Traveling wave solutions of a generalized curvature flow equation in the plane

We study a generalized curvature flow equation in the plane: $V =F(k,$ n, $x)$, where for a simple plane curve $\Gamma$ and for any $P \in \Gamma, k$ denotes the curvature of $\Gamma$ at $P$, n denotes the unit normal vector at $P$ and $V$ denotes the velocity in direction n, $F$ is a smooth function which is 1-periodic in $x$. For any given $\alpha \in ( - \pi/2, \pi/2)$, we prove the existence and uniqueness of a planar-like traveling wave solution of $V = F(k,$n,$x)$, that is, a curve: $y = v$*$(x) + c$*$t$ traveling in $y$-direction in speed $c$*, the graph of $v$*$(x)$ is in a bounded neighborhood of the line $x$tan$\alpha$. Also, we show that the graph of $v$*$(x)$ is periodic in the direction (cos$\alpha$, sin$\alpha$).