d'Alembert's Solution

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)