= Seasonal phase lag in linked cavity drainage
{title2=$Q=[J(\tau)-J(\tau-\kappa x)]/\kappa,\quad J'=M$}
A linear storage law gives $\kappa Q_\tau+Q_x=M(\tau)$ with zero upstream water flux. Integration along <characteristic curves> yields the displayed boundary-generated solution. For $M=1-\delta\cos\tau$, $J=\tau-\delta\sin\tau$ and $J^{-1}(\theta)=\theta+\delta\sin\theta+O(\delta^2)$. A fixed-flux contour therefore obeys
$$
x=Q+\frac{2\delta}{\kappa}\sin(\kappa Q/2)\cos(\tau-\kappa Q/2)+O(\delta^2).
$$
Its extrema have phase $\tau=\kappa Q/2\pmod\pi$ when the first-order amplitude is nonzero. The phase lag arises from water storage and downstream propagation, even if the ice thickness is approximately frozen during one fast seasonal cycle.
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