= Rotating shallow-water height equation with variable depth
{title2=$\partial_t[\eta_{tt}+f^2\eta-g\nabla_h\cdot(H\nabla_h\eta)]+fgH_y\eta_x=0$}
For linear rotating <shallow water equations> with fixed depth $H(y)$ and constant <Coriolis parameter> $f$, elimination of the horizontal velocity gives
$$
\partial_t\left[\eta_{tt}+f^2\eta-g\nabla_h\cdot(H\nabla_h\eta)\right]+fgH_y\eta_x=0.
$$
At a depth step this is interpreted distributionally, with continuity of surface elevation and normal volume flux. Because velocity elimination raises the time order, initial height alone is insufficient; the original velocity and continuity equations determine the remaining initial derivatives.
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