In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and specifically
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to how ...
(PDEs), d´Alembert's formula is the general solution to the one-dimensional
wave equation
The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light ...
:
:
for
It is named after the mathematician
Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert ( ; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Encyclopé ...
, who derived it in 1747 as a solution to the problem of a
vibrating string.
Details
The
characteristics of the PDE are
(where
sign states the two solutions to quadratic equation), so we can use the change of variables
(for the positive solution) and
(for the negative solution) 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
.
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
Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
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 forms. This equation is one of these three cases:
Elliptic partial differential equation
In mathematics, an elliptic partial differential equation is a type of partial differential equation (PDE). In mathematical modeling, elliptic PDEs are frequently used to model steady states, unlike parabolic PDE and hyperbolic PDE which gene ...
,
Parabolic partial differential equation
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena in, for example, engineering science, quantum mechanics and financial ma ...
and
Hyperbolic partial differential equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n - 1 derivatives. More precisely, the Cauchy problem can ...
.
The only difference between a
homogeneous
Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
and an
inhomogeneous
Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
(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.
See also
*
D'Alembert operator
In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of ...
*
Mechanical wave
In physics, a mechanical wave is a wave that is an oscillation of matter, and therefore transfers energy through a material medium.Giancoli, D. C. (2009) Physics for scientists & engineers with modern physics (4th ed.). Upper Saddle River, N.J. ...
*
Wave equation
The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light ...
Notes
{{reflist
Partial differential equations