- Details
- Generalization for inhomogeneous canonical hyperbolic differential equations
- See also
- Notes
- External links
In mathematics, and specifically partial differential equations (PDEs), d'Alembert's formula is the general solution to the one-dimensional wave equation
(where subscript indices indicate partial differentiation, using the d'Alembert operator, the PDE becomes: 
).
The solution depends on the boundary conditions at 
: 
and 
.
It consists of separate terms for the boundary conditions and 
:
It is named after the mathematician Jean le Rond d'Alembert, who derived it in 1747 as a solution to the problem of a vibrating string.[1]
Details
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
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
Now, using
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:
for 
.
All second order differential equations with constant coefficients can be transformed into their respective canonic form
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 
.
See also
- D'Alembert operator
- Mechanical wave
- Wave equation
Notes
External links
- An example of solving a nonhomogeneous wave equation from www.exampleproblems.com
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