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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2025

Paper 1

Choose membrane displacement $\eta$ positive into the fluid. The linear <boundary conditions> at $z=0$ are
$$
\phi_z=\eta_t,
\qquad
p'=-\mu\eta.
$$
A wave localized near the membrane has complex form
$$
\phi=\operatorname{Re}\left\{\Phi e^{ik(x-ct)}e^{-\alpha z}\right\}.
$$
The wave equation requires
$$
\alpha=k\sqrt{1-\frac{c^2}{c_0^2}},
$$
so localization requires $0<c<c_0$. If $\eta=\operatorname{Re}\{H e^{ik(x-ct)}\}$, the kinematic condition and $p'=-\rho_0\phi_t$ give
$$
-\alpha\Phi=-ikcH,
\qquad
i\rho_0kc\Phi=-\mu H.
$$
Eliminating $H$ yields
$$
\rho_0k^2c^2=\mu\alpha.
$$
On squaring and writing $s=c/c_0$, we obtain
$$
\boxed{As^4+s^2-1=0,
\qquad
A=\left(\frac{\rho_0kc_0^2}{\mu}\right)^2.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 40A / c / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 40A / c / i / Solution

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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 40A / c / i / Solution

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