Wave-driven ice-band compaction ($-{\partial }_{x}F=\kappa F$)

If forward-going surface-gravity-wave energy decays as $a^2(x)=a_0^2{e}^{-\kappa x}$ and local wave force is $F=C{\rho }_{w}g{R}^2{a}^2$, its spatial decrease is $-{\partial }_{x}F=C{\rho }_{w}g{R}^2\kappa a^2$. In a weak-reflection independent-layer approximation, $\kappa \simeq p{R}^2/d$. Standard deep-water wave radiation stress gives $C=1/2$, hence a compaction-force density ${\rho }_{w}gp{R}^4{a}^2/(2d)$. Different amplitude or force conventions change the coefficient; the force formula alone does not fix $\kappa$. This quantity is force per horizontal area, not automatically the three-dimensional stress in the ice skeleton.