You are mixing up two different (but related) terms. The wave equation is a second-order partial differential equation which has some solutions in the form of waves. A QM wave function is a particular case of a wave equation, yielding a probability amplitude.

One significant application of the wave equation in pre-quantum physics occurred when Maxwell was able to derive, from the electromagnetic theory he was developing, a wave equation having solutions in which electromagnetic waves propagate at the speed of light.