= Ray turning in a wall shear layer
{title2=$k_y^2=(\omega-Uk_x)^2/c_0^2-k_x^2$}
In a stationary, horizontally homogeneous wall <boundary layer>, <Hamiltonian ray-tracing equations> conserve $\omega$ and $k_x$. If $U$ increases away from the wall, downstream-oriented $k_x>0$ waves have decreasing positive intrinsic <frequency> and decreasing $k_y^2$. At $U=\omega/k_x-c_0$, the transverse <wavenumber> vanishes and the acoustic ray turns. For $U(0)=0$, a downstream launch angle $\beta_0$ above the wall can reach a uniform stream of <Mach number> $M_\infty$ only if $\cos\beta_0\le(1+M_\infty)^{-1}$. This uses the positive acoustic branch, constant <sound speed> and no intervening critical layer; a turning point itself requires a uniform local approximation rather than the ordinary <WKB method>.
Back to article page
