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.[1]
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,
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: