Problem is 3-D, write down Laplacian in cylindrical coordinates, separate out Z(z) first, introducing one constant, k.
Separate R(r) and Theta(theta) next. Single valuedness of Theta requires the new separation constant to be an integer, m.
Radial equation is Bessel's equation (after some simple manipulation)
Theta independence of b'dry conditions forces m=0, radial b'dry condition, u(r=1)=0, forces k to be a zero of the zero'th Bessel function.
B'dry condition at base requires a "Fourier-Bessel" series, which we can solve numerically or analytically
Brief discussion of analogy to circular membrane/drum, and why the higher "harmonics" of the drum aren't really harmonics, i.e. they are not just integer multiples of the lowest frequency.
I will be teaching Physics 2170 (an introduction to modern physics) next semester (spring, '96) You can always find information on whatever class I'm currently teaching here.
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