Small-amplitude two-dimensional waves disturb the free surface of an incompressible, irrotational fluid with a pressure p(x, y, t) and a velocity potential $φ(x, y, t)$ which satisfies Laplace’s equation. The free surface, given by $y = η(x,t)$, is at constant atmospheric pressure pa and the fluid is of infinite depth. In terms of $φ$ and $η$, what is the kinematic boundary condition at the free surface? Starting from the Euler equations, show that φ may be chosen such that $$\frac{∂φ}{∂t}+\frac{1}{2}|∇φ|^2+gy+\frac{p−p_a}{\rho} =0$$ You are given that the jump in pressure across the free surface is given by $$p−p_a=-T\frac{∂^2η}{∂x^2}/{(1+(\frac{∂η}{∂x})^2)}^\frac{3}{2} $$ on y = η, where T is the surface tension. Show that, when the problem is linearised by neglecting quadratic terms, the boundary conditions are simplified to $$\frac{∂φ}{∂y}=\frac{∂η}{∂t}, \frac{∂φ}{∂t} + gη − T \frac{∂^2η}{∂x^2} =0$$ on $y = 0$.
I understand the first boundary condition but how to do the second one, obviously using the free surface pressure jump.
The kinematic boundary condition simply states that the substantial derivative of the function $f = y - \eta(x,t)$ is zero (that is, the interface is a fluid surface). Therefore, it is required:
$$ Df/Dt = -\eta_t - \varphi_x \eta_x + \varphi_y = 0 $$
at the free surface.