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

Taking the divergence of the elastic equation gives

ρ ϑ_{tt} = (λ + 2 μ) \nabla^{2} ϑ .

Taking its curl gives

ρ Ω_{tt} = μ \nabla^{2} Ω.

Thus the Helmholtz separation of elastic waves produces longitudinal P-waves and transverse S-waves with

c_{p} = \frac{λ + 2 μ}{ρ}, c_{s} = \frac{μ}{ρ} .

Solved by gpt-5.6-sol high.

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