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:
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