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The characteristics of the PDE are , so we can 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 let us consider this solution with the Cauchy data .
Using we get .
Using we get .
We can integrate the last equation to get
Now we can solve this system of equations to get
d'Alembert's formula becomes:
Generalization for inhomogeneous canonical hyperbolic differential equations
The general form of an inhomogeneous canonical hyperbolic type differential equation takes the form of:
All second order differential equations with constant coefficients can be transformed into their respective canonic forms. This equation is one of these three cases: Elliptic partial differential equation, Parabolic partial differential equation and Hyperbolic partial differential equation.
The only difference between a homogeneous and an inhomogeneous (partial) differential equation is that in the homogeneous form we only allow 0 to stand on the right side ( ), while the inhomogeneous one is much more general, as in could be any function as long as it's continuous and can be continuously differentiated twice.
The solution of the above equation is given by the formula:
If , the first part disappears, if , the second part disappears, and if , the third part disappears from the solution, since integrating the 0-function between any two bounds always results in 0.
This means, that the homogeneous equation ( ) gives back our original formula for the case of .
- D'Alembert (1747) "Recherches sur la courbe que forme une corde tenduë mise en vibration" (Researches on the curve that a tense cord forms [when] set into vibration), Histoire de l'académie royale des sciences et belles lettres de Berlin, vol. 3, pages 214-219. See also: D'Alembert (1747) "Suite des recherches sur la courbe que forme une corde tenduë mise en vibration" (Further researches on the curve that a tense cord forms [when] set into vibration), Histoire de l'académie royale des sciences et belles lettres de Berlin, vol. 3, pages 220-249. See also: D'Alembert (1750) "Addition au mémoire sur la courbe que forme une corde tenduë mise en vibration," Histoire de l'académie royale des sciences et belles lettres de Berlin, vol. 6, pages 355-360.
- An example of solving a nonhomogeneous wave equation from www.exampleproblems.com