  We consider an evolving plane curve with two endpoints that can move freely
on the $x$-axis with generating constant contact angles. We discuss the
asymptotic behavior of global-in-time solutions when the evolution of this
plane curve is governed by area-preserving curvature flow equation. The main
result shows that any moving curve converges to a traveling wave if the moving
curve starts from an embedded convex curve and remains bounded in global time.
