Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 2 / c

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- Solution c

Solution

For zero forcing, the given equation for

v_{y}

is

(\frac{D ^{2}}{D t ^{2}} + κ_{r}^{2} - v_{s}^{2} \nabla^{2}) v_{y} = 0.

Use the shearing-wave ansatz

v_{y} = ℜ {v (t) e^{i [k_{x} (t) x + k_{y} y]}} .

The explicit

x

dependence cancels from the material derivative when

\dot{k}_{x} = S k_{y}, k_{x} (t) = k_{x} (0) + S k_{y} t .

The amplitude then obeys the time-dependent harmonic oscillator equation

\ddot{v} + g (t) v = 0, g (t) = κ_{r}^{2} + v_{s}^{2} [k_{x} (t)^{2} + k_{y}^{2}] .

This evolving wavevector is the characteristic signature of a shearing wave.

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