Traveling waves of a curvature flow in almost periodic media

In a plane media with almost periodic vertical striations, we study a curvature flow and construct two kinds of traveling waves, one having a straight line like profile and the other having a V shaped profile. For each of the first-kind traveling waves, its profile is the graph of a function whose derivative is almost periodic. For each of the second-kind traveling waves, its profile is like a pulsating cone, whose two tails approach asymptotically the profiles of the first-kind traveling waves. Also we consider a homogenization problem and provide an explicit formula for the homogenized traveling speed.