Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 74 / 3 / i / Solution

Let $K^2=|\mathbf{k}{|}^2=k_x^2+k_z^2+{\Omega }^2{t}^2{k}_x^2$. For the complex wave before taking its real part,since ${\dot{k}}_y=-\Omega k_x$. The asserted velocity equation therefore follows directly from $\dot{\mathbf{v}}+\Omega v_y\hat{\mathbf{x}}=-i\mathbf{k}q$. To verify the pressure amplitude, solenoidality is $\mathbf{k}\cdot \mathbf{v}=0$. Its derivative requiresand hence $q=2i\Omega k_x{v}_y/K^2$, as claimed.

The transverse amplitude equations areBoth coefficients are real. Differentiating $v_y{v}_z^{\ast }-v_z{v}_y^{\ast }$ makes the terms proportional to $b|v_y{|}^2$ cancel, while the remaining terms give $a(v_y{v}_z^{\ast }-v_z{v}_y^{\ast })$. Therefore the helicity invariant of an inviscid shearing wave obeysHere $H$ is real because the difference in its definition is purely imaginary. Its sign is preserved, and shear reduces its magnitude away from $t=0$.