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 fromandtoandin 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 ofandas derivatives in terms ofandHenceand

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 toto give whereis an arbitrary function ofThen integrate with respect to to obtain

whereis an arbitrary function ofand Finally replaceandby their expressions in terms ofand

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 byThus, we have

(1)

We now need to calculate

but atwe haveandThus,andare obtained by replacing

byandbyThat is

The initial speedevaluated atis then

Integrating with respect toas bothandare functions ofwhenwe 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