In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and specifically
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
(PDEs), d'Alembert's formula is the general solution to the one-dimensional
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
(where subscript indices indicate
partial differentiation
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
, using the
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 ...
, the PDE becomes:
).
The solution depends on the
initial condition
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...
s at
:
and
.
It consists of separate terms for the initial conditions
and
:
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 ''Encyclopédie ...
, who derived it in 1747 as a solution to the problem of a
vibrating string
A vibration in a strings (music), string is a wave. Acoustic resonance#Resonance of a string, Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch (music), pitch. If the length or tension of the strin ...
.
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 often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
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
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form
:Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\,
wher ...
,
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, including heat conduction, particle diffusion, and pricing of derivati ...
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 be ...
.
The only difference between a
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
and an
inhomogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
(partial)
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
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
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** 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 medium. While waves can move over long distances, the movement of the medium of transmission—the material—is limited. Therefor ...
*
Wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
Notes
{{reflist
External links
An exampleof solving a nonhomogeneous wave equation from www.exampleproblems.com
Partial differential equations
https://www.knowledgeablegroup.com/2020/09/equations%20change%20world.html