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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 40D / c / i / Solution

Seek an SH displacement

u = (0, Y (z), 0) e^{i (κ x - ω t)} .

The S-wave equation gives

Y^{''} + q^{2} Y = 0, q^{2} = \frac{ω ^{2}}{c _{s}^{2}} - κ^{2} .

Rigidity at

z = 0

gives

Y (0) = 0

, while zero tangential traction at the free surface gives

μ Y^{'} (h) = 0

. Thus

Y (z) = C sin (q z), cos (q h) = 0,

so

q = q_{n} = \frac{( 2 n + 1 ) π}{2 h}, ω^{2} = c_{s}^{2} (κ^{2} + q_{n}^{2}) .

These are the guided SH modes between rigid and free planes.

Solved by gpt-5.6-sol high.

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