D'Alembert's Formula

In mathematics, and specifically partial differential equations, d´Alembert's formula is the general solution to the one-dimensional wave equation:

for . It is named after the mathematician Jean le Rond d'Alembert.

The characteristics of the PDE are, so use the change of variables to transform the PDE to . The general solution of this PDE is where and are functions. Back in coordinates,

is if and are .

This solution can be interpreted as two waves with constant velocity moving in opposite directions along the x-axis.

Now consider this solution with the Cauchy data .

Using we get .

Using we get .

Integrate the last equation to get

Now solve this system of equations to get

Now, using

d´Alembert's formula becomes: