d Alembert's Solution to the Wave Equation

A particularly neat solution to the wave equation, valid when the string is  so long it may be approximated by one of infinite length, was obtained by  d’Alembert. The idea is to change coordinates from and to and in order to  simplify the equation. Anticipating the final result, we choose the following linear  transformation and Solutions of the wave equation are a linear superpositions of waves with speed  c and -c. Thus, and we must use the chain rule to express derivatives in terms of and as derivatives in terms of and Hence and The second derivatives require a bit of care. and similarly for  Thus, the wave equation becomes which simplifies to This equation is much simpler and can be solved by direct integration.  Integrate with respect to to give where is an arbitrary function of Then integrate with respect to to obtain where is an arbitrary function of and Finally replace and by their expressions in terms of and  D'Alembert's solution is a complete solution to the wave equation, with initial conditions and is given by Proof: Recall that the general solution is given by Thus, we have (1)

We now need to calculate but at we have and Thus, and are obtained by replacing by and by That is The initial speed evaluated at is then Integrating with respect to as both and are functions of when we get (2)

Subtracting from the initial condition (1) gives Hence Adding (1) and (2) gives Hence Hence, d’Alembert’s solution that satisfies the initial conditions is 