Let’s consider the d’Alembert’s solution

u(x, t) := [f(x−ct) + f(x+ct)] + ∫ x−ctx+ctg(s) ds

(1)

of the wave equation in one dimension in the special case when the other initial condition is

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]:

Thus we may write

Adding these equations and dividing by 2 yield

u(x, t) = [f(x−ct) + f(x+ct)] = ∑ n=1∞A n cos sin ,

(3)

which indeed is the solution of D. Bernoulli in the case g(x) ≡ 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

u(x, 0) := f(x) ≡ sin .

(4)

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

u(x, t) = cos sin .

(5)

Remark. In the case of quantum systems one has Schrödinger’s wave equation whose solutions are different from the above.