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

Let

η = \hat{k} \cdot x - c t

. Then

\nabla^{2} u = f^{''} (η), \nabla (\nabla \cdot u) = \hat{k} (\hat{k} \cdot f^{''} (η)),

and substitution gives

ρ c^{2} f^{''} = (λ + μ) \hat{k} (\hat{k} \cdot f^{''}) + μ f^{''} .

For a P-wave,

c = c_{P}

and the varying part of

f

is longitudinal:

f (η) = \hat{k} F (η), θ = F^{'} (η), ω = 0.

For an S-wave,

c = c_{S}

and

f

is transverse:

\hat{k} \cdot f = 0, θ = 0, ω = \hat{k} \times f^{'} (η) .

Solved by gpt-5.6-sol high.

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