D'alembert's wave equation solution

Question

I'm trying to understand D'alembert's solution to the wave equation. I'm stuck on the change of variables: $$\xi = x - ct, \eta = x + ct$$ which allows $$u(x,t) = u(\xi(x, t), \eta(x, t)).$$ How is it that we're allowed to substitute for x (and t) a linear function of x and t? I'm sure this concept has a name but I'm rusty on the topic, thanks.

Answer 1

Pretty much any substitution is allowed (up to invertibility, and even that can be circumvented). The substitution you cite is nice because it causes the differential operator to become the product of two operators, each a differential operator in one variable only. This leads to "separation of variables". However, knowing that this change of variables works requires studying the method of characteristics, or already knowing the solution so that the substitution is made a posteriori from the author's point of view.