Two membranes

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Special functions and orthogonal polynomials

Mathematical Methods for Physics, Ch Separation of variables in Cartesian, cylindrical and spherical geometry

Problem

Two membranes, each with wave speed

c=120\,\mathrm{m\,s^{-1}}

, are clamped along every edge. The first is a quarter of a circular disc,

0\le\rho\le a

0\le\phi\le\pi/2

, with

a=0.20\,\mathrm{m}

, clamped along its curved rim and along both straight edges. The second is a square of side

0.40\,\mathrm{m}

with a circular hole of radius

0.10\,\mathrm{m}

cut from its centre, clamped along its four outer edges and around the rim of the hole. ((a)) For each membrane, decide whether separation of variables applies, and if so in which coordinates, naming the feature of the boundary that decides it. ((b)) For the membrane that separates, find the angular factors

H(\phi)

its straight edges allow, and hence the allowed orders

m

of Bessel's equation. ((c)) Find its lowest frequency, and compare it with the fundamental of a full circular drum of the same radius and wave speed.

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