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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.

The free surface is a material surface, so its kinematic boundary condition isWith constant atmospheric pressure $p_a$ and no surface tension, the dynamic boundary condition for an inviscid interface follows from Unsteady Bernoulli equation:Write $\varphi =Ux+\phi$, where $\phi$ and $\eta$ are small. Choose the time-dependent potential gauge so the undisturbed right-hand side is $U^2/2$. Retaining first-order terms, evaluated at $y=0$, givesThe incompressibility condition also gives Laplace's equation for $\phi$.

Set $\Omega =\omega -kU$. Substitution of the specified normal mode into the two linearized boundary conditions yields $-i\Omega a=ikb$ and $\Omega b+ga=0$. Eliminating $b$ gives the dispersion relationThe decay factor $e^{ky}$ in a fluid extending to $y=-\infty$ uses $k>0$. With signed wavenumber the decaying factor is $e^{|k|y}$ and the right-hand side is $g|k|$. The frequency shift is the advection of a deep-water gravity wave by the uniform current.

Differentiating the velocity potential gives $u=a\omega e^{kz}\cos (kx-\omega t)$ and $w=a\omega e^{kz}\sin (kx-\omega t)$. Let $A=a{e}^{kz}$ and $\vartheta =kx-\omega t$. To first order in the wave steepness $ka$, evaluate these velocities at the initial parcel position and integrate from time zero:These expressions satisfy both initial conditions and distinguish initial and mean parcel labels in a surface wave. Their actual short-time limits areThe printed expressions omit the integration constants and therefore cannot be the small-time displacement from the prescribed initial point. For example, at $x=0$, they give $Z(0)-z=a{e}^{kz}\ne 0$. This is a genuine inconsistency, rather than a missing step in the calculation.

The intended oscillatory orbit is recovered by using mean parcel coordinates instead. Set $x_c=x+A\sin (kx)$ and $z_c=z-A\cos (kx)$ to this order. ThenThus a deep-water gravity wave produces approximately circular parcel orbits with radius $a{e}^{k{z}_c}$, decaying exponentially with depth. The approximation is one of small amplitude over a wave cycle, not an assertion that these oscillatory coordinates vanish at time zero.

Write $w(x,t)$ for the vertical displacement of the sea ice, and take the undisturbed water surface as $z=0$, with water occupying $z<0$. The elastic plate has areal mass $m={\rho }_{i}h$ and bending stiffnessHere $E$ is Young's modulus and $\eta$ is Poisson's ratio. We neglect in-plane prestress, viscosity and plate shear deformation, and linearize about hydrostatic equilibrium. These are important assumptions: perfect elasticity alone does not specify every term in a floating-plate model.

For a plane wave $w=\hat{w}{e}^{i(kx-\omega t)}$, $k>0$, potential flow in deep water has velocity potential $\varphi =\hat{\varphi }{e}^{kz}{e}^{i(kx-\omega t)}$. This solves Laplace's equation and decays downwards. The kinematic boundary condition gives $k\hat{\varphi }=-i\omega \hat{w}$. Linearizing the water pressure at the displaced interface gives an upward excess loadThe elastic plate equation is $m{w}_{tt}+D{w}_{xxxx}=p$. Substitution and multiplication by $k$ therefore give the flexural-gravity wave dispersion relationEquivalently, with $\alpha ={\rho }_{i}h/{\rho }_w$ and $\beta =D/{\rho }_w$,This determines the positive wavenumber implicitly for prescribed positive angular frequency. It is unambiguous: the derivative of ${\omega }^2$ is $(g+5\beta k^4+4\alpha \beta k^5)/(1+\alpha k{)}^2>0$, while ${\omega }^2$ runs from zero to infinity.

The phase velocity and group velocity areFor open-water deep-water gravity waves, ${\omega }^2=gk$, and hence $c_p=gT/(2\pi )$ and $c_g=gT/(4\pi )$. At large period the ice-covered curves approach these straight lines. At shorter period, plate bending raises the speeds, so both ice-covered curves turn upward as period decreases. In the bending regime with negligible plate inertia, $\omega \propto k^{5/2}$ and $c_g\simeq (5/2)c_p$; in the formal plate-inertia-dominated limit, $\omega \propto k^2$ and $c_g\simeq 2c_p$. The latter extrapolation eventually leaves thin-plate validity and should not be read as a prediction at arbitrarily small wavelength.

The plotted parameters are illustrative rather than measured at the observation site. They give a group-velocity minimum of a flexural-gravity wave near $15.1\mathrm{m}{\mathrm{s}}^{-1}$ at period $17.6\mathrm{s}$. There is no arbitrarily slow wave-energy branch under a continuous elastic sheet. Energy put into a localized disturbance travels away at at least this minimum group velocity; a slowly moving wind system cannot retain a wave packet indefinitely beneath itself. This reduces the opportunity for sustained local growth compared with slow, short open-water waves, and the continuous cover also prevents direct wind forcing of an exposed water surface. Incoming long swell can still propagate.

A group-velocity minimum of a flexural-gravity wave is not by itself a universal minimum wind speed for wave generation. A steadily translating forcing pattern requires a phase velocity matching its translation speed, so the minimum of $c_p$, not $c_g$, supplies the corresponding resonance threshold. Random wind forcing, dissipation and aerodynamic coupling must be specified before making an absolute generation claim.

Use a common emission time $t_0$ and a path length $L$, and treat each observed period as a narrow wave packet. For deep-water gravity waves, its group velocity is $gT/(4\pi )$, not the phase velocity $gT/(2\pi )$. Therefore the dispersive swell source inversion givesThus period decreases hyperbolically, while frequency $1/T$ increases linearly. The elapsed time between the two observations is $40\mathrm{h}40\min =146400\mathrm{s}$, soThe frequency slope is $2.6021\times 10^{-7}\mathrm{Hz}{\mathrm{s}}^{-1}$. Initially $dT/dt=-0.843\mathrm{s}{\mathrm{h}}^{-1}$; at 14 s it has slowed to $-0.184\mathrm{s}{\mathrm{h}}^{-1}$. The inferred emission was $C/30=35\mathrm{h}35\min$ before the first arrival, namely 19 March at 1710 UT. Both observed endpoints give this same time.

A northward-propagating swell with this distance scale points to a remote energetic storm south of the high Arctic rather than local wind acting on continuous sea ice. Along a meridional route, 3,000 km is about $27^{\circ }$ of latitude, putting the source on the scale of the northern North Atlantic and adjacent open seas. A route through Fram Strait is plausible, but longitude and refraction are not supplied, so no particular storm or unique source position follows. Long swell arriving first and steadily increasing frequency are the expected signatures of remote wave dispersion. The distance and time are conditional on an approximately impulsive source and open-water propagation speeds; passage through ice, currents and finite storm duration produce corrections.

The sea ice acts as a frequency-selective filter. Over a long path, wave attenuation in sea ice is generally much stronger for shorter surface gravity waves, through repeated scattering, internal ice losses and water-side dissipation. Their amplitude can fall below the tiltmeter's detection level even if the source initially generated them. Thus the long-period swell survives while the shorter-period tail does not.

The disappearance near 14 s is an attenuation and detectability limit, not a forbidden-frequency interval of the ideal elastic plate. The flexural-gravity wave dispersion relation admits a real positive wavenumber for every positive angular frequency. In an impulsive-source picture, shorter-period deep-water gravity waves also arrive later because their group velocity is smaller, but delayed arrival alone does not explain a persistent observed cutoff.

Let $\varphi$ be the velocity potential. The linearized deep-water equations are ${\nabla }^2\varphi =0$ for $z<0$, ${\eta }_t={\varphi }_z$ and ${\varphi }_t+g\eta =0$ at $z=0$, with zero normal velocity at the four side walls. The normal modes arewhere $m,n\ge 0$ are not both zero. The free-surface conditions give the dispersion relation for a deep-water gravity wave:

The initial displacement is the $(m,n)=(3,4)$ mode, so $k=5\pi /a$. Release from rest selects a cosine in time:Velocity decays as $e^{kz}$, so it falls to $1/e$ of its surface value at depth $a/(5\pi )$.

Substitution into ${\nabla }^2\varphi =0$ givesSetThen $Z=A{e}^{kz}+B{e}^{-kz}$. Since the fluid occupies $z<0$, decay as $z\to -\infty$ forces $B=0$. Absorbing $A$ into $F$ and choosing $Z(0)=1$ givesThis is the usual exponential depth dependence of a deep-water gravity wave.

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.