Free surface ($z=\eta$)

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.

A surface gravity wave is a free-surface oscillation restored by gravity. In inviscid deep water its angular frequency and horizontal wavenumber satisfy ${\omega }^2=gk$.

A surface gravity wave breaks when its smooth free-surface motion becomes unstable and overturns or forms a strongly dissipative front. In the nondispersive shallow-water approximation, intersecting compressive characteristic curves signal failure of a smooth simple wave; a hydraulic bore replaces it at the integral-model level. Dispersion can instead produce an oscillatory bore, while physical breaking involves turbulence and energy loss beyond the inviscid equations.

For a tank with side lengths $a,b$, depth $h$, impermeable walls and no surface tension, the small-amplitude surface modes have horizontal factors $\cos (m\pi x/a)\cos (n\pi y/b)$ and $k_{mn}^2=(m\pi /a{)}^2+(n\pi /b{)}^2$. The velocity potential has vertical factor $\cosh (k_{mn}(z+h))$. Combining the linearized kinematic and dynamic conditions on the undisturbed surface gives ${\omega }_{mn}^2=g{k}_{mn}\tanh (k_{mn}h)$. The uniform $m=n=0$ elevation is excluded when the total water volume is fixed.

For depth $h=\alpha x^p$ with $p>0$, stationary surface gravity waves conserve frequency $\omega$ and alongshore wavenumber $K$. Near shore their dispersion becomes ${\omega }^2\sim g{\kappa }^{2}h$, so $\kappa \sim \omega x^{-p/2}/\sqrt{g\alpha }$. The shoreward ray has $y^{\prime }\sim -K\sqrt{g\alpha }{x}^{p/2}/\omega$, giving the displayed integrated scaling. Rays approach normally, while equal-phase crests approach parallel to the shore and their shore-normal spacing shrinks as $x^{p/2}$. These are asymptotics within the ray approximation; for $p<2$, its slow-variation assumption eventually fails arbitrarily close to the endpoint.

Wave fetch is the distance over which wind acts on an uninterrupted water surface to generate waves. Growth also depends on wind speed and duration. A polynya can provide a local fetch inside otherwise extensive sea ice.

The excess depth-integrated, time-averaged momentum flux of a surface gravity wave is its radiation stress. In deep water $c_g/c_p=1/2$, so $S_{xx}=\mathcal{E}/2={\rho }_{w}g{a}^2/4$. The force per transverse width on a reflecting object is the incident plus reflected minus transmitted radiation stress. Reflection increases force because the reflected wave carries momentum in the opposite direction.

A linear deep-water gravity wave of surface-displacement amplitude $a$ has equal mean kinetic energy and potential energy, and total mean energy per horizontal area $\mathcal{E}={\rho }_{w}g{a}^2/2$. Its energy transport speed is the group velocity. This amplitude is half the crest-to-trough height.

The dimensionless product of surface-wave amplitude and wavenumber, $ka$, measures orbital displacement relative to the wavelength. Small wave steepness permits a Taylor expansion of the velocity sampled by moving parcels. For a deep-water gravity wave, orbital amplitudes and the local steepness decrease exponentially with depth.

Airy wave theory linearizes the motion of small-amplitude surface gravity waves in an inviscid, incompressible, irrotational fluid. At depth $h$, its dispersion relation is ${\omega }^2=gk\tanh (kh)$; deep water is the limit $kh\to \infty$.

A small-amplitude deep-water wave with surface elevation $\eta =A{e}^{i(kx-\omega t)}$ has dispersion relation ${\omega }^2=gk$ and a velocity potential that decays as $e^{kz}$ below the surface.

For an impulsive distant source of deep-water gravity waves, a component of period $T$ travels at group velocity $gT/(4\pi )$. Its arrival at distance $L$ obeys the displayed law, so inverse period increases linearly with arrival time. Two arrivals determine both $L$ and $t_0$. Finite source duration, currents, refraction and passage through sea ice can invalidate the simple inversion.

For small kinematic viscosity $\nu$, the mean viscous dissipation per unit horizontal area is $2\rho \nu g{k}^2|A{|}^2$. Since the mean wave energy is $\rho g|A{|}^2/2$, the amplitude obeys

For small-amplitude potential flow beneath a mean surface $z=0$, the linearized kinematic and dynamic conditions are

In a rectangular box with Neumann side walls, a horizontal modehas horizontal wavenumber $k=[(m\pi /L_x{)}^2+(n\pi /L_y{)}^2{]}^{1/2}$, depth dependence $e^{kz}$ in infinitely deep fluid, and natural frequency $\Omega =\sqrt{gk}$.

Pressure forcing of one rectangular free-surface mode reduces its amplitude toAt $\omega =\sqrt{gk}$, the inviscid undamped response contains $t\sin (\omega t)$ and grows without bound in the linear model.