I will discuss a simple-looking isoperimetric problem for curves in the plane where length is measured with respect to a degenerate metric. One motivation for the study is that geodesics for this problem, appropriately parametrized, lead to traveling waves associated with a Hamiltonian system based on a bi-stable potential. This is joint work with Stan Alama, Lia Bronsard, Andres Contreras and Jiri Dadok.