= Characteristic-coordinate domain of dependence
{title2=$a+|s-s_0|\leq r\leq b-|s-s_0|$}
For the <one-dimensional wave equation> $U_{ss}=U_{rr}$ with <Cauchy data> on $s=s_0$, $a\leq r\leq b$, 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 $|r-(a+b)/2|+|s-s_0|\leq(b-a)/2$. 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 $y=1$, $1/4\leq x\leq1$, 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.
\Image[/past-exam-of-the-differential-equations-1-course-of-the-university-of-oxford/2017-characteristic-domain.png]
{title=An invertible nonlinear coordinate change curves a wave-equation <domain of dependence>}
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