=================

As mentioned in the introduction, our idea is to use collisions of unidirectional Stokes (i.e. traveling) waves as starting guesses to find time-periodic and relative periodic solutions of the Euler equations. We begin by computing traveling waves of varying wave height and record their periods. This is easily done in the framework of §\[sec:method\]. We set $\theta=\pi/64$ (or any other small number) and minimize $\tilde f$ in (\[eq:f1\]). The resulting “period” $T$ will give the wave speed via $c=\theta/T$. Below we report $T=2\pi c$, i.e. $T$ is rescaled as if $\theta$ were $2\pi$. We control the amplitude by specifying $\hat\eta_1(0)$, which is the second option listed in §\[sec:method\] for defining the first component $r_1$ of the residual. A more conventional approach for computing traveling waves is to substitute $\eta(x-ct)$, $\varphi(x-ct)$ into (\[eq:ww\]) and solve the resulting stationary problem (or an equivalent integral equation) by Newton’s method [@chen80a; @chandler:93; @milewski:11]. Note that the wave speed $c$ here is unrelated to the vector $c$ of unknowns in (\[eq:init:trav\]).

![\[fig:bif:stokes\] Plots of wave height and first Fourier mode versus period for Stokes waves with wavelength $2\pi$ and fluid depth $h=0.05$. The temporal periods are $6T_A=137.843\approx
  137.738 = 5T_C$.](figs/bif_stokes){width="3.3in"}

![\[fig:align:stokes\] Collision of two 