  We prove the existence of a global semigroup for conservative solutions of
the nonlinear variational wave equation $u_{tt}-c(u)(c(u)u_x)_x=0$. We allow
for initial data $u|_{t=0}$ and $u_t|_{t=0}$ that contain measures. We assume
that $0<\kappa^{-1}\le c(u) \le \kappa$. Solutions of this equation may
experience concentration of the energy density $(u_t^2+c(u)^2u_x^2)dx$ into
sets of measure zero. The solution is constructed by introducing new variables
related to the characteristics, whereby singularities in the energy density
become manageable. Furthermore, we prove that the energy may only focus on a
set of times of zero measure or at points where $c'(u)$ vanishes. A new
numerical method to construct conservative solutions is provided and
illustrated on examples.
