Someone checks the orthogonality of two axisymmetric radial modes of the clamped drum

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

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

Problem

Someone checks the orthogonality of two axisymmetric radial modes of the clamped drum,

R_1(r)=J_0(j_{0,1}r/a)

and

R_3(r)=J_0(j_{0,3}r/a)

, where

j_{0,1}=2.405

and

j_{0,3}=8.654

are zeros of

J_0

(Chapter 12). Reasoning that “both solve the same radial equation, so their product must integrate to zero”, then computes

\int_0^aR_1(r)\,R_3(r)\,\mathrm dr

and finds it numerically equal to

0.1168\,a

, not zero, and concludes that Chapter 13's orthogonality theorem fails for these two modes. Identify the error in the reasoning, trace the correct weight to the separated Sturm-Liouville problem built earlier in this chapter, and show that the correctly weighted integral does vanish.

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