d'Alembert's Solution

d'Alembert's solution solves the one-dimensional wave equation

(1)

that models vibrations of a string, where is the wave speed.

The general solution can be obtained by introducing new variables and , and applying the chain rule to obtain

			(2)
			(3)

Using these operators to compute the second partial derivatives gives

			(4)
			(5)

respectively, so plugging in and expanding then gives

(6)

This partial differential equation has general solution

			(7)

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 (7),

(8)

Taking the derivative with respect to then gives

			(9)

and integrating, with the additive constants in and chosen so that , gives

(10)

Solving (8) and (10) simultaneously for and immediately gives

			(11)
			(12)

so plugging these into (7) then gives the solution to the wave equation with specified initial conditions as

(13)