A particularly neat solution to the wave equation, that is valid when
the string is so long that 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

Thus, and we must use the chain rule to express derivatives in terms of and as derivatives in terms of and . Hence,

The second derivatives require a bit of care.

Thus,

and in a similar manner

Thus, (2.6) becomes

This equation is much simpler and can be solved by direct integration. First
of all 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, we replace and by their expressions in terms of and
to obtain

__Example 2. .14__Verify that
is a solution to the wave
equation. Here
and
.

Thus

Therefore,

Note that we may use the trigonometric identities for
to obtain

and we remark that this is the product of a function of with a function of . This result will be used later.

D'Alembert's solution involves two arbitrary functions that are determined
(normally) by two initial conditions. If the initial conditions are

Then, from (2.7), we have

Note that is a function of but at . The tricky part is calculating

but at and . Thus, the functional form of and are obtained by replacing by and .

__Example 2. .15__If, for example,
, then

The initial speed, that is
evaluated at
, is then

Integrating with respect to , as both and are functions of when , we get

Subtracting (2.10) from (2.9) gives

Hence, d'Alembert's solution that satisfies the initial conditions (2.8) is

__Example 2. .16__ can be thought of as a
``shape", defined by the function , moving to the right. We can see
this if we consider the value . Say

Then at when

when

when

Thus, we are moving to the * right* to larger values of . Similarily
corresponds to a wave propagating to the * left*.

__Example 2. .17__
Thus, the solution to the wave equation is

corresponding to a wave travelling to the right (increasing ).

Here is the amplitude,

is the wavenumber - ,

is the wavelength - distance from one crest to the next,

is the angular frequency ,

is the period of the wave - time from one crest to the next to pass afixed point.

__Example 2. .18__
This gives the solution

The solution is a standing wave as shown in Figure 2.2. The maximum and minimum displacements are at and the nodes (where ) are at

__Example 2. .19__

where is a constant. In Figure 2.3, .

From d'Alembert's solution we have

where is a constant. Adding the two solutions we have

Hence, the solution is

and the arbitrary constant, , does not appear in the solution. Thus, we may, in fact, take without any loss of generality. So we have 2 equal triangles of height going to the right and the left. This is shown in Figure 2.4.