|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
d'Alembert and D. Bernoulli solutions of wave equation
|
(Derivation)
|
|
|
Let's consider the d'Alembert's solution
![$\displaystyle u(x,\,t) \,:=\, \frac{1}{2}[f(x\!-\!ct)+f(x\!+\!ct)]+\frac{1}{2c}\int_{x-ct}^{x+ct}g(s)\,ds$](http://images.planetmath.org:8080/cache/objects/11031/js/img1.png "$\displaystyle u(x,\,t) \,:=\, \frac{1}{2}[f(x\!-\!ct)+f(x\!+\!ct)]+\frac{1}{2c}\int_{x-ct}^{x+ct}g(s)\,ds$") |
(1) |
of the wave equation in one dimension in the special case when the other initial condition is
 |
(2) |
We shall see that the solution is equivalent with the solution of D. Bernoulli.
We expand the given function $f$ to the Fourier sine series on the interval $[0,\,p]$ : $$ f(y) \,=\, \sum_{n=1}^\infty A_n\sin\frac{n\pi y}{p} \quad \mbox{with}\;\; A_n = \frac{2}{p}\int_0^pf(x)\sin\frac{n\pi x}{p}\,dx \quad (n = 1,\,2,\,\ldots) $$ Thus we may write
Adding these equations and dividing by 2 yield
![$\displaystyle u(x,\,t) = \frac{1}{2}[f(x\!-\!ct)+f(x\!+\!ct)] = \sum_{n=1}^\infty A_n\cos\frac{n\pi ct}{p}\sin\frac{n\pi x}{p},$](http://images.planetmath.org:8080/cache/objects/11031/js/img4.png "$\displaystyle u(x,\,t) = \frac{1}{2}[f(x\!-\!ct)+f(x\!+\!ct)] = \sum_{n=1}^\infty A_n\cos\frac{n\pi ct}{p}\sin\frac{n\pi x}{p},$") |
(3) |
which indeed is the solution of D. Bernoulli in the case $g(x) \equiv 0$ .
Note. The solution (3) of the wave equation is especially simple in the special case where one has besides (2) the sine-formed initial condition
 |
(4) |
Then $A_n = 0$ for every $n$ except 1, and one obtains
 |
(5) |
Remark. In the case of quantum systems one has Schrödinger's wave equation whose solutions are different from the above.
|
Anyone with an account can edit this entry. Please help improve it!
"d'Alembert and D. Bernoulli solutions of wave equation" is owned by pahio. [ full author list (2) ]
|
|
(view preamble | get metadata)
Cross-references: quantum systems, equations, interval, Fourier sine series, function, equivalent, solution, initial condition, dimension, wave equation
There is 1 reference to this entry.
This is version 8 of d'Alembert and D. Bernoulli solutions of wave equation, born on 2008-09-15, modified 2008-10-20.
Object id is 11031, canonical name is DAlembertAndDBernoulliSolutionsOfWaveEquation.
Accessed 824 times total.
Classification:
| AMS MSC: | 35L05 (Partial differential equations :: Partial differential equations of hyperbolic type :: Wave equation) | | | 35L15 (Partial differential equations :: Partial differential equations of hyperbolic type :: Initial value problems for second-order, hyperbolic equations) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
