A cylindrical resonant cavity has radius and height

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Special functions and orthogonal polynomials

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

Problem

A cylindrical resonant cavity has radius

a=0.10\,\mathrm{m}

and height

L=0.20\,\mathrm{m}

. A field

u

inside it obeys the Helmholtz equation

\nabla^2u+k^2u=0

and vanishes on the whole boundary: the curved wall

r=a

and the two flat ends

z=0

z=L

. ((a)) Starting from the general Laplacian in cylindrical coordinates,

\nabla^2u = \frac{1}{r}\frac{\partial}{\partial r}\!\left(r\frac{\partial u}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2u}{\partial\phi^2} + \frac{\partial^2u}{\partial z^2}

(Chapter 4), substitute

u=R(r)\Phi(\phi)Z(z)

and separate the equation into three ordinary equations, one for each factor. Give

p

q

and

w

for the radial factor's Sturm-Liouville form. ((b)) State the boundary condition each factor carries, and find the lowest value of

k^2

that these boundary conditions allow.

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