Exact (nonlinear) governing equations for surface gravity waves

assuming potential theory

Unknown variables:

$$\begin{align}& \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpo... ...t( {x,y,z,t}\right) \notag \\ & p\left( {x,y,z,t} \right) \notag \end{align}$$

Field equation:

$$\begin{align}\mbox{Continuity:} & \nabla ^2\phi = 0 \ \ \ y < \eta \mbox{\ or\ }... ...pheric}}}-\underbrace{\rho gy}_{\mbox{\tiny {hydrostatic}}} \notag \end{align}$$

Boundary Conditions:

1.

On an impervious boundary $B\left( {x,y,z,t} \right) = 0$, we have KBC:

$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightha... ...arpoonup$ }}\over {x}} ,t} \right) = U_n \mbox{\ on\ } B = 0$$

Alternatively: a particle P on $B$ remains on $B$, i.e. $B$ is a material surface; e.g. if P is on $B$ at $t = t_{0}$, i.e.

$$B(\vec{x}_{P},t_{0}) = 0, \mbox{\ then\ } B(\vec{x}_{P}(t),t_{0}) = 0 \mbox{\ for all \ } t,$$

so that, following P, $B = 0$.

$$% latex2html id marker 897 \therefore \frac{DB}{Dt} = \frac{\... ... {\nabla \phi \cdot \nabla } \right)B = 0 \mbox{\ on\ } B = 0$$

For example, flat bottom at $y = -h$:

$${\partial \phi } \mathord{\left/ {\vphantom {{\partial \phi }... ...artial y} = 0 \mbox{\ on\ } y = -h \mbox{\ or\ } B:y + h = 0$$

2.

On the free surface, $y = \eta$ or $F = y - \eta (x,z,t) = 0$.
KBC: free surface is a material surface, no perpendicular relative velocity to free surface, particle on free surface remains on free surface:

$$\frac{DF}{Dt} = 0 = \frac{D}{Dt}\left( {y - \eta } \right) = ... ...}\mbox{\tiny {still}} \\ \mbox{\tiny {unknown}}\end{array}}$$

DBC: $p = p_{a}$ on $y = \eta$ or $F = 0$. Apply Bernoulli equation at $y = \eta$:

$$\frac{\partial \phi }{\partial t} + \frac{1}{2}\left\vert {\n... ...eta = 0 \mbox{\ on\ } y = \eta \mbox{\ (assuming} p_{a} = 0)$$