  Suppose that $L(\partial_t,\partial_x)$ is a homogeneous constant coefficient
strongly hyperbolic partial differential operator on ${\mathbb R}^{1+d}$ and
$H$ is a characteristic hyperplane. Suppose that in a conic neighborhood of the
conormal variety of $H$, the characteristic variety of $L$ is the graph of a
real analytic function $\tau(\xi)$ with ${\rm rank}\,\tau_{\xi\xi}$ identically
equal to zero or the maximal possible value $d-1$. Suppose that the source
function $f$ is compactly supported in $t\ge 0$ and piecewise smooth with
singularities only on $H$. Then the solution of $Lu=f$ with $u=0$ for $t<0$ is
uniformly bounded on ${\mathbb R}^{1+d}$. Typically when ${\rm
rank}\,\tau_{\xi\xi}\ne 0$ on the conormal variety, the sup norm of the the
jump in the gradient of $u$ across $H$ grows linearly with $t$.
