= Solution
Let $\phi$ be the <velocity potential>. The linearized deep-water equations are $\nabla^2\phi=0$ for $z<0$, $\eta_t=\phi_z$ and $\phi_t+g\eta=0$ at $z=0$, with zero normal velocity at the four side walls. The <normal modes> are
$$
\phi=C e^{kz}\cos\frac{m\pi x}{a}\cos\frac{n\pi y}{a}e^{-i\omega t},
\quad
k=\frac\pi a\sqrt{m^2+n^2},
$$
where $m,n\geq0$ are not both zero. The free-surface conditions give the dispersion relation for a <deep-water gravity wave>:
$$
\boxed{\omega^2=gk.}
$$
The initial displacement is the $(m,n)=(3,4)$ mode, so $k=5\pi/a$. Release from rest selects a cosine in time:
$$
\boxed{\eta=\varepsilon\cos\frac{3\pi x}{a}\cos\frac{4\pi y}{a}
\cos\left(\sqrt{\frac{5\pi g}{a}}\,t\right).}
$$
Velocity decays as $e^{kz}$, so it falls to $1/e$ of its surface value at depth \b[$a/(5\pi)$].
Back to article page
