A square drumhead of side has one corner rounded off by removing a quarter-disc of…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Function spaces and operator theory

Conditions of Validity, Ch Separation of variables as a chain of Sturm-Liouville problems

Problem

A square drumhead of side

L = 0.30\,\mathrm{m}

has one corner rounded off by removing a quarter-disc of radius

0.10\,\mathrm{m}

centred at that corner, and is clamped along its whole remaining boundary. Its wave speed is

c = 180\,\mathrm{m\,s^{-1}}

. The largest square that fits inside the notched drum, clear of the missing corner, has side

0.229\,\mathrm{m}

. ((a)) For the full, un-notched

0.30\,\mathrm{m}

square, separating the clamped Helmholtz problem in Cartesian coordinates gives, for the

x

-factor,

-X'' = k_x^2 X

with

X(0)=X(L)=0

. Identify

p

q

w

for this Sturm-Liouville problem and confirm with the vanishing-boundary-term criterion that the clamped condition is self-adjoint. ((b)) Using the domain monotonicity of the Dirichlet Rayleigh quotient, bracket the notched drum's fundamental frequency between the fundamentals of the full square and the inscribed square.

Browse the problem library