 Solution

Assume $0<\epsilon <1$ and setThe travel-time coordinate for a one-dimensional variable-speed wave equationsends the initial interval $I_{\epsilon }=(-a,a)$ to $(-S_{\epsilon },S_{\epsilon })$. By the characteristic curves for speed one minus y squared, the two characteristic coordinates are $s-x$ and $s+x$. The finite propagation speed and uniqueness theorem for hyperbolic partial differential equations therefore give the maximal characteristic diamondEquivalently,In the $(x,s)$ plane this is a diamond with vertices $(0,\pm S_{\epsilon })$ and $(\pm S_{\epsilon },0)$; transforming back bends its four sides into the characteristic curves found in part 3(c). Beyond any one of those sides, a point's backward characteristics meet $x=0$ outside $I_{\epsilon }$, where no Cauchy data were prescribed, so uniqueness cannot be extended farther.