Download Wave Equation--1

Page 1 of 7
Calculus and Analysis
INDEX
Algebra
Differential Equations
Partial Differential Equations
Wave Equation--1-Dimensional
Applied Mathematics
Calculus and Analysis
Discrete Mathematics
Foundations of Mathematics
The one-dimensional wave equation is given by
Geometry
History and Terminology
(1)
Number Theory
Probability and Statistics
Recreational Mathematics
Topology
In order to specify a wave, the equation is subject to boundary conditions
(2)
(3)
Alphabetical Index
DESTINATIONS
About MathWorld
About the Author
and initial conditions
Headline News (RSS)
(4)
New in MathWorld
MathWorld Classroom
(5)
Interactive Entries
Random Entry
CONTACT
The one-dimensional wave equation can be solved exactly by d'Alembert's solution, using a Fourier transform method, or via
separation of variables.
Contribute an Entry
Send a Message to the Team
d'Alembert
devised his solution in 1746, and Euler
MATHWORLD - IN PRINT
subsequently expanded the method in 1748. Let
(6)
(7)
Order book from Amazon
By the chain rule,
file://C:\DOCUME~1\STEVEN~1\LOCALS~1\Temp\OG81EYUJ.htm
9/12/2005
Page 2 of 7
(8)
(9)
The wave equation then becomes
(10)
Any solution of this equation is of the form
(11)
where and are any functions. They represent two waveforms traveling in opposite directions,
and in the positive direction.
in the negative
direction
The one-dimensional wave equation can also be solved by applying a Fourier transform to each side,
(12)
which is given, with the help of the Fourier transform derivative identity, by
(13)
where
(14)
This has solution
(15)
file://C:\DOCUME~1\STEVEN~1\LOCALS~1\Temp\OG81EYUJ.htm
9/12/2005
Page 3 of 7
Taking the inverse Fourier transform gives
(16)
(17)
(18)
(19)
where
(20)
(21)
This solution is still subject to all other initial and boundary conditions.
The one-dimensional wave equation can be solved by separation of variables using a trial solution
(22)
This gives
(23)
(24)
So the solution for
is
(25)
Rewriting ( ) gives
file://C:\DOCUME~1\STEVEN~1\LOCALS~1\Temp\OG81EYUJ.htm
9/12/2005
Page 4 of 7
(26)
so the solution for
is
(27)
where
. Applying the boundary conditions
to ( ) gives
(28)
where
is an integer. Plugging ( ), ( ) and ( ) back in for
in ( ) gives, for a particular value of
,
(29)
(30)
The initial condition
then gives
, so ( ) becomes
(31)
The general solution is a sum over all possible values of
, so
(32)
Using orthogonality of sines again,
(33)
where
is the Kronecker delta defined by
(34)
file://C:\DOCUME~1\STEVEN~1\LOCALS~1\Temp\OG81EYUJ.htm
9/12/2005
Page 5 of 7
gives
(35)
(36)
(37)
so we have
(38)
The computation of
s for specific initial distortions is derived in the Fourier sine series section. We already have found that
, so the equation of motion for the string ( ), with
(39)
is
(40)
where the
coefficients are given by ( ).
A damped one-dimensional wave
(41)
file://C:\DOCUME~1\STEVEN~1\LOCALS~1\Temp\OG81EYUJ.htm
9/12/2005
Page 6 of 7
given boundary conditions
(42)
(43)
initial conditions
(44)
(45)
and the additional constraint
(46)
can also be solved as a Fourier series.
(47)
where
(48)
(49)
(50)
SEE ALSO: d'Alembertian, d'Alembert's Solution, Korteweg-de Vries Equation, Laplacian, Telegraph Equation, Wave Equation,
Wave Equation--Disk, Wave Equation--Rectangle, Wave Equation--Triangle. [Pages Linking Here]
file://C:\DOCUME~1\STEVEN~1\LOCALS~1\Temp\OG81EYUJ.htm
9/12/2005
Page 7 of 7
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). "Wave Equation in Prolate and Oblate Spheroidal Coordinates." §21.5 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 752-753, 1972.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 124-125 and 271, 1953.
Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 417, 1995.
Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 130, 1997.
CITE THIS AS:
Eric W. Weisstein. "Wave Equation--1-Dimensional." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/WaveEquation1Dimensional.html
© 1999-2005 Wolfram Research, Inc. | Terms of Use
file://C:\DOCUME~1\STEVEN~1\LOCALS~1\Temp\OG81EYUJ.htm
9/12/2005