constructed from “displaced” standard solutions are useful in understanding the modes of vibration of membranes and plates of a variety of shapes. Hopefully, they can provide a basis for perturbation calculations. Constructions have yielded enticing suggestions of relations between mathematical functions. Most work to date has been with Bessel function solutions, but other solutions, such as those using the cylindrical functions, can also be combined.