A complete group classification is given of both the wave equation (I) \(c^ 2(x)u_{xx}-u_{tt}=0\) and its equivalent system (II) \(v_ t=u_ x\), \(c^ 2(x)v_ x=u_ t\), when the wave speed c(x)\(\neq const\). Equations (I) and (II) admit either a two-or four-parameter group. For the exceptional case, \(c(x)=(Ax+B)^ 2\), 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 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 - \(\infty\) to \(+\infty\).