On invariance properties of the wave equation

A complete group classification is given of both the wave equation c 2(x)u x x −u t t =0 (I) and its equivalent system v t =u x , c 2(x)v x =u t (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. Separately for (I) and (II), all possible c(x) that admit a four‐parameter group are found explicitly. The corresponding invariant (similarity) solutions are considered. Some of these wave speeds have realistic physical properties: c(x) varies monotonically from one positive constant to another positive constant as x goes from −∞ to +∞.