The dynamic boundary condition at the interface requires the pressure to be continuous across the interface (when surface tension is neglected). The unsteady Bernoulli equations above and below the layer are $$ \frac{\partial \tilde{\phi}_1}{\partial t} +\frac{1}{2}\left|\nabla \tilde{\phi}_1\right|^2 +\frac{p_1}{\rho_1}+gz=C_1, \qquad \frac{\partial \tilde{\phi}_2}{\partial t} +\frac{1}{2}\left|\nabla \tilde{\phi}_2\right|^2 +\frac{p_2}{\rho_2}+gz=C_2 $$

So pressure matching requires $$ p_1=\rho_1\!\left(C_1-\frac{\partial \tilde{\phi}_1}{\partial t}-\frac{1}{2}\left|\nabla \tilde{\phi}_1\right|^{2}-gz\right) =\rho_2\!\left(C_2-\frac{\partial \tilde{\phi}_2}{\partial t}-\frac{1}{2}\left|\nabla \tilde{\phi}_2\right|^{2}-gz\right) =p_2 \quad \text{on } z=\zeta $$ $\rightarrow$ In the undisturbed state $(\phi_1=\phi_2=0,\ \zeta=0)$ $$ (p_1)_{\text{undisturbed}} =\rho_1\!\left(C_1-\tfrac{1}{2}U_1^{2}\right) =\rho_2\!\left(C_2-\tfrac{1}{2}U_2^{2}\right) =(p_2)_{\text{undisturbed}} $$

Subtracting the pressure expression from the undisturbed pressure state $$ \rho_1\!\left(C_1-\tfrac{1}{2}U_1^{2}\right) -\rho_1\!\left(C_1-\frac{\partial \tilde{\phi}_1}{\partial t}-\frac{1}{2}\left|\nabla \tilde{\phi}_1\right|^{2}-gz\right) = \rho_2\!\left(C_2-\tfrac{1}{2}U_2^{2}\right) -\rho_2\!\left(C_2-\frac{\partial \tilde{\phi}_2}{\partial t}-\frac{1}{2}\left|\nabla \tilde{\phi}_2\right|^{2}-gz\right) $$ $$ \xrightarrow{\text{Insert }\tilde{\phi}_1=U_1x+\phi_1,\ \tilde{\phi}_2=U_2x+\phi_2\ \text{and simplify}} \rho_1\!\left( \frac{\partial \phi_1}{\partial t} +U_1\frac{\partial \phi_1}{\partial x} +\frac{1}{2}\left|\nabla \phi_1\right|^{2}+gz \right) = \rho_2\!\left( \frac{\partial \phi_2}{\partial t} +U_2\frac{\partial \phi_2}{\partial x} +\frac{1}{2}\left|\nabla \phi_2\right|^{2}+gz \right) $$ Linearizing by dropping quadratic terms and evaluating derivatives at $z=0$ $$ \rho_1\!\left( \frac{\partial \phi_1}{\partial t} +U_1\frac{\partial \phi_1}{\partial x} +g\zeta \right) = \rho_2\!\left( \frac{\partial \phi_2}{\partial t} +U_2\frac{\partial \phi_2}{\partial x} +g\zeta \right) \quad \text{on } z=0 $$ Thus, the field equations $ \nabla^{2}\phi_{1}=0, \quad \nabla^{2}\phi_{2}=0 $ , together with the boundary conditions $ \phi_{1}\to 0 \ \text{as}\ z\to +\infty,\quad \phi_{2}\to 0 \ \text{as}\ z\to -\infty $ and the relations $ - U_{1}\frac{\partial \zeta}{\partial x} + \frac{\partial \phi_{1}}{\partial z} = \frac{\partial \zeta}{\partial t} = - U_{2}\frac{\partial \zeta}{\partial x} + \frac{\partial \phi_{2}}{\partial z} $, $ \rho_{1}\!\left( \frac{\partial \phi_{1}}{\partial t} + U_{1}\frac{\partial \phi_{1}}{\partial x} + g\zeta \right) = \rho_{2}\!\left( \frac{\partial \phi_{2}}{\partial t} + U_{2}\frac{\partial \phi_{2}}{\partial x} + g\zeta \right) $ specify the linear stability of a velocity discontinuity between uniform flows of different speeds and densities

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