= Solution
<Basal sliding>[Glacier sliding] is motion of the <ice> base relative to its bed, distinct from internal <ice> deformation. Near a melting bed, basal water, <pressure>-melting/refreezing around obstacles and <material creep>[creep] past bed <surface roughness>[roughness] can permit motion; a cold frozen bed can suppress it. A simple sliding law is increasing in basal traction, for example $u_b\propto\tau_b^m$ on the positive-traction branch, with resistance also affected in richer models by <surface roughness>[roughness] and <basal water pressure>.
Slow <ice> motion has negligible inertial acceleration. In a long shallow <glacier>, leading vertical stress is hydrostatic and longitudinal stress gradients are neglected relative to transverse shear gradients. Resolving gravity downslope and normally yields the two stated balances. This approximates a two-dimensional slab/valley centreline; lateral drag, rapidly varying geometry and strong longitudinal-stress regions need additional terms.
With a stress-<free surface>,
$$
\boxed{p=\rho g\cos\alpha(h-z),\qquad\tau=\rho g\sin\alpha[1-\cot\alpha\,h_x](h-z).}
$$
Put $K=\rho g\sin\alpha$ and $D=1-\cot\alpha h_x$, assuming positive driving shear for the stated power law. Integrating from the bed gives
$$
\boxed{u(z)=u_b+\frac{A(KD)^n}{n+1}[h^{n+1}-(h-z)^{n+1}],\quad
q=hu_b+\frac{A(KD)^n h^{n+2}}{n+2}.}
$$
The surface speed is $u_s=u_b+A(KD)^nh^{n+1}/(n+1)$. <Mass conservation>, with $a$ measured as <ice>-equivalent thickness <ice accumulation>[accumulation], is $\boxed{h_t+q_x=a}$.
Choose $h=H\hat h$, $x=l\hat x$, $u_b=U\hat u_b$ and $t=(l/U)\hat t$, with $U=(KH/R)^m$. Scale <ice accumulation>[accumulation] with $HU/l$. The <dimensionless> basal law and parameters are
$$
\boxed{u_b=h^m(1-\mu h_x)^m,\qquad\mu=\frac{H\cot\alpha}{l},\qquad\delta=\frac{A K^nH^{n+1}}U=A R^mK^{n-m}H^{n+1-m}.}
$$
The resulting flux is exactly $hu_b+\delta(1-\mu h_x)^nh^{n+2}/(n+2)$.
When internal deformation is small and slope corrections are uniformly small, $u_b\simeq h^m$ and
$$
\boxed{h_t+(h^{m+1})_x=a.}
$$
For a steady <ice accumulation>[accumulation] profile and prescribed upstream flux $q_{\rm in}$, $\boxed{h_*(x)=[q_{\rm in}+\int_0^xa(\xi)\,d\xi]^{1/(m+1)}}$. For constant $a$ and zero upstream flux this is $(ax)^{1/(m+1)}$ on the region of positive flux.
A small thickness disturbance $\zeta$ satisfies $\zeta_t+[c(x)\zeta]_x=0$ with $c=(m+1)h_*^m$. Hence it travels down-<glacier> along $dx/dt=c>0$, with amplitude changing as $d\zeta/dt=-c'\zeta$. Since surface speed is $u_s\simeq u_b=h_*^m$ in this sliding-dominated limit,
$$
\boxed{c=(m+1)u_s.}
$$
The thickness wave therefore outruns individual <ice> particles. This is the <sliding-dominated glacier kinematic wave>.
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