A "racetrack" drumhead is a membrane shaped like a rectangle with a semicircular cap…

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 "racetrack" drumhead is a membrane shaped like a

0.30\,\mathrm{m}\times0.20\,\mathrm{m}

rectangle with a semicircular cap of radius

0.10\,\mathrm{m}

(matching the rectangle's width) attached to each of the two short ends, clamped along its whole boundary. Its wave speed is

c = 200\,\mathrm{m\,s^{-1}}

. ((a)) The racetrack lies entirely inside a disc of radius

0.25\,\mathrm{m}

centred on the racetrack's centre. For that disc's clamped

m=0

radial problem, identify

p(r)

q(r)

and

w(r)

, and use the vanishing-boundary-term criterion for a self-adjoint Sturm-Liouville problem to confirm that pairing regularity at

r=0

with the clamped condition at the rim is self-adjoint. ((b)) The

0.30\,\mathrm{m}\times0.20\,\mathrm{m}

rectangle fits inside the racetrack, its straight sides coinciding with part of the racetrack's boundary. Using the domain monotonicity of the Dirichlet Rayleigh quotient, bracket the racetrack's fundamental frequency between the fundamentals of the enclosing disc and the inscribed rectangle.

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