Circular mountain-wave ray in linear shear ($(x-x_t{)}^2+(z+H{)}^2=(H/q_0{)}^2$)

For a stationary internal gravity wave in constant buoyancy frequency $N$ and wind $U=U_0(1+z/H)$, set $q_0=k{U}_0/N<1$. Its upward group velocity has ratio $dx/dz=k/m=q/\sqrt{1-q^2}$, where $q=q_0(1+z/H)$. Integrating from the origin gives a circular ray with center $(H\sqrt{1-q_0^2}/q_0,-H)$ and radius $H/q_0$. The ascending ray becomes horizontal at the turning level of a stationary internal wave; its reflected branch continues downwind along the same circle.