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:
Famous quotes containing the word formula:
“Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the beings which compose it, if moreover this intelligence were vast enough to submit these data to analysis, it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom; to it nothing would be uncertain, and the future as the past would be present to its eyes.”
—Pierre Simon De Laplace (1749–1827)