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Let $z_r$ be either endpoint at which $Z_r$ is prescribed, and put $d_z=z-z_r$, $q=\mu p_{\beta }$ and $\kappa =\omega p_{\beta }$. Integrating the constant-coefficient velocity–traction system givesTaking the ratio $-T/V$ yields the uniform-layer SH impedance update and its accompanying velocity:The initial value $V_r$ sets an arbitrary normalization; specifying impedance alone cannot specify the overall wave wave amplitude. Positive $d_z$ propagates data from $z_0$, while negative $d_z$ propagates a specified load from $z_1$, so the same expressions solve both requested initial-value problems.

Expanding sine and cosine givesAbsorbing the reference-point phases into the two constants gives the requested forward/backward exponential superposition. Equivalently, wherever the ratios are defined,The exceptional values $Z_r=q,-q$ give the pure directional solutions. If a denominator in the impedance formula vanishes, propagate the finite pair $(V,T)$ through that point; this is an impedance-chart pole, not a failure of the physical solution. The upper-half-plane assumptions ensure $q\ne 0$.

Set the lower half-space load to $q_b={\mu }_b{p}_{\beta ,b}$ using the causal vertical wavenumber branch, which excludes a wave arriving from depth infinity. Apply the uniform-layer SH impedance update upwards through every layer, keeping particle velocity and traction continuous. For upper half-space impedance $q_a$, incident velocity $V_0$ and reflected velocity $R{V}_0$, the top values are $V=V_0(1+R)$ and $T=-q_a{V}_0(1-R)$. Their ratio gives the displayed reflection coefficient, incorporating every underlying interface and multiple reflection.