= Solution
In the lower triangular region, the characteristic coordinates obey $-1\leq\xi<\eta\leq1$. Integrating the canonical <wave equation> $u_{\xi\eta}=0$ gives
$$
u(x,y)=F(\xi)+G(\eta).
$$
On the right characteristic edge, $\eta=1$ and $x=(\xi+1)/2$, so its prescribed value gives $F(\xi)+G(1)=-\sin(\pi\xi)$. On the left edge, $\xi=-1$ and $x=(\eta-1)/2$, giving $F(-1)+G(\eta)=\sin(\pi\eta)$. At the common vertex the two values agree and equal zero; hence $F(-1)+G(1)=0$.
Adding the two edge identities and subtracting this vertex identity determines the entire solution:
$$
\boxed{u(x,y)=\sin\bigl(\pi(x-y)\bigr)-\sin\bigl(\pi(x+y)\bigr)
=-2\cos(\pi x)\sin(\pi y),\quad y<0.}
$$
The apparent freedom to add a constant to $F$ and subtract it from $G$ does not change $u$. Every classical lower-region solution has the displayed separated form, so these two characteristic boundary traces determine it uniquely. Substitution into $u_{xx}-u_{yy}=0$ and into both edge values directly verifies existence.
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