On p. 10 of Stein and Shakarchi's Fourier Analysis book, they consider a standing wave $u(x, t)$ and make the change of variables $\xi = x + t, \eta = x - t$, and define $v(\xi, \eta) = u(x, t)$. The change of variables formula shows that
$\displaystyle\frac{\partial^2 v}{\partial \xi \partial \eta} = 0$
They then claim that "integrating this relation twice gives $v(\xi, \eta) = F(\xi) + G(\eta)$". I don't see why this last claim is true: couldn't we for example have $v(\xi, \eta) = \xi \eta$ which satisfies the same relation?