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For a slowly varying parallel shear flow $U(y){\mathbf{e}}_x$, the leading WKB method neglects mean-velocity gradients while retaining local advection. The continuity equation and linearized Euler equations then give the displayed acoustic dispersion relation. On the positive intrinsic-frequency branch, $\omega =U{k}_x+c_0|\mathbf{k}|$ and the group velocity is $U{\mathbf{e}}_x+c_0\mathbf{k}/|\mathbf{k}|$. A wavefront normal is parallel to $\mathbf{k}$; the acoustic energy-ray direction generally differs because of the advection term.

Write $\mathbf{X}=ϵ\mathbf{x}$, $\mathbf{k}={\nabla }_{\mathbf{X}}\theta$, and denote the slow density and velocity field amplitudes by $a_{\rho }$ and ${\mathbf{a}}_u$. Acting on the rapid phase, ${\partial }_t+\mathbf{U}\cdot \nabla$ gives $i\Omega$, where $\Omega =\omega -\mathbf{U}\cdot \mathbf{k}$. Spatial differentiation gives $-i\mathbf{k}$, while derivatives of the amplitudes and the mean shear flow are $O(ϵ)$. The leading continuity equation and linearized Euler equations therefore becomeEliminating ${\mathbf{a}}_u$ on the nonzero acoustic branch gives the acoustic eikonal equation in a shear flow:The mean-shear term is small here because $\nabla \mathbf{U}=O(ϵ)$; retaining advection does not require retaining that gradient in the leading phase equation. The positive intrinsic-frequency acoustic dispersion relation is $\omega =U{k}_x+c_0|\mathbf{k}|$, with group velocity ${\mathbf{v}}_g=U{\mathbf{e}}_x+c_0\mathbf{k}/|\mathbf{k}|$.

In a stationary wall boundary layer, horizontal homogeneity preserves $\omega$ and $k_x$ along a Hamiltonian ray-tracing equations trajectory. For a two-dimensional ray,Suppose $U$ increases from zero at the wall to a positive free-stream value. For a downstream-directed wave normal, $k_x>0$, increasing $U$ decreases $\Omega$ and $k_y$. The wave normal bends towards the wall, and the ray becomes more nearly parallel to it. If $U$ reaches $\omega /k_x-c_0$, then $k_y=0$: the ray turns and returns towards the wall rather than reaching the free stream. With launch angle ${\beta }_0$ above the wall and free-stream Mach number $M_{\infty }=U_{\infty }/c_0$, the escape condition isFor upstream-directed wave normals, $k_x<0$, $\Omega$ and $|k_y|$ increase through the layer; there is no such downstream turning barrier on this branch. A wavefront normal is parallel to $\mathbf{k}$, while the energy ray follows the group velocity: advection makes their directions different. In particular, a wall-normal wave normal with $k_x=0$ can still have a ray drifting downstream. The ordinary WKB method needs a local turning-point continuation at $k_y=0$; the ray sketch depicts that qualitative continuation.

For photon-dominated inertia, $R\gg 1$ and $c_s^2\simeq 1/3$. The baryon expansion and pressure terms are suppressed by $1/R$. Ignoring metric driving as stipulated, the common-velocity Euler equation is ${\theta }^{\prime }=-{\delta }_{\gamma }/4+{\sigma }_{\gamma }$. Differentiating the photon continuity equation givesThis limit is compatible with a matter-dominated background: cold dark matter can dominate the expansion while photons dominate the inertia of the photon-baryon subsystem.

In the first nonzero quadrupole approximation, neglect ${\sigma }_{\gamma }^{\prime }$ relative to its collision damping. The shear equation givesSubstitution yields the shear-only photon diffusion damping equationThe sign of the first-derivative term is positive, so finite photon mean free time damps acoustic energy.

Put $\omega =k/\sqrt{3}$ and $\gamma (\tau )=4k^2{\tau }_c(\tau )/15$. For constant ${\tau }_c$, the characteristic roots are $-\gamma /2\pm i\sqrt{{\omega }^2-{\gamma }^2/4}$. The frequency shift begins at second order in the mean free time. To the requested order the general solution iswhere the error describes the frequency expansion at fixed $k$ and time interval. Keeping the exponential retains the accumulated damping rather than expanding away a possibly important effect.

For a general time-dependent coefficient, a first-order solution with prescribed initial conditions can also be written aswhere ${\delta }_0=A\cos [\omega (\tau -{\tau }_i)]+B\sin [\omega (\tau -{\tau }_i)]$. Substitution verifies that the integral solves the oscillator forced by $-\gamma {\delta }_0^{\prime }$, and its value and first derivative vanish initially. This gives the general first-order result without assuming arbitrary time dependence may be absorbed exactly into an exponential.

To obtain the slowly varying subhorizon form, set ${\delta }_{\gamma }=e^{-G}y$ with $G^{\prime }=\gamma /2$. The exact transformed equation isDropping the quadratic term alone leaves ${\omega }^2-{\gamma }^{\prime }/2$, so slow variation is also needed. When $k\gg \mathcal{H}$, $k{\tau }_c\ll 1$, and the mean free time changes on an expansion timescale, $|{\gamma }^{\prime }|/{\omega }^2\ll 1$. The WKB method solution has local frequency $\Omega =({\omega }^2-{\gamma }^{\prime }/2{)}^{1/2}$ and prefactor ${\Omega }^{-1/2}$. At leading acoustic order both reduce to constants; the small derivative corrections to phase and prefactor can be neglected. This slowly varying shear-damped photon oscillator therefore givesA change in the phase origin is absorbed into $A,B$. At zero mean free time the damping vanishes; as time passes the diffusion length grows and $k_D$ decreases. The coefficient is that of the question's isotropic-scattering, shear-only truncation, not the polarization-inclusive full Silk damping formula.

If ionization remains approximately constant during matter domination, $n_e\propto a^{-3}$, so ${\tau }_c\propto a^2\propto {\tau }^4$. Away from the lower limit this givesNear recombination the rapidly changing ionization fraction and eventual failure of tight coupling require the full hierarchy; the integral formula specifies the damping accumulated during the regime used here.

Assume the medium varies on scales much longer than the wavelength and period, so the leading-order WKB method gives the local eikonal equation. For the phase $\theta$ in the stated ansatz, defineThe local dispersion relation becomes the Hamilton--Jacobi equationDifferentiating it with respect to $x_i$, using equality of mixed derivatives, and choosing a ray whose velocity isgivesHere the dot, or $d/dt$, is the total derivative along a ray,and not merely a partial time derivative at fixed position.

Finally, differentiate $\omega =\Omega (\mathbf{k};\mathbf{x},t)$ along the same trajectory:because the last two terms cancel. The Hamiltonian ray-tracing equations are therefore

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.