Linerization of Free-surface Conditions
On earth the gravitational acceleration is large enough that the restoring role it plays leads to small wave slopes in most, but not all, cases.
So it is often a very good assumption to set
[math]\displaystyle{ | \nabla \zeta | = O (\epsilon) }[/math]
Let:
[math]\displaystyle{ \epsilon = \epsilon_1 + \epsilon_2 + \epsilon_3 + \cdots }[/math]
[math]\displaystyle{ \Phi = \Phi_1 + \Phi_2 + \Phi_3 + \cdots }[/math]
And derive boundary value problems for [math]\displaystyle{ \epsilon_i,\Phi_i }[/math]. Rarely we need to go beyond [math]\displaystyle{ i = 3 }[/math].
Here we will derive the free-surface conditions up to second order.
Kinematic condition
[math]\displaystyle{ \left ( \frac{\partial \zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right )_{Z=\zeta} = \left ( \frac{\partial \Phi}{\partial Z} \right )_{Z=\zeta} }[/math]
[math]\displaystyle{ \left( \frac{\partial\zeta}{\partial t} + \nabla \Phi \cdots \nabla \zeta \right)_{Z=0} + \zeta \frac{\partial}{\partial Z} \left( \frac{\partial\zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right)_{Z=0} + \cdots }[/math]
[math]\displaystyle{ = \left( \frac{\partial\Phi}{\partial Z} \right)_{Z=0} + \zeta \left( \frac{\partial^2 \Phi}{\partial Z^2} \right)_{Z=0} + \cdots }[/math]
