Wave Equation--1-Dimensional

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1 Page 1 of 7 INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Calculus and Analysis Differential Equations Partial Differential Equations Wave Equation--1-Dimensional The one-dimensional wave equation is given by In order to specify a wave, the equation is subject to boundary conditions (1) Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News (RSS) New in MathWorld MathWorld Classroom Interactive Entries Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon and initial conditions The one-dimensional wave equation can be solved exactly by d'alembert's solution, using a Fourier transform method, or via separation of variables. d'alembert devised his solution in 1746, and Euler subsequently expanded the method in Let (2) (3) (4) (5) (6) (7) By the chain rule, file://c:\docume~1\steven~1\locals~1\temp\og81eyuj.htm

2 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, in the negative direction and in the positive 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)

3 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

4 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)

5 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)

6 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]

7 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 , Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp and 271, Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 417, Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 130, CITE THIS AS: Eric W. Weisstein. "Wave Equation--1-Dimensional." From MathWorld--A Wolfram Web Resource. Dimensional.html Wolfram Research, Inc. Terms of Use

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