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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2026

Paper 1

Let $H=k\beta_0h$, $\beta_0=\sqrt{1-V^2/c_0^2}$, and
$\beta_1=\sqrt{V^2/c_1^2-1}$. Match <pressure> and normal <velocity> between an evanescent lower potential and an outgoing upper wave. The transmitted amplitude has
$$
|A_1|^2=\frac{V^2\epsilon^2}
{\beta_1^2\cosh^2H+(\rho_1/\rho_0)^2\beta_0^2\sinh^2H}.
$$
Therefore
$$
\langle I_y\rangle=
\frac{\rho_1k^2V^3\epsilon^2\beta_1}
{2[\beta_1^2\cosh^2H+(\rho_1/\rho_0)^2\beta_0^2\sinh^2H]}.
$$
For $H\gg1$ this is
$$
\frac{2\rho_1k^2V^3\epsilon^2\beta_1e^{-2H}}
{\beta_1^2+(\rho_1/\rho_0)^2\beta_0^2},
$$
showing tunnelling suppression. For $H\ll1$ it tends to
$\rho_1k^2V^3\epsilon^2/(2\beta_1)$, as if the upper medium met the wall.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 40C / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 40C / b / Solution

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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 40C / b / Solution

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