= Solution
<Basal sliding> allows ice to pass over bed obstacles by enhanced deformation and <regelation>: pressure-induced melting on the upstream side and refreezing downstream transfer ice around small obstacles. <Cavitation> opens water-filled gaps on the downstream side. Their size and the remaining ice-bed contact depend on <effective pressure in subglacial drainage> $N=p_{\rm ice}-p_{\rm water}$ as well as basal <shear stress> $\tau$. At fixed $\tau$, reducing $N$ generally enlarges cavities and reduces resistance, giving a sliding law $u(\tau,N)$ with $u_N<0$ in this regime.
A <linked cavity drainage system> connects these cavities through passages along the bed. Under the prescribed decreasing relation $N(Q)$, the <chain rule> gives $\boxed{u_Q=u_NN'(Q)>0}$. This conclusion uses that hydraulic relation; it need not apply to a different drainage regime.
With $t=\varepsilon\tau$, the ice continuity equation becomes $H_\tau+\varepsilon(uH)_x=\varepsilon s'(x)$. For bounded spatial gradients over $\tau=O(1)$, $H$ is independent of the fast seasonal time to leading order. Meanwhile $S'(Q)=\varepsilon\kappa$ gives $S_t=\kappa Q_\tau$, so
$$
\boxed{\kappa Q_\tau+Q_x=M(\tau).}
$$
Let $J(\tau)=\int_0^\tau M(r)\,dr$. Direct differentiation proves that
$$
\boxed{Q(x,\tau)=\frac{J(\tau)-J(\tau-\kappa x)}\kappa}
$$
satisfies the equation and $Q(0,\tau)=0$. This is the boundary-generated solution, or a periodic-forcing solution with the forcing continued into its past. An independently prescribed initial profile would also need matching in the region whose <characteristic curves> have not yet reached the upstream boundary.
If $M>0$, $J$ is invertible and rearranging gives $x=[\tau-J^{-1}(J(\tau)-\kappa Q)]/\kappa$. For $M=1-\delta\cos\tau$ with $|\delta|<1$,
$$
\boxed{J(\tau)=\tau-\delta\sin\tau,\qquad J^{-1}(\theta)=\theta+\delta\sin\theta+O(\delta^2).}
$$
Substitution, followed by the sine-difference identity, gives
$$
\boxed{x=Q+\frac{2\delta}{\kappa}\sin\frac{\kappa Q}{2}\cos\left(\tau-\frac{\kappa Q}{2}\right)+O(\delta^2).}
$$
For fixed $Q$ and nonzero first-order amplitude, the extrema occur at $\tau=\kappa Q/2+n\pi+O(\delta)$. Since $x=Q+O(\delta)$, the <seasonal phase lag in linked cavity drainage> gives the stated phase relation $\boxed{\tau\approx\kappa x/2\pmod\pi}$. Maxima and minima alternate; when $\sin(\kappa Q/2)=0$, the first-order approximation is flat and does not locate individual extrema.
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