d’Alembert and D. Bernoulli solutions of wave equation
Let’s consider the d’Alembert’s solution (http://planetmath.org/WaveEquation)
We shall see that the solution is equivalent with the solution of D. Bernoulli.
We the given function to the Fourier sine series on the interval :
Thus we may write
Adding these equations and dividing by 2 yield
which indeed is the solution of D. Bernoulli (http://planetmath.org/SolvingTheWaveEquationByDBernoulli) in the case .
Note. The solution (3) of the wave equation is especially in the special case where one has besides (2) the sine-formed initial condition
Then for every except 1, and one obtains
Remark. In the case of quantum systems one has Schrödinger’s wave equation (http://planetmath.org/SchrodingersWaveEquation) whose solutions are different from the above.
|Title||d’Alembert and D. Bernoulli solutions of wave equation|
|Date of creation||2013-03-22 18:23:15|
|Last modified on||2013-03-22 18:23:15|
|Last modified by||pahio (2872)|