  The nonlinear wave equation $u_{tt}-c(u)(c(u)u_x)_x=0$ determines a flow of
conservative solutions taking values in the space $H^1(\mathbb{R})$. However,
this flow is not continuous w.r.t. the natural $H^1$ distance. Aim of this
paper is to construct a new metric which renders the flow uniformly Lipschitz
continuous on bounded subsets of $H^1(\mathbb{R})$. For this purpose, $H^1$ is
given the structure of a Finsler manifold, where the norm of tangent vectors is
defined in terms of an optimal transportation problem. For paths of piecewise
smooth solutions, one can carefully estimate how the weighted length grows in
time. By the generic regularity result proved in [7], these piecewise regular
paths are dense and can be used to construct a geodesic distance with the
desired Lipschitz property.
