Surface gravity wave

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

For the subcritical travelling-pressure response of finite-depth surface gravity waves, put $c_{\ast }^2=g\tanh (kh)/k$. When $0\le c<c_{\ast }$, the amplitude isThe inequality follows immediately because the denominator belongs to $(0,1]$. Equality holds precisely at $c=0$, where hydrostatic equilibrium gives $\rho g\eta =-p_1\cos (kx)$. If only strictly positive $c$ is allowed, $p_1/(\rho g)$ is the infimum, approached as $c\downarrow 0$, and is not an attained minimum. This bound applies to the travelling forced component, rather than an arbitrary superposition with free surface gravity waves.

For an inviscid flow of constant mass density $\rho$ and depth $h$, impose excess pressure $p_1\cos (k(x-ct))$ with $k>0$ on a free surface. Neglect surface tension. A travelling solution of the linearized free-surface boundary conditions isThe velocity potential satisfies the Laplace equation and the bed no-penetration boundary condition. Its surface time derivative is $-A{c}^{2}k\coth (kh)\cos (k(x-ct))$, so the dynamic condition gives the displayed signed response. The nonnegative amplitude is $|A|$. It is finite except atthe squared phase speed of a free surface gravity wave at that wavenumber. Equality produces resonance. This describes a travelling forced component; free surface gravity waves must be added to meet general initial conditions. At exact resonance, a solution of the initial value problem generally grows with time rather than having a bounded periodic amplitude.

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 negligible surface tension, the small-amplitude surface gravity wave normal 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 $\cosh (k_{mn}(z+h))$, satisfying the bottom Neumann boundary condition. Combining the linearized free-surface boundary conditions gives ${\omega }_{mn}^2=g{k}_{mn}\tanh (k_{mn}h)$. The uniform $m=n=0$ surface elevation is excluded when the total water volume is fixed. Taking one horizontal index zero gives the two-dimensional tank modes.

For $k_{mn}h\ll 1$, the shallow-water approximation gives ${\omega }_{mn}\sim \sqrt{gh}{k}_{mn}$; for $k_{mn}h\gg 1$, the deep-water gravity wave relation ${\omega }_{mn}^2\sim g{k}_{mn}$ follows. An oscillating-wall forcing of a rectangular tank couples to these same normal modes and becomes resonant at their natural frequencies.

For a tank $0<x<L$, $-h<z<0$, let the left wall move by $\mathrm{Re}(c{e}^{-i\omega t})$ with small amplitude, while the right wall and bottom remain fixed. A periodic velocity potential $\mathrm{Re}(f{e}^{-i\omega t})$ has $f_x(0,z)=-i\omega c$. Add the harmonic functionto obtain $G=f+Q$ with homogeneous wall and bottom Neumann boundary conditions. Thus $G=a_0/2+{\sum }_{n\ge 1}{a}_n\cos (k_{n}x)\cosh (k_n(z+h))$, where $k_n=n\pi /L$. The linearized free-surface boundary conditions give $G_z-\lambda G=H$ at $z=0$, with $\lambda ={\omega }^2/g$ andIts Fourier cosine coefficients $H_n=(2/L){\int }_0^{L}H(x)\cos (k_{n}x)dx$ give the title expression for $n\ge 1$ and $a_0=-H_0/\lambda$. In particular $H_n=2i\omega c\lambda /(L{k}_n^2)$ for $n\ge 1$. The denominator vanishes at the corresponding normal modes of surface gravity waves in a rectangular tank, giving resonance. Exact undamped resonance produces secular growth instead of a bounded periodic response. Free transients do not generally decay in this inviscid model.

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 $kh\to \infty$.

Consider fluid at rest with mean depth $d$ and a bottom displacement $a\sin (k(x-ct))$. The linearized free-surface boundary conditions at $y=0$ and moving-bottom kinematic boundary condition at $y=-d$ give a periodic particular response $\eta =b\sin (k(x-ct))$ and velocity potentialThe bottom derivative fixes the signed response coefficient to the title formula. Here $k>0$, $c\ne 0$, and the forcing must avoid $k^2{c}^2=gk\tanh (kd)$. This equality is resonance with the natural finite-depth surface gravity wave; nonzero undamped resonant forcing has no bounded periodic solution. The response reverses phase below resonance. General initial data can add free surface gravity waves, which do not decay in this inviscid model. Near resonance the resulting wave steepness and particle excursions, rather than the bottom displacement alone, determine whether the linearization remains valid.

For a bottom-forced gravity wave with signed surface coefficient $b$, evaluate the first-order velocity field at a parcel's mean position $(\bar{x},\bar{y})$ and integrate, with $\Theta =k(\bar{x}-ct)$. The horizontal and vertical displacement coefficients areThus the pathline is an ellipse with horizontal and vertical semiaxes $|A|,|B|$ whenever both are nonzero, and a line segment if one vanishes. The identity $A^2-B^2=b^2[(g/(k{c}^2){)}^2-1]$ determines which axis is longer. Also $B(0)=b$ and $B(-d)=a$, verifying the surface and bottom displacements. Closed first-order orbits do not rule out a higher-order mean drift.

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.

A deep-water gravity wave has dispersion $\omega (k)=\sqrt{g|k|}$. An observer moving with nonzero speed $V$ samples Fourier phases $kV\pm \omega (k)$. The stationary phase method selects the points where their derivatives vanish, which gives the displayed wavenumbers. Their group velocity matches the observer's velocity. For real even initial displacement with a nonzero Fourier coefficient there, the resulting signal has leading size $t^{-1/2}$ and phase $gt/(4|V|)-\pi /4$. If the initial coefficient vanishes, that leading term is absent; a stationary observer does not select a finite nonzero wavenumber by this rule.

For two wave periods emitted together at one location, deep-water group velocity is $gT/(4\pi )$. A distant observer sees the shorter period $T_2<T_1$ later, with $\Delta t=(4\pi D/g)(1/T_2-1/T_1)$. This yields the displayed range estimate. Different emission times, a moving storm, finite-depth dispersion, currents and refraction are possible confounders; the propagation calculation alone does not establish a unique source range without these assumptions.

For elevation $h_0\sin (\omega t)\sin (kx)$, depth-integrated kinetic energy per surface area is $\rho h_0^2{\omega }^2{\cos }^2(\omega t)/(4k)$. Wavelength-averaged excess gravitational energy is $\rho g{h}_0^2{\sin }^2(\omega t)/4$. Conservation gives ${\omega }^2=gk$ in the inviscid gravity-only model.

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 a small-amplitude surface gravity wave described by velocity potential $\varphi$ and surface elevation $\eta$, linearizing the kinematic boundary condition for a free-surface graph and the Bernoulli equation at the mean surface $z=0$ givesHere $p$ is the applied excess pressure above constant atmospheric pressure. With $p=0$, differentiate the second equation in time and use the first to obtain ${\varphi }_{tt}+g{\varphi }_z=0$. These surface conditions apply at finite depth as well as in the deep-water gravity wave limit; depth enters through the bottom condition. The quadratic kinetic energy term is omitted at this order, and surface tension is neglected.

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.