D'Alembert's formula
Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, and specifically partial differential equations, d´Alembert's formula is the general solution to the one-dimensional wave equation
Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...

:
for . It is named after the mathematician Jean le Rond d'Alembert
Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert was a French mathematician, mechanician, physicist, philosopher, and music theorist. He was also co-editor with Denis Diderot of the Encyclopédie...

.

The characteristics
Method of characteristics
In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation...

 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

  • An example of solving a nonhomogeneous wave equation from www.exampleproblems.com
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