Normal-mode form $$ \phi_1(x,z,t)=A_1(z)\,e^{\,ik(x-ct)}, \qquad \phi_2(x,z,t)=A_2(z)\,e^{\,ik(x-ct)} $$ Insert into Laplace equations $$ -\,k^{2}A_1+\frac{d^{2}A_1}{dz^{2}}=0, \qquad -\,k^{2}A_2+\frac{d^{2}A_2}{dz^{2}}=0 $$ Solve for vertical structure $$ A_{1,2}(z)=A_{\pm}\,e^{\pm kz} $$ Apply boundary conditions $$ \phi_1=A_{-}\,e^{\,ik(x-ct)-kz}\quad(z > 0), \qquad \phi_2=A_{+}\,e^{\,ik(x-ct)+kz}\quad(z < 0) $$ Interface shape (matching form) $$ \zeta=\zeta_0\,e^{\,ik(x-ct)} $$ Insert $\phi_1,\phi_2,\zeta$ into the kinematic and dynamic conditions $$ -\,iU_1k\zeta_0-kA_{-}=-ikc\zeta_0 = -\,iU_2k\zeta_0+kA_{+}, $$ $$ \rho_1\!\left(-ikcA_{-}+ikU_1A_{-}+g\zeta_0\right) =\rho_2\!\left(-ikcA_{+}+ikU_2A_{+}+g\zeta_0\right) $$ Solve the kinematic relation for $A_{\pm}$ in terms of $\zeta_0$ $$ kA_{-}=-(ikU_1-ikc)\,\zeta_0, \qquad kA_{+}=(ikU_2-ikc)\,\zeta_0 $$