This exercise comes from A First Course in Partial Differential Equations - H.F. Weinberger
8.2) Show that if the operator $L[u] = A(\frac{d^2u}{dt^2})+B(\frac{d^2u}{dxdt})+C(\frac{d^2u}{dx^2})$ is hyperbolic and $A$ does not equal 0, the transformation to moving coordinates $x' = x-(\frac{B}{2A})t$, $t' = t$, takes $L$ into a multiple of the wave operator.
The textbook gives no example of when or where to substitute these new coordinates into the operator, or what would justify a proof of this. Any help or direction would be appreciated.