A complete group classification is given of both the wave equation c 2 (x) uxx− utt= 0 (I) and
its equivalent system vt= ux, c 2 (x) vx= ut (II) when the wave speed c (x)≠ const. Equations
(I) and (II) admit either a two‐or four‐parameter group. For the exceptional case, c (x)=(A x+
B) 2, equation (I) admits an infinite group. Equations (I) and (II) do not always admit the same
group for a given c (x): The group for (I) can have more parameters or fewer parameters than
that for (II); moreover, the groups can be different with the same number of parameters …