= Solution
In the characteristic coordinates, the <chain rule> gives $\partial_x=\partial_\xi+\partial_\eta$ and $\partial_t=c(\partial_\xi-\partial_\eta)$. Hence
$$
\zeta_{tt}-c^2\zeta_{xx}=-4c^2\zeta_{\xi\eta}=0.
$$
Integrating this mixed derivative gives $\zeta=F(\xi)+G(\eta)$, a sum of oppositely traveling profiles. The initial data require $F+G=u_0$ and $c(F'-G')=v_0$, so
$$
F'=\frac12\left(u_0'+\frac{v_0}c\right),\qquad G'=\frac12\left(u_0'-\frac{v_0}c\right).
$$
Integrating the profiles and matching their total additive constant gives the <D'Alembert formula with initial velocity>:
$$
\boxed{\zeta(x,t)=\frac{u_0(x+ct)+u_0(x-ct)}2+\frac1{2c}\int_{x-ct}^{x+ct}v_0(s)\,ds.}
$$
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.
Back to article page
