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An ice-edge band is an elongated concentration of ice floes separated from neighbouring ice by open water. In one mechanism, an off-ice wind opens polynyas; short wind waves then exert wave radiation stress on their downwind floes. Wave forcing compacts floes into bands, opposed by incoming longer-period swell. Collisions and mergers can produce fewer, larger bands.

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.

Pancake ice consists of approximately disk-shaped young ice floes, often with raised edges formed by collisions. Wave-driven collisions and motion in the surrounding frazil ice can dissipate energy.

It is important to distinguish loss of forward-going surface-gravity-wave energy from conversion of mechanical energy into heat. Scattering attenuation by ice floes redirects energy; it can attenuate a coherent transmitted wave without dissipating the total energy.

For fixed floe geometry, increasing frequency usually increases attenuation over the relevant swell range: shorter wavelengths respond more strongly to the contrast between water and the elastic plate, and to repeated floe edges. Long surface gravity waves have weak curvature and often penetrate much farther. This is a trend over a specified frequency range, not a theorem excluding resonances.

The diameter dependence is governed by $d/\lambda$. An ice floe much smaller than the wavelength moves nearly with the water and scatters weakly. Scattering becomes appreciable when floe size is comparable with the wavelength, and interference between its two edges can give maxima and minima. At fixed ice concentration, larger ice floes also mean fewer edges per unit propagation distance, roughly proportional to $p/d$. Consequently the attenuation coefficient need not increase monotonically with diameter: the single-floe reflection and the number of encounters must both be considered. Thickness increases areal inertia as ${\rho }_{i}h$ and bending stiffness as $h^3$, generally increasing wave mismatch and reflection, although detailed frequency-dependent resonances again prevent a universal monotonic law.

When $d\ll \lambda$, particularly for frazil ice and pancake ice, weak individual scattering leaves other processes dominant. Relative crystal and water motion causes viscous dissipation; an aggregate layer can behave as a viscous or viscoelastic material, and pancake ice collisions, rubbing and overwash remove energy. Their importance depends on concentration and wave amplitude.

For a uniform continuous sheet with horizontal dimensions much greater than the wavelength, there are no repeated floe edges in its interior. A perfectly elastic sheet over inviscid water supports undamped flexural-gravity waves, so internal scattering is not an explanation of decay there. Real attenuation can instead arise from internal ice anelasticity or viscoelasticity, a dissipative sub-ice viscous boundary layer, turbulence, cracks and brine-related processes. Small-floe mixtures and continuous sheets require dissipation models beyond the isolated-floe scattering picture.

Take $x$ positive in the off-ice wind direction, from the compact pack towards the open sea. The wind initially separates the outer ice floes, creating irregular polynyas. A larger opening gives more open-water fetch, so stronger short wind waves develop before reaching its downwind edge. Their reflection supplies a positive force on the downwind ice floes; these catch their neighbours and compact into an ice-edge band. Incoming longer swell exerts force in the opposite direction. The short waves can exert substantial force despite their smaller amplitude because a small ice floe reflects them much more effectively than it reflects the long swell.

The surface-gravity-wave energy of the short waves grows with fetch in the windward polynya, then decays rapidly across the band. Swell enters from the seaward side and usually attenuates more slowly. This gives inward forcing from the two sides: short-wave forcing is largest at the windward face and swell forcing is largest at the seaward face. In the next open polynya, short wind waves regrow from the weak transmitted component. Reflected waves also enhance energy locally on the incident side, with interference that is omitted from a smooth, phase-averaged sketch.

Each plotted component is normalized by its own incident energy; the curves do not assert equal absolute wind-wave and swell energies. The band is partially transmitting. For a perfectly opaque reflector, the transmitted component would instead vanish.

The initial bands have unequal floe inventories, widths and forcing. Differential drift and collisions merge some into composite bands. Larger bands tend to shield smaller downstream accumulations from the short-wave forcing needed to keep them separate, while sufficiently wide intervening polynyas can generate fresh wind waves and maintain separation. Finite available ice, available wave fetch, the opposing swell, wind strength and duration, floe size and thickness, and subsequent mergers limit the number of persistent bands. The wave-force formula alone does not select a universal count. This mechanism and its merger interpretation are supported by the original ice-band study.

There is a normalization issue in the requested stress calculation. Let $\mathcal{E}={\rho }_{w}g{a}^2/2$ be standard linear surface-gravity-wave energy for crest amplitude $a$. The deep-water wave radiation stress is $S_{xx}=\mathcal{E}/2$. Hence standard momentum balance givesFor lossless reflection with amplitude reflection coefficient $R$, $r^2=R^2{a}^2$ and $t^2=(1-R^2)a^2$, so $F_{\mathrm{phys}}={\rho }_{w}g{R}^2{a}^2/2$.

In the usual independent-floe, weak-reflection approximation, there are about $p/d$ effective layers per unit distance. Each removes a fraction $R^2$ of the forward energy, givingThis yields the intended wave-driven ice-band compaction:The attenuation approximation retains the leading term in $R^2$; an independent discrete-layer model instead gives an energy coefficient $-(p/d)\log (1-R^2)$. Neither coefficient follows from the single-object force formula without this additional scattering closure.

For ordinary crest amplitudes, the force printed in the paper is $4F_{\mathrm{phys}}$. With that printed normalization and the same attenuation law, its derivative is four times the requested stress. More generally, if $a^2=a_0^2{e}^{-\kappa x}$, the printed lossless force gives $-F^{\prime }=2{\rho }_{w}g{R}^2\kappa a^2$. Recovering the stated stress from it requires $\kappa =p{R}^2/(4d)$ instead, a different unspecified attenuation convention. The supplied force and stress cannot both be derived from standard amplitudes and the usual floe-layer attenuation law. Restoring the factor $1/4$ in the force gives a consistent intended model.

Using the printed force for the numerical question, perfect reflection gives $r=a$, $t=0$, and thereforeUsing the requested stress formula for partial reflection givesFor comparison, the consistently normalized perfect-reflection force is $50.3\mathrm{N}{\mathrm{m}}^{-1}$, and the printed force with standard attenuation would give $0.628\mathrm{N}{\mathrm{m}}^{-2}$ for the partial-reflection derivative. The approximate stress is a force per horizontal area, not a direct measure of the three-dimensional ice-skeleton stress.

The inward force gradient helps maintain a coherent, close-packed band, especially near the incident-wave faces. It is modest enough that wind, current, swell changes or mergers can disrupt the arrangement; it does not guarantee permanent mechanical stability. Perfect reflection estimates a bounding force, whereas the much smaller partial-reflection stress varies as $R^4$. No mechanical-strength law is supplied, so stability can be assessed only qualitatively.

A sea-ice pressure ridge forms where horizontal convergence compresses sea ice. The initially thinner sheet or colliding ice floes fracture and ride over or under one another. Continued convergence piles broken blocks into an emergent sail and a submerged keel. The submerged volume is normally larger because buoyancy supports the pile. Pores and brine-filled gaps initially make the rubble unlike a solid intact sheet; refreezing can consolidate it.

A sea-ice shear ridge develops along a fracture where neighbouring ice moves tangentially in opposite directions or at different speeds. Rough edges interlock, crush and locally converge, producing chains of piled blocks along the shear boundary. Thus the large-scale strain is mainly shear, but the actual production of ridge rubble involves local compression. Pure sliding of perfectly smooth parallel surfaces need not create a ridge. Pressure ridging is driven by convergence; shear ridging is driven by relative tangential motion with local crushing and convergence.

Interfaces between ice floes and open water reflect and redirect surface gravity waves. The effective attenuation depends on floe thickness, diameter, spatial arrangement and wave period. For very small diameter relative to wavelength, a floe responds nearly with the water and scatters weakly. Individual-floe interference and the number of interfaces per unit distance prevent a universal monotonic diameter law.