I understand the idea behind the method of characteristics as applied to first-order PDEs: watching how $u(x,y)$ changes along special curves $(x(s),y(s))$ simplifies the problem to a set of coupled ODEs.
What does the method of characteristics mean for second-order (hyperbolic) PDEs? All I see is that we perform a smart change of coordinates $(x,y) \to (\xi,\eta)$ to obtain a simpler PDE. What is the intuition behind this change of variables?