No normal flow at a rigid wall gives $\boxed{\phi(\pm1)=0}$. At a free surface $y=1+\eta$, the linearized <kinematic boundary condition> is
$$
(\partial_t+U\partial_x)\eta=v=-\partial_x\psi,
$$
so $(U-c)\eta=-\phi$. The linearized tangential momentum equation gives the pressure amplitude $p=(U-c)\phi'-U'\phi$. Constant surface pressure therefore requires
$$
\boxed{(U-c)\phi'-U'\phi=0}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 331 / 2 / i / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-331/2/i/b/solution