 Solution

Applying ${\mathcal{D}}_0$ to the shearing-wave ansatz produces an extra phase termThe perturbation equations have spatially uniform amplitude coefficients only when this term vanishes. HenceFor $k_y\ne 0$, the radial wavenumber changes linearly. A leading disturbance with $k_x/k_y<0$ first opens until $k_x=0$, then becomes an increasingly tightly wound trailing disturbance with $k_x/k_y>0$. This is the geometric swing of a shearing wave.