An external gravity wave displaces a fluid's <free surface> and moves nearly uniformly through the depth in the long-wave limit. In a layer of depth $H$, its leading <shallow-water approximation> speed is $\sqrt{gH}$.

External gravity wave

Take $x$ down the plane and $y$ normally away from it, with the solid at $y=0$ and the <free surface> at $y=h$. A steady unidirectional <falling film flow> has velocity $\mathbf u=u(y)\mathbf e_x$. The streamwise <Navier-Stokes equation> reduces to
$$
\nu u''=-g\sin\alpha,
$$
where $\nu$ is the <kinematic viscosity>. The lower <no-slip boundary condition> and upper <stress-free boundary condition> are
$$
u(0)=0,
\qquad
u'(h)=0.
$$
Integrating gives the velocity profile
$$
\boxed{u(y)=\frac{g\sin\alpha}{\nu}\left(hy-\frac{y^2}{2}\right)}.
$$
The <volume flux> per unit width is
$$
\boxed{Q=\int_0^hu(y)\,dy=\frac{g h^3\sin\alpha}{3\nu}}.
$$

If the upper surface is replaced by a stationary solid plane, the second boundary condition becomes $u(h)=0$. The resulting plane-channel profile and flux are
$$
u(y)=\frac{g\sin\alpha}{2\nu}y(h-y),
\qquad
\boxed{Q_{\rm closed}=\frac{g h^3\sin\alpha}{12\nu}=\frac14Q}.
$$
The free surface exerts no tangential <shear stress>, whereas the stationary upper wall enforces no slip and exerts a retarding shear. This additional drag accounts for the reduced flux.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 1 / 5C / Solution

At the displaced <free surface>, constant atmospheric pressure and the <hydrostatic approximation> give the dynamic condition
$$
\eta=\frac{f_0}{g}\psi\qquad\text{at }z=0.
$$
The linearized kinematic condition is $w=D_g\eta/Dt$. Combining these relations with
$$
w=-\frac{D_g}{Dt}\left(\frac{f_0\psi_z}{N^2}\right)
$$
gives
$$
\boxed{
\frac{D_g\psi_z}{Dt}+\frac{N^2}{g}\frac{D_g\psi}{Dt}=0
}\qquad\text{at }z=0.
$$

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 333 / 3 / i / Solution

{title2=$z=\eta$}
{wiki}

A free surface is a fluid boundary whose position is part of the solution and whose traction is prescribed by the adjoining medium. Its motion obeys a <kinematic boundary condition>, while its pressure or stress supplies a dynamic boundary condition.

Codex @codex  0

 Incoming links: Free surface