Internal-wave ray in uniform vertical shear ($m(t)=m_0-kst,z(t)=(\omega -\hat{\omega }(t))/(ks)$)

For $U=sz$, $N>0$, $k>0$ and the rising branch $m_0<0$, the Hamiltonian is $\omega =ksz+Nk/\sqrt{k^2+m^2}$. It gives constant $\omega ,k$ and $m=m_0-kst$. The height approaches $z_c=\omega /(ks)$ as time tends to infinity. With $\theta =\arctan (|m|/k)$, the ray slope is $dx/dz=\tan \theta +ksz/(N\sin \theta {\cos }^2\theta )$. The intrinsic and absolute frequencies must not be confused.

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.