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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 $kh\to \infty$.

The capillary length measures when gravity and surface tension are comparably important in shaping a liquid interface. For a liquid of uniform mass density $\rho$, beneath a gas whose mass density is negligible, with uniform gravitational acceleration $g>0$ and constant surface tension $\gamma >0$,A deformation of height $a$ and horizontal length scale $L$ has hydrostatic pressure variation of order $\rho ga$ and small-slope capillary pressure of order $\gamma a/L^2$. Equating these gives $L\sim {\ell }_c$: surface tension smooths small features, whereas gravity matters for broad ones. For two resting fluids, replace $\rho$ by their positive mass density difference $\Delta \rho$ in this static balance. This change alone does not give the two-fluid wave dynamics, because both fluids can have inertia. ^[ref]

With the stipulated illustrative parameters $\gamma =0.072\mathrm{N}{\mathrm{m}}^{-1}$, $\rho =1000\mathrm{kg}{\mathrm{m}}^{-3}$, and $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, the capillary length is ${\ell }_c=2.71\mathrm{mm}$. These parameters define the ideal calculations below; they are not a measurement of a particular sample.

For a sinusoid with wavenumber $k>0$, the relevant horizontal scale is $L=1/k$, while its wavelength is $\lambda =2\pi /k$. Thus the Bond number for this scale and the reciprocal restoring-term ratio areEquality is at $k_c=1/{\ell }_c$, hence ${\lambda }_c=2\pi {\ell }_c=17.02\mathrm{mm}$ for these parameters. Using $L=\lambda$ instead produces a different numerical Bond number; state the length convention. The capillary length itself is not the equality wavelength. This distinction helps choose whether a measured ripple should be modeled as a surface gravity wave, a nearly pure capillary wave, or a gravity-capillary wave. ^[ref]

A dispersion relation specifies which angular frequencies and wavevectors a linear wave model admits. Fix the plane wave conventionHere $a$ is a constant complex amplitude, $\mathbf{k}$ is the wavevector, and $\omega$ is the angular frequency. Substitution into a constant-coefficient linear partial differential equation replaces ${\partial }_t$ by $-i\omega$ and $\nabla$ by $i\mathbf{k}$, turning the wave problem into an algebraic condition. A system can have several roots; $\omega =\omega (\mathbf{k})$ means one chosen branch, rather than a promise that the full dispersion relation is single-valued. For real $\mathbf{k}$, $\mathrm{Im}\omega >0$ means temporal growth in this convention, while $\mathrm{Im}\omega <0$ means damping. The real-branch discussion of phase velocity and group velocity below assumes negligible loss.

For the one-dimensional wave equation $u_{tt}=c^2{u}_{xx}$, the dispersion relation is ${\omega }^2=c^2{k}^2$. The branch $\omega =ck$ carries right-moving travelling waves for $c>0$; both its phase velocity and group velocity are $c$. One may instead use the positive-angular frequency branch $\omega =c|k|$, with right-moving components at $k>0$ and left-moving components at $k<0$. A real wave pairs each $(k,\omega )$ with $(-k,-\omega )$; that conjugate pair represents the same travelling pattern. Branch and sign choices therefore matter when interpreting a dispersion diagram.

A particularly tangible dispersive example is a small-amplitude surface gravity wave on still water of uniform depth $h$. Neglect surface tension and viscosity, assume incompressible flow and irrotational flow, and impose a flat impermeable bottom. With $k>0$ and positive angular frequency,Here $g$ is gravitational acceleration and $\tanh$ is the hyperbolic tangent. Deep water means $kh\gg 1$: $\omega \simeq \sqrt{gk}$ and longer wavelengths have larger phase speed. Shallow water means $kh\ll 1$: $\omega \simeq \sqrt{gh}k$, an approximately nondispersive wave. Thus depth relative to wavelength, rather than depth alone, determines whether crests and wave packets travel together.^[ref]

The dispersion diagram below also includes the ideal deep-water capillary wave limit $\omega =\sqrt{\gamma /\rho }{k}^{3/2}$, where $\gamma$ is surface tension and $\rho$ is mass density. This limit requires the surface tension restoring term to dominate gravity; it reverses the trend of the surface gravity wave, giving larger phase speed at shorter wavelength. The combined gravity-capillary wave has ${\omega }^2=gk+(\gamma /\rho )k^3$.^[ref]

The curves use dimensionless normalizations chosen separately for each model; they are analytical diagrams, not experimental data. Their beauty is that a single dispersion relation encodes several observable behaviours: its value gives the angular frequency, its origin-to-point slope gives the one-dimensional phase velocity, its tangent slope gives the group velocity, and variation of that tangent slope predicts distortion of a wave packet. A straight line through the origin describes a nondispersive wave; an affine branch $\omega =ck+b$ with $b\ne 0$ has constant group velocity but phase velocity $c+b/k$, so those properties must be distinguished.

Frazil ice consists of small suspended ice crystals formed in supercooled turbulent water. A layer of crystals can damp surface gravity waves through viscous dissipation, even when individual crystals scatter very little wave energy.

A gravity-capillary wave has restoring contributions from both gravity and surface tension. Consider an initially resting, horizontally unbounded liquid layer of constant depth $h$, with uniform mass density $\rho$, incompressible flow, inviscid flow and irrotational flow. The upper gas exerts constant pressure and has negligible inertia. Assume a clean interface with constant surface tension $\gamma$: no surfactant stresses, imposed current or wind. The wave amplitude $a$ must satisfy $ka\ll 1$ and $a\ll h$, so linearized boundary conditions apply. Deep water means $kh\gg 1$, rather than a particular depth in metres. ^[ref]

Let $z=0$ be the undisturbed free surface, with liquid below it. Potential flow gives a velocity potential $\varphi$ with $\mathbf{u}=\nabla \varphi$, and incompressible flow gives the Laplace equation. The linearized kinematic boundary condition for a free-surface graph and the Unsteady Bernoulli equation, including gravity and the Young–Laplace equation, giveHere $\eta (x,t)$ is the interface displacement; the signed small-slope pressure jump is $p_{\mathrm{liquid}}-p_{\mathrm{gas}}=-\gamma {\eta }_{xx}$. A wave crest has ${\eta }_{xx}<0$, so its capillary pressure is positive. Choose a right-travelling Fourier mode of positive wavenumber $k$ and angular frequency $\omega$:The factor $e^{kz}$ decays into the deep liquid, satisfies the Laplace equation, and makes ${\varphi }_z={\eta }_t$ at $z=0$. Substituting in the dynamic boundary condition leavesThis derivation displays the two restoring contributions with the same sign. Dimensional analysis also checks the result: $\gamma /\rho$ has units ${\mathrm{m}}^3{\mathrm{s}}^{-2}$, so both $gk$ and $(\gamma /\rho )k^3$ have units ${\mathrm{s}}^{-2}$. ^[ref]

At finite depth, a flat impermeable bottom at $z=-h$ requires ${\varphi }_z=0$. Replacing $e^{kz}$ by the normalized vertical factor $\cosh (k(z+h))/\cosh (kh)$ makes the surface ratio ${\varphi }_z/\varphi =k\tanh (kh)$. Eliminating the mode amplitudes then givesThus the same model supplies its own depth check: use the deep-water formula only when $\tanh (kh)$ is sufficiently close to one for the required accuracy. For example, $kh=3$ reduces $\omega$ by about $0.248\%$ relative to deep water, while $kh=1$ reduces it by about $12.7\%$. These percentages are calculations from this factor. ^[ref]

In deep water the phase velocity magnitude follows a wave crest, while the group velocity magnitude follows the wave envelope of a narrow wave packet. Differentiating the positive-frequency dispersion relation givesFor $k{\ell }_c\ll 1$, the surface gravity wave limit has $c_g\simeq c_p/2$. For $k{\ell }_c\gg 1$, the deep-water capillary-wave dispersion relation has $c_g\simeq 3c_p/2$: a wave crest slips backward relative to the faster wave envelope, even while both advance. Neither speed is the oscillating fluid element's own velocity. ^[ref]

There is a particularly neat coincidence at the crossover. Differentiate $c_p^2=g/k+\gamma k/\rho$: its unique minimum is at $k_c=\sqrt{\rho g/\gamma }=1/{\ell }_c$, exactly where the restoring terms are equal. ThereforeFor the stipulated parameters in capillary length, the minimum is $0.231\mathrm{m}{\mathrm{s}}^{-1}$. At $\lambda =3\mathrm{mm}$, direct substitution gives $f=\omega /(2\pi )\simeq 131\mathrm{Hz}$, $c_p\simeq 0.394\mathrm{m}{\mathrm{s}}^{-1}$ and $c_g\simeq 0.580\mathrm{m}{\mathrm{s}}^{-1}$. This short wavelength is capillary dominated, but its group velocity is still slightly below the pure-capillary ratio $3c_p/2$ because gravity has not vanished.

A gravity wave is a fluid disturbance restored by gravity or buoyancy. A surface gravity wave displaces a free surface, while an internal gravity wave displaces density surfaces within a stratified fluid. Rotation can additionally contribute to restoration, giving an inertia-gravity wave.

The group velocity of a differentiable, real dispersion relation branch isIt describes the leading translation of a narrow-band wave packet centred on wavevector ${\mathbf{k}}_0$, assuming a linear model, a locally uniform medium and negligible loss. Narrow-band means that the appreciably excited wavevectors occupy a small neighbourhood on which one smooth branch can be linearized. Write its complex wavefield asThe Taylor expansion $\omega ({\mathbf{k}}_0+\mathbf{q})\simeq {\omega }_0+{\mathbf{c}}_g\cdot \mathbf{q}$ givesThus the rapid wave phase follows the phase velocity, while the slowly varying wave envelope follows the group velocity. This Fourier analysis argument is an outline of the approximation, rather than a statement that every wave packet translates rigidly forever. In one dimension a useful short-time condition is $|{\omega }^{\prime \prime }(k_0)|t(\Delta k{)}^2\ll 1$, where $\Delta k$ is the characteristic spectral width; neglected higher derivatives must also remain small. Curvature of the dispersion relation eventually changes the wave envelope.^[ref]

Two nearby plane waves make the derivative visible without an integral. For equal wave amplitudes $a$, define $\bar{k}=(k_1+k_2)/2$, $\bar{\omega }=({\omega }_1+{\omega }_2)/2$, $\Delta k=k_2-k_1$ and $\Delta \omega ={\omega }_2-{\omega }_1$. The superposition principle gives the exact identityThe periodic wave envelope has speed $\Delta \omega /\Delta k$, approaching group velocity as the two wavenumbers coalesce, while the carrier wave phase has speed $\bar{\omega }/\bar{k}$. Two tones alone form an infinite beat train, not a localized wave packet. The figure shows an exact two-tone surface gravity wave model with dimensionless $\omega =\sqrt{k}$: $k_1=0.9$, $k_2=1.1$. Its carrier phase velocity is approximately $0.99875$, its finite-difference envelope velocity approximately $0.50063$, and ${\omega }^{\prime }(1)=0.5$ is the limiting group velocity. The marked carrier phase line need not coincide exactly with a maximum of the full modulated wave.

The deep-water surface gravity wave therefore has $c_g=c_p/2$, whereas the deep-water capillary wave limit has $c_g=3c_p/2$: crests can drift either forward or backward relative to a wave envelope. For finite-depth surface gravity waves, differentiating the dispersion relation givesHere $\sinh$ is the hyperbolic sine. The shallow-water limit has $c_g\simeq c_p\simeq \sqrt{gh}$; the deep-water limit has $c_g\simeq c_p/2$.^[ref]

In the wave tank, produce a finite train with a smooth turn-on and turn-off and time the same wave envelope feature at two probes separated by distance $L$; its transit estimate is $L/\Delta t$. Independently follow wave crests through that train. Techet's 2003 lab asks students to observe a deep-water train's foremost crest disappearing as the group moves forward, and to compare deep- and shallow-water cases. The interpretation is that a wave crest overtakes the slower wave envelope, dwindling at its front while other crests become visible at its rear. This is a qualitative expected observation in the teaching protocol, not a published set of measured curves.^[ref]

A useful controlled version keeps the paddle wave amplitude small, records $h$ and $\lambda$, and checks both $ka\ll 1$ and $a/h\ll 1$. Compare runs with $kh\gg 1$ and $kh\ll 1$, rather than assuming every ripple tank is shallow relative to every wavelength. Increase train duration at fixed central frequency to narrow its spectrum, and check that the envelope transit estimate stabilizes. Calibrate both probes, sample fast enough to resolve the carrier, and compare records before reflected waves return; an absorbing beach reduces this contamination. A current changes laboratory velocities, large wave amplitudes introduce nonlinear effects, and very short wavelengths require surface tension. Strong damping or noticeable reshaping makes the chosen threshold or peak an unreliable marker of simple group velocity. These are proposed controls and model limits, not claims that the cited lab measured every correction.

Three velocities should consequently be kept separate. The group velocity predicts narrow-band wave envelope motion. An energy-transport velocity is energy flux divided by energy density, using the appropriate time average; it agrees with group velocity in standard lossless linear wave models when all stored energy is included.^[ref] A signal-front velocity concerns the earliest response to a new disturbance and requires a causal model over its full spectrum. A strictly band-limited wave packet has no sharply bounded onset, and a mathematical two-tone envelope cannot test front arrival. In strongly absorbing or broadband propagation, a single group velocity need not describe the observed peak, energy transport, or first detectable response. A complex $\omega$ or $k$ also prevents interpreting its derivative as an ordinary real transport velocity without further analysis.

In particular, an unusually large or negative group velocity from pulse reshaping does not establish superluminal information transmission. For the causal dielectric model analysed by Fitzpatrick, the field ahead of the front vanishes for $t<x/c$, even when its phase velocity exceeds the vacuum speed of light $c$. That result concerns the electromagnetic model's causality; the ideal water-wave diagrams above contain no relativistic signal-front claim.^[ref]

Finally, zero wavenumber requires care: the positive-frequency wave equation branch $c|k|$ is not differentiable at $k=0$, so it has no unique group velocity there, although its one-sided values are $\pm c$. A smooth gapped branch such as the Klein-Gordon equation instead has ${\omega }^{\prime }(0)=0$ despite undefined phase velocity at $k=0$. The same signed branch must be used in both derivatives and ratios; a negative group velocity means an envelope moving in the negative coordinate direction, not a negative elapsed time. The practical value of group velocity is precisely this translation between a local slope on a dispersion diagram and a measurable packet transit, with its assumptions made explicit.

The Gaussian envelope in a Klein-Gordon medium supplies a quantitative next step: the first derivative translates the wave envelope, the second derivative spreads it, and higher derivatives limit that approximation. At finite spectral width, a peak's measured speed need not equal the group velocity at the central wavenumber. Neither a peak nor a fixed fraction of its height generally identifies front velocity.

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.

An ice floe is a connected floating piece of sea ice. Its diameter relative to wavelength and its thickness affect the reflection, transmission and dissipation of surface gravity waves.

Zhu, Miao, Xu and Cao's 2007 experiment drove a triangular metal frame electromagnetically with a sinusoidal generator. In a roughly $3\mathrm{cm}$-deep, temperature-regulated pool, it produced nearly plane capillary waves. A $632.8\mathrm{nm}$ laser illuminated the surface obliquely; the moving ripples acted as a reflection diffraction grating. One split beam monitored laser stability, while a CCD recorded diffraction orders on a screen $6.76\mathrm{m}$ away. An air-supported optical table reduced ambient vibration. ^[ref]

The frequency sweep was $100$–$290\mathrm{Hz}$ in $10\mathrm{Hz}$ steps, using distilled water and kerosene at $20{}^{\circ }\mathrm{C}$. For each setting they averaged at least three measurements of first-order spot separation. Known optical geometry and the grating equation yielded $k=2\pi /\lambda$; generator readings yielded $\omega =2\pi f$. Agreement with the predicted log–log slope $3/2$ tests the capillary law, and its intercept gives surface tension if mass density is known. The authors identify surface contamination as a major error source; noncontact optical detection reduces additional disturbance. ^[ref]

The optical measurement can be understood from the grating equation. Let ${\lambda }_L$ be the laser wavelength, $\alpha$ the incident grazing angle above the mean surface, and ${\beta }_m$ the outgoing grazing angle of diffraction order $m$, with the sign of $m$ chosen so thatThus calibrated spot positions determine outgoing angles, which determine the ripple wavelength $\lambda$, independently of the mechanical dispersion relation. This optical geometry treats the surface as a reflection diffraction grating at the laser wavelength. ^[ref]

The inferred surface tension in the full deep-water model isThis formula makes useful experimental controls explicit. Measure mass density at the sample's temperature; check depth with $kh$; keep the driven wave amplitude small enough that $ka\ll 1$; and distinguish $f$ from angular frequency $\omega$. For the stipulated parameters, the $100$–$290\mathrm{Hz}$ window corresponds to calculated wavelengths about $3.62$–$1.76\mathrm{mm}$ and $kh\simeq 52$–$107$ at $h=30\mathrm{mm}$. Depth corrections are negligible here, but the gravity contribution is about $4.5\%$–$1.1\%$ of the capillary contribution. These are ideal predictions, not digitized results from the experiment.

Omitting gravity would infer ${\gamma }_{\mathrm{cap}}=\rho {\omega }^2/k^3=\gamma +\rho g/k^2$, biasing surface tension upward. In the capillary limit, the differential estimate $d\gamma /\gamma \simeq d\rho /\rho +2d\omega /\omega -3dk/k$ also shows why careful spot-spacing calibration matters. A clean surface and weak viscosity are modeling assumptions: appreciable damping, surfactant stresses or large wave amplitudes call for additional physics. The experiment tests the short-wave regime; its drive window does not establish the crossover or the long-wave surface gravity wave limit.

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.

The marginal ice zone is the transition between open ocean and compact sea ice, where interacting ice floes and surface gravity waves can dominate the mechanical evolution.

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.

A fixed wave phase satisfies $kx-\omega t=$ a constant. Differentiating gives the phase velocity, called the wave speed here:It is the speed of an individual crest or other fixed-phase point. The group velocity isevaluated on the relevant dispersion relation. To see its meaning, superpose two equal-amplitude waves with nearby $(k,\omega )$:The slowly varying wave envelope moves with speed $\Delta \omega /\Delta k$, tending to $d\omega /dk$ as the two components approach one another. Thus group velocity describes a narrow wave packet and, for conservative linear surface gravity waves in still water, their energy transport. A single time-harmonic wave has no distinguished packet envelope. When $c=c(k)$, the equivalent relation is $c_g=c+kdc/dk$.

Seek the periodic response of a surface gravity wave forced by a traveling bottom, with the same wavenumber $k$ and angular frequency $kc$ as the bottom. The linear equations have constant coefficients in $x,t$, so they preserve each such harmonic. There is no dissipative phase lag in this model; away from resonance, the surface response is in phase or in opposition with the bottom. Thus write $\eta =b\sin \theta$, $\theta =k(x-ct)$, with a signed coefficient $b$. Its physical amplitude is $|b|$. Arbitrary additional free surface gravity waves depend on initial data; the paper's expression selects the forced traveling component.

For $k>0$, $c\ne 0$, use $\varphi =F(y)\cos \theta$. The Laplace equation gives $F^{\prime \prime }-k^{2}F=0$. At $y=0$, the two linearized free-surface boundary conditions implyThese determine both constants in the vertical solution:Its vertical derivative at the bottom isMatching it to $-akc\cos \theta$ and multiplying by $c$ givesand thereforeAn equivalent expression for the velocity potential, which makes the bottom condition immediate, isLet ${\omega }_f=kc$ be the forcing angular frequency, and let ${\omega }_0^2=gk\tanh (kd)$ be the natural dispersion relation from Airy wave theory. ThenThe denominator vanishes at $c^2=(g/k)\tanh (kd)$, when the forcing travels at the free-wave phase velocity. For nonzero forcing there is no bounded periodic forced solution at resonance; the undamped linear response develops secular growth. Below that speed $b<0$, giving a phase reversal, while above it $b>0$. At $c=0$ the bottom is stationary: the rest solution has $\varphi =0$ and $\eta =0$, consistent with the zero limit of $b$, though the formula containing $1/c$ must be interpreted through that limit.

The denominator of the travelling response vanishes atFor the positive speeds in the paper, a finite travelling response exists whenever $c\ne c^{\ast }$. At $c=c^{\ast }$, the dynamic condition would reduce to $p_1/\rho =0$, which is impossible. The finite-depth dispersion relation of a surface gravity wave with wavenumber $k$ is ${\omega }^2=gk\tanh (kh)$. At $k=1$, its phase speed $\omega /k$ equals $c^{\ast }$: the moving pressure pattern drives a natural surface gravity wave at its own speed, producing resonance.

The restriction concerns a bounded travelling response. To distinguish this from existence of a solution of an initial value problem, set $T=\tanh h$, $P=p_1/\rho$, and ${\omega }_{\ast }^2=gT$. Ifthe linearized free-surface boundary conditions reduce to the forced harmonic oscillatorFor example, if the forcing is switched on at $t=0$ from a flat surface with zero initial velocity, then, for $c\ne {\omega }_{\ast }$,This satisfies $H(0)=\dot{H}(0)=0$ and contains both free oscillations and the forced component. At $c={\omega }_{\ast }$ a solution of the initial value problem still exists, but its particular solution containswhich grows linearly with time. Thus the ideal inviscid flow model predicts resonant growth rather than a finite constant amplitude. Additional free oscillations can set the initial conditions but cannot remove that secular growth.

Seek the travelling forced component of the surface elevation in the form $\eta =A\cos \chi$, where $\chi =x-ct$ and $A$ is a signed coefficient. The forcing has wavenumber one. Separation of variables in the Laplace equation, followed by the bed no-penetration boundary condition, givesSince ${\eta }_t=Ac\sin \chi$, the surface kinematic boundary condition requires $B\sinh h=Ac$, soAt $y=0$, ${\varphi }_t=-A{c}^2\coth h\cos \chi$. The dynamic member of the linearized free-surface boundary conditions therefore becomesConsequently the signed response and the nonnegative amplitude areThe travelling-pressure response of finite-depth surface gravity waves is opposite in phase to the imposed pressure below the resonant speed and in phase above it. These expressions describe the travelling forced component; freely propagating surface gravity waves can be superposed to meet additional initial conditions.

Let $\chi =kx-\omega t$, with $k>0$. The surface elevation is $\eta =\alpha \cos \chi$, so ${\eta }_t=\alpha \omega \sin \chi$. Use separation of variables in the interior Laplace equation:The $\sinh (k(z+h))$ term is excluded by the bed no-penetration boundary condition. The linear surface kinematic boundary condition determinesAn irrelevant additive constant is omitted. At $z=0$,The linear dynamic boundary condition then reduces toConsequently the dispersion relation of a floating elastic plate with zero plate inertia isThe negative-frequency branch gives propagation in the opposite direction. For signed wavenumber, replace $k$ by $|k|$ in the squared-frequency formula. Setting $B=0$ recovers the finite-depth surface gravity wave relation, a useful sign check.

Define the dimensionless parameterThe stipulated hierarchy $kh\ll \delta \ll 1$ implies both $kh\ll 1$ andThus the finite-depth correction uses $\tanh (kh)\sim kh$, while the restoring effect of bending stiffness is small compared with gravity. The dispersion relation becomesHence, to leading order, these surface gravity waves are nondispersive waves withFor clarity, the first corrections to the phase speed areThey exhibit the weak residual wave dispersion at finite wavenumber. The claimed nondispersive behavior is the asymptotic leading-order result, not exact equality for the complete relation.

Wave dispersion means that different frequencies propagate with different phase velocities, so a wave packet generally changes shape. Equivalently the dispersion relation is not proportional to $k$ over the relevant band. Three natural examples are the colour separation in a rainbow, where the refractive index of water depends on wavelength; deep-water surface gravity waves, for which ${\omega }^2=gk$ and long waves have larger phase velocity; and capillary waves, whose short-wave limit is ${\omega }^2=(\gamma /\rho )k^3$ and whose shorter waves have larger phase velocity. Here $g$ is gravitational acceleration, $\gamma$ is surface tension, and $\rho$ is fluid mass density, with positive $k$ and negligible viscosity.

A distinct application is an optical prism spectrometer: the wavelength-dependent refractive index produces different refraction angles, separating spectral components for measurement. This exploits wave dispersion deliberately.