A drum builder wants a circular drumhead of radius to sound its fundamental note at

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

A drum builder wants a circular drumhead of radius

a = 0.25\,\mathrm{m}

to sound its fundamental note at

f_{01} = 150\,\mathrm{Hz}

. The membrane is uniform, has no bending stiffness, and its rim is clamped rigidly, so the bounded radial mode shapes are the Bessel functions

J_m(k\rho)

and the rim condition

J_m(ka)=0

fixes

k

at the tabulated zeros

\alpha_{01}=2.405

\alpha_{11}=3.832

\alpha_{21}=5.136

. The mode frequencies are

f_{mn}=\alpha_{mn}c/(2\pi a)

, where

c

is the transverse wave speed set by the head's tension. ((a)) Find the wave speed

c

needed to give the stated fundamental frequency. ((b)) Predict the frequencies of the

(1,1)

and

(2,1)

modes. ((c)) A builder who mistakenly assumes the head behaves like a string predicts overtones at

2f_{01}

and

3f_{01}

. By what percentage do the true

(1,1)

and

(2,1)

frequencies found in (b) differ from these string-overtone guesses? ((d)) State, in one sentence, why this ideal membrane's pitch is harder for the ear to pin down precisely than a plucked string's, using the mode frequencies found.

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