  The paper is concerned with conservative solutions to the nonlinear wave
equation $u_{tt} - c(u)\big(c(u) u_x\big)_x = 0$. For an open dense set of
$C^3$ initial data, we prove that the solution is piecewise smooth in the
$t$-$x$ plane, while the gradient $u_x$ can blow up along finitely many
characteristic curves. The analysis is based on a variable transformation
introduced in [7], which reduces the equation to a semilinear system with
smooth coefficients, followed by an application of Thom's transversality
theorem.
