with Cantor-like gaps [@plotnikov01], and the long simulation times that occur between collisions in the unidirectional case. Our shooting method has proven robust enough to succeed in finding time-periodic solutions, when they exist, with a poor starting guess. False positives are avoided by resolving the solutions spectrally to machine accuracy and overconstraining the minimization problem. Much of the challenge is in determining the form of the initial condition and the objective function to avoid wandering off in the wrong direction and falling into a nonzero local minimum before locking onto a nearby relative-periodic solution.

Equations of motion {#sec:eqm}
===================

The equations of motion of a free surface $\eta(x,t)$ evolving over an ideal fluid with velocity potential $\phi(x,y,t)$ may be written [@whitham74; @johnson97; @craik04; @craik05] $$\begin{aligned}
  \label{eq:ww}
    \eta_t &= \phi_y - \eta_x\phi_x, \\[-3pt]
    \notag
    \varphi_t &= P\left[\phi_y\eta_t - \frac{1}{2}\phi_x^2 -
  \frac{1}{2}\phi_y^2 - g\eta\right],
$$ where subscripts denote partial derivatives, $\varphi(x,t) =
\phi(x,\eta(x,t), t)$ is the restriction of $\phi$ to the free surface, $g=1$ is the acceleration of gravity, $\rho=1$ is the fluid density, and $P$ is the projection $$Pf = f - \frac{1}{2\pi}\int_0^{2\pi} f(x)\,dx,$$ where we assume a $2\pi$-periodic domain. The velocity components $u=\phi_x$ and $v=\phi_y$ at the free surface can 