- D'Alembert's formula
In
mathematics , and specificallypartial differential equations , d´Alembert's formula is the general solution to the one-dimensional wave equation: :for . It is named after the mathematicianJean 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:
:
External links
* [http://www.exampleproblems.com/wiki/index.php/PDE27 An example] of solving a nonhomogeneous wave equation from www.exampleproblems.com
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