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
The second derivatives require a bit of care.
This equation is much simpler and can be solved by direct integration. First
of all integrate with respect to to give
Finally, we replace and by their expressions in terms of and
Example 2. .14Verify that is a solution to the wave equation. Here and .
Note that we may use the trigonometric identities for
D'Alembert's solution involves two arbitrary functions that are determined
(normally) by two initial conditions. If the initial conditions are
Example 2. .15If, for example,
The initial speed, that is
, is then
Adding (2.9) and (2.10) 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
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
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
Example 2. .19