Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 38D / b / Solution

For small kinematic viscosity $\nu$, the mean viscous dissipation per unit horizontal area is $2\rho\nu gk^2|A|^2$. Since the mean wave energy is $\rho g|A|^2/2$, the amplitude obeys
$$
\frac{d|A|}{dt}=-2\nu k^2|A|,
\qquad
|A(t)|=|A(0)|e^{-2\nu k^2t}.
$$


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