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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2025

Paper 2

Let $\eta=\hat{\mathbf k}\cdot\mathbf x-ct$. Then
$$
\nabla^2\mathbf u=\mathbf f''(\eta),
\qquad
\nabla(\nabla\cdot\mathbf u)
=\hat{\mathbf k}(\hat{\mathbf k}\cdot\mathbf f''(\eta)),
$$
and substitution gives
$$
\rho c^2\mathbf f''
=(\lambda+\mu)\hat{\mathbf k}
(\hat{\mathbf k}\cdot\mathbf f'')+\mu\mathbf f''.
$$

For a P-wave, $c=c_P$ and the varying part of $\mathbf f$ is longitudinal:
$$
\mathbf f(\eta)=\hat{\mathbf k}F(\eta),
\qquad
\theta=F'(\eta),
\qquad
\boldsymbol\omega=0.
$$
For an S-wave, $c=c_S$ and $\mathbf f$ is transverse:
$$
\boxed{\hat{\mathbf k}\cdot\mathbf f=0,
\qquad
\theta=0,
\qquad
\boldsymbol\omega=\hat{\mathbf k}\times\mathbf f'(\eta).}
$$


Solution

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

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

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

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