be computed from $\varphi$ via $$\label{eq:uv:from:G}
  \begin{pmatrix} \phi_x \\ \phi_y \end{pmatrix} =
  \frac{1}{1+\eta'(x)^2}\begin{pmatrix}
    1 & -\eta'(x) \\ \eta'(x) & 1 \end{pmatrix}
  \begin{pmatrix}
    \varphi'(x) \\
    {\mathcal{G}}\varphi(x)
  \end{pmatrix},$$ where a prime denotes a derivative and ${\mathcal{G}}$ is the Dirichlet-Neumann operator [@craig:sulem:93] $$\label{eq:DNO:def}
  {\mathcal{G}}\varphi(x)
  = \sqrt{1+\eta'(x)^2}\,\, {\frac{\partial \phi}{\partial n}}(x+i\eta(x))
  = \phi_y - \eta_x\phi_x$$ for the Laplace equation, with periodic boundary conditions in $x$, Dirichlet conditions ($\phi=\varphi$) on the upper boundary, and Neumann conditions ($\phi_y=0$) on the lower boundary, assumed flat. We have suppressed $t$ in the notation since time is frozen in the Laplace equation. We compute ${\mathcal{G}}\varphi$ using a boundary integral collocation method [@lh76; @baker:82; @krasny:86; @mercer:92; @baker10] and advance the solution in time using an 8th order Runge-Kutta scheme [@hairer:I] with 36th order filtering [@hou:li:07]. See [@water2] for details.

Computation of relative-periodic solutions {#sec:method}
==========================================

Traveling waves have the symmetry that $$\label{eq:init}
  \eta(x,0) \, \text{ is even}, \qquad \varphi(x,0) \, \text{ is odd.}$$ This remains true if $x$ is replaced by $x-\pi$. As a starting guess for a new class of time-periodic and relative-periodic solutions, we have in mind superposing two traveling waves, one centered at $x=0$ and the other at $x=\pi$. Doing so will preserve the property (\[eq:init\]), but the 