exists a $\theta$ such that (\[eq:ts:def\]) holds.

Objective function {#sec:obj:fun}
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We adapt the overdetermined shooting method of [@water1; @water2] to compute solutions of (\[eq:init\])–(\[eq:even:odd\]). This method employs the Levenberg-Marquardt method [@nocedal] with delayed Jacobian updates [@water2] to solve the nonlinear least squares problem described below.

For (\[eq:init\]), we build the symmetry into the initial conditions over which the shooting method is allowed to search: we choose an integer $n$ and consider initial conditions of the form $$\label{eq:init:trav}
  \hat\eta_k(0) = c_{2|k|-1}, \qquad\quad
  \hat\varphi_k(0) = \pm ic_{2|k|},$$ where $k\in\{\pm1,\pm2,\dots,\pm \frac{n}{2}\}$ and $\hat\eta_k(t)$, $\hat\varphi_k(t)$ are the Fourier modes of $\eta(x,t)$, $\varphi(x,t)$. The numbers $c_1,\dots,c_n$ are assumed real and all other Fourier modes (except $\hat\eta_0$) are zero. We set $\hat\eta_0$ to the fluid depth so that $y=0$ is a symmetry line corresponding to the bottom wall. This is convenient for computing the Dirichlet-Neumann operator [@water2]. In the formula for $\hat\varphi_k$, the minus sign is taken if $k<0$ so that $\hat\varphi_{-k} =\overline{\hat\varphi_k}$. We also solve for the period, $$\label{eq:T:theta}
  T=c_{n+1}.
$$ The phase shift $\theta$ is taken as a prescribed parameter here. Alternatively, in a study of traveling-standing waves [@trav:stand], the author defines a traveling parameter $\beta$ and varies $\theta=c_{n+2}$ as part of the 