The method of d'Alembert provides a solution to the one-dimensional wave equation
(1)
|
that models vibrations of a string.
The general solution can be obtained by introducing new variables and , and applying the chain rule to obtain
(2)
| |||
(3)
| |||
(4)
| |||
(5)
|
Using (4) and (5) to compute the left and right sides of (3) then gives
(6)
| |||
(7)
| |||
(8)
| |||
(9)
|
respectively, so plugging in and expanding then gives
(10)
|
This partial differential equation has general solution
(11)
| |||
(12)
|
where and are arbitrary functions, with representing a right-traveling wave and a left-traveling wave.
The initial value problem for a string located at position as a function of distance along the string and vertical speed can be found as follows. From the initial condition and (12),
(13)
|
Taking the derivative with respect to then gives
(14)
| |||
(15)
|
and integrating gives
(16)
|
Solving (13) and (16) simultaneously for and immediately gives
(17)
| |||
(18)
|
so plugging these into (13) then gives the solution to the wave equation with specified initial conditions as
(19)
|