d’Alembert and D. Bernoulli solutions of wave equation

Let’s consider the d’Alembert’s solution (http://planetmath.org/WaveEquation)

u(x,t):=12[f(x-ct)+f(x+ct)]+12cx-ctx+ctg(s)𝑑s (1)

of the wave equationMathworldPlanetmath in one dimension in the special case when the other initial conditionMathworldPlanetmath is

ut(x, 0):=g(x) 0. (2)

We shall see that the solution is equivalent with the solution of D. Bernoulli.

We the given function f to the Fourier sine seriesMathworldPlanetmath on the interval  [0,p]:

f(y)=n=1AnsinnπypwithAn=2p0pf(x)sinnπxpdx(n=1, 2,)

Thus we may write


Adding these equations and dividing by 2 yield

u(x,t)=12[f(x-ct)+f(x+ct)]=n=1Ancosnπctpsinnπxp, (3)

which indeed is the solution of D. Bernoulli (http://planetmath.org/SolvingTheWaveEquationByDBernoulli) in the case  g(x)0.

Note.  The solution (3) of the wave equation is especially in the special case where one has besides (2) the sine-formed initial condition

u(x, 0):=f(x)sinπxp. (4)

Then  An=0  for every n except 1, and one obtains

u(x,t)=cosπctpsinπxp. (5)

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
Canonical name DAlembertAndDBernoulliSolutionsOfWaveEquation
Date of creation 2013-03-22 18:23:15
Last modified on 2013-03-22 18:23:15
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Derivation
Classification msc 35L15
Classification msc 35L05
Related topic AdditionFormulasForSineAndCosine
Related topic SchrodingersWaveEquation