= Sliding-dominated glacier kinematic wave
{title2=$h_t+(h^{m+1})_x=a,\quad c=(m+1)u_s$}
For a shallow <glacier> with basal traction proportional to thickness and sliding law $u_b\propto\tau_b^m$, negligible internal-deformation flux gives $u_b\propto h^m$. After scaling, the thickness flux is $h^{m+1}$. <Linearization> around a steady thickness $h_*(x)$ gives $\zeta_t+[c(x)\zeta]_x=0$, $c=(m+1)h_*^m$. Thickness disturbances therefore propagate faster than the surface material speed $u_s\simeq h_*^m$ by factor $m+1$.
A steady <ice accumulation>[accumulation] profile and upstream discharge $q_{\rm in}$ give $h_*^{m+1}=q_{\rm in}+\int_0^xa(\xi)\,d\xi$. Along a disturbance ray, $dx/dt=c$ and $d\zeta/dt=-c'\zeta$, so a spatially varying background also changes amplitude. The surface-speed ratio is specific to sliding-dominated flow; internal deformation and significant surface-slope corrections change the flux <derivative>.
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