The general solution.

  The most general solution of the wave equation (10) is of the form

 

u(x,t) = u_R(x-ct) + u_L(x+ct).

The contribution u_R is a function of x that maintains a fixed shape but propagates towards the right with speed c, while u_L propagates to the left with that speed. In other words, u(x,t) always consists of two travelling wave moving in opposite directions. In case the initial velocity is zero, half of the initial displacement will move toward the right and half toward the left.

It is possible to take the wave equation apart into two equations, each of which is a one-way wave equation. One of these will describe the left going wave and the other the right going one.