Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 2 / 16C / b

Solution

For an axisymmetric separated amplitude

ϕ (r, z) = R (r) Z (z)

\frac{1}{R} \frac{1}{r} (r R^{'})^{'} = - \frac{Z ^{''}}{Z} = - k^{2} .

The radial equation is

R^{''} + \frac{1}{r} R^{'} + k^{2} R = 0.

Its solution regular at the axis is the Bessel function

J_{0} (k r)

. The vertical equation is

Z^{''} - k^{2} Z = 0

, and the bottom condition

Z^{'} (- h) = 0

selects

Z (z) = cosh (k (z + h))

up to an irrelevant constant multiplier. Finally, the sidewall condition gives

0 = R^{'} (R) = k J_{0}^{'} (k R) .

Therefore the allowed positive wavenumbers are

k_{n} = \frac{x _{n}}{R}, J_{0}^{'} (x_{n}) = 0,

and

ϕ (r, z) = A J_{0} (k_{n} r) cosh (k_{n} (z + h)) .

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