Causal envelope of stationary internal waves ($(x-Ut/2{)}^2+z^2=(Ut/2{)}^2$)

In uniform positive flow $U$, upward stationary internal gravity waves have laboratory group velocity $U({\cos }^2\theta ,\sin \theta \cos \theta )$. Packets emitted by a localized source at startup lie on $(x-Ut/2{)}^2+z^2=(Ut/2{)}^2$, with later emissions inside. This envelope belongs to the permanent stationary component, not every possible startup transient.

A lower ray reaches $H$ after ${\tau }_1=H/[U_1\sin {\theta }_1\cos {\theta }_1]$, at $x_H=H\cot {\theta }_1$. Conserved horizontal wavenumber gives $\cos {\theta }_2=(U_2{N}_1/U_1{N}_2)\cos {\theta }_1$. The transmitted front is $(x_H,H)+(t-{\tau }_1){\mathbf{c}}_{g2}$ on propagating branches; reflection reverses the vertical group velocity.

A broad-spectrum stationary source below height $H$ first sends an upward packet there after $2H/U$, at $x=H$. This uses the maximum vertical group velocity $U/2$ at $\theta =\pi /4$. A source lacking the corresponding wavenumber has a later actual arrival.