Solution
= Solution
Write $k_{z+}=+iq_+$ above the interface and $k_{z-}=-iq_-$ below it, with $q_\pm=|k_{z\pm}|>0$, so both waves decay away from $z=0$. A common interface displacement $\xi_0$ gives
$$
\delta\Pi_\pm
=-i\rho_\pm\frac{k_x^2}{k_{z\pm}}
(v^2-v_{A\pm}^2)\xi_0.
$$
Pressure continuity therefore requires
$$
\boxed{\frac{\rho_+}{q_+}(v^2-v_{A+}^2)
+\frac{\rho_-}{q_-}(v^2-v_{A-}^2)=0}.
$$