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
$$
\boxed{\frac{\rho_+}{q_+}(v^2-v_{A+}^2)
+\frac{\rho_-}{q_-}(v^2-v_{A-}^2)=0}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 314 / 3 / e / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-314/3/e/ii/solution