 Solution

In the characteristic coordinates, the chain rule gives ${\partial }_x={\partial }_{\xi }+{\partial }_{\eta }$ and ${\partial }_t=c({\partial }_{\xi }-{\partial }_{\eta })$. HenceIntegrating this mixed derivative gives $\zeta =F(\xi )+G(\eta )$, a sum of oppositely traveling profiles. The initial data require $F+G=u_0$ and $c(F^{\prime }-G^{\prime })=v_0$, soIntegrating the profiles and matching their total additive constant gives the D'Alembert formula with initial velocity:Here $u_0$ names the initial surface displacement, rather than the water velocity $u$ appearing in the original shallow water equations. The formula verifies both initial conditions directly.