Wave speed and d'Alembert's formula

[j5rp]

Does d'Alembert's formula hold if the wave speed is not constant?
For example, if the speed c in the link is not a constant, but instead a function of the coordinate x. Is the formula still valid?

 

[HallsofIvy]

No, it is not. d'Alembert's formula is based on the fact that the wave equation is a "hyperbolic" equation with two "characteristic lines". If c is variable that may not be true. (If it is true, we can use the same idea but with not have functions of x- ct and x+ ct.)