 Characteristic-coordinate domain of dependence

For the one-dimensional wave equation $U_{ss}=U_{rr}$ with Cauchy data on $s=s_0$, $a\le r\le b$, the solution is determined where both characteristic footpoints $r\pm (s-s_0)$ lie in $[a,b]$. This is the diamond displayed in the title, or $|r-(a+b)/2|+|s-s_0|\le (b-a)/2$. In characteristic coordinates, it is a rectangle of known arguments for the two travelling-wave functions. A smooth invertible coordinate transformation maps it to a region bounded by the original equation's characteristic curves; it need not remain a Euclidean diamond. Both forward and backward determination can be described this way. Outside the region, changing an unspecified part of one travelling-wave function can change the solution without changing the supplied finite data segment.

For $r=\sqrt{x}$, $s=y^{3/2}/3$ and data on $y=1$, $1/4\le x\le 1$, the transformed segment is $s_0=1/3$, $a=1/2$, $b=1$. The known characteristic arguments lie in $[5/6,4/3]$ and $[1/6,2/3]$. The corresponding original region has curved boundaries despite being a rectangle in characteristic coordinates and a diamond in $(r,s)$ coordinates.