[*planer strings*]{}. The solutions for are $$\begin{aligned}
     x= \rho_{max} \sin \left(\frac{\Omega z}{q} \right),
\label{Plasol}
\end{aligned}$$ where the amplitude of the waving string $\rho_{max}$ is given by the parameters $\Omega$ and $q$ as $$\rho_{max}=\frac{\sqrt{1-q^2}}{\Omega}.$$ The shape of planer strings is depicted in Fig.\[FigPla\].

In the inertial Cartesian coordinate, the planar strings are described by $$\begin{aligned}
    \bar x= \rho_{max} \sin \left(\frac{\Omega \bar z}{q}\right)\cos\Omega \bar t, \\
    \bar y= \rho_{max} \sin \left(\frac{\Omega \bar z}{q}\right)\sin\Omega \bar t.
\end{aligned}$$ These strings consist of a standing wave which is superposition of an up-moving wave and a down-moving wave in the equal amplitude. In the limit $q=1$ the solution becomes the straight string solution.

![Planar strings: $\ell  =0$. The three-dimensional snapshots and projected curves are shown for $q=1/5$, $95/100$, and $1$ as same as Fig.3.\
[]{data-label="FigPla"}](planer.eps){width="16cm"}

### General cases

In general, when $\ell\neq 0$, two first order differential equations and are integrated in the form $$\begin{aligned}
    &{\rho}^2 
        = \frac{1}{2} \left\{
            (\rho_{max}^2+\rho_{min}^2) 
                -(\rho_{max}^2-\rho_{min}^2)\cos\left(2\Omega\sigma\right) 
        \right\} , 
\label{sol_rho} \\
    &\frac{2\ell}{\Omega^2}\tan\left({\varphi} - \varphi_0 
        + \ell\Omega\sigma\right)
        =   
    (\rho_{max}^2+\rho_{min}^2)
        \tan\left(\Omega\sigma+\frac{\pi}{4} \right) 
        - (\rho_{max}^2-\rho_{min}^2),
\label{sol_phi} \end{aligned}$$ where $\varphi_0$ is a constant. (see also ref[@BurdenSRS].)

In the case of $q=0$, leads that strings are confined in a $ z=const.$ plane ($x-y$ plane). In this case, 