Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 333 / 3 / e

Table of contents
- Solution e

Solution

Multiply

w_{zz} + (l^{2} - k^{2}) w = 0

by

w

and integrate from the rigid bottom to an upper endpoint

z

at which

w w_{z} = 0

. The bottom term also vanishes because

w (0) = 0

. Integration by parts gives

- \int_{0}^{z} w_{z^{'}}^{2} d z^{'} + \int_{0}^{z} (l^{2} - k^{2}) w^{2} d z^{'} = 0.

Therefore the horizontal wavenumber has the Rayleigh quotient

k^{2} = \frac{\int _{0}^{z} ( l ^{2} w ^{2} - w _{z^{'}}^{2} ) d z ^{'}}{\int _{0}^{z} w ^{2} d z ^{'}} .

For a trapped wave the upper endpoint may be taken to infinity because the eigenfunction decays.

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