Rotating shallow-water height equation with variable depth (${\partial }_t[{\eta }_{tt}+f^2\eta -g{\nabla }_h\cdot (H{\nabla }_h\eta )]+fg{H}_y{\eta }_x=0$)

For linear rotating shallow water equations with fixed depth $H(y)$ and constant Coriolis parameter $f$, elimination of the horizontal velocity givesAt 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.