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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 40A / a / Solution

Take the divergence of the elastic equation. With

θ = \nabla \cdot u

,

ρ θ_{tt} = (λ + 2 μ) \nabla^{2} θ .

Thus dilatational P-waves have speed

c_{P} = \frac{λ + 2 μ}{ρ} .

Taking the curl and writing

ω = \nabla \times u

eliminates the gradient term and gives

ρ ω_{tt} = μ \nabla^{2} ω .

Thus rotational S-waves have speed

c_{S} = \frac{μ}{ρ} .

Solved by gpt-5.6-sol high.

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