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 ofandas 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 toto give 
where
is an arbitrary function ofThen integrate with respect to to obtain
where
is 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 by
Thus, we have
(1)
We now need to calculate
but at
we have
and
Thus,
and
are obtained by replacing
byandby
That is
The initial speed
evaluated at
is then
Integrating with respect to
as 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
