A circular drumhead of radius

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Special functions and orthogonal polynomials · Function spaces and operator theory

Conditions of Validity, Ch Eigenfunction expansions, convergence and the Rayleigh quotient

Problem

R=1.00\,\mathrm{m}

, clamped at its rim, is released from the shape

f(r)=y_0\big[1-(r/R)^2\big]

with

y_0=1.00\,\mathrm{mm}

. Its radial modes are the eigenfunctions of

-\tfrac1r(ry')'=\lambda y

[0,R]

with

y(R)=0

, regularity at

r=0

, and weight

w(r)=r

: these are

J_0(j_{0n}r/R)

, where

j_{0n}

is the

n

th positive zero of the Bessel function

J_0

. The displacement is expanded as

f(r)=\sum_{n=1}^\infty c_nJ_0(j_{0n}r/R)

with

c_n=\langle J_0(j_{0n}\cdot/R),f\rangle_w/\|J_0(j_{0n}\cdot/R)\|_w^2

. The needed weighted integrals have been evaluated. For

n=1,2,3

j_{0n}=2.4048,\ 5.5201,\ 8.6537

;

\int_0^R rf(r)J_0(j_{0n}r/R)\,\mathrm{d}r=0.14931,\ -0.008092,\ 0.0016755\ \mathrm{mm\,m^2}

; and

\|J_0(j_{0n}\cdot/R)\|_w^2=0.13476,\ 0.05789,\ 0.036843\ \mathrm{m^2}

. ((a)) Compute

c_1

c_2

c_3

(in mm). ((b)) Use Parseval's identity to find the percentage of

\int_0^R rf(r)^2\,\mathrm{d}r

carried by the first term of the expansion alone.

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