How do we know that the solution to the electromagnetic wave equation is sinusoidal?

Question

Since the equation is in the form of d^2u/dx^2 = c^2 * d^2u/dt^2, which has the general solution u = F(x-ct) + G(x+ct), how do we know that the solution is sinusoidal?

Answer 1

it isn't necessarily sinusoidal. As you have stated, the solution is any function of x +- ct.

Sinusoidal functions, however, are the eigenfunctions of the equation.

Answer 2

Those solutions are not sinusoidal. Plug in a*sin(bx+ct) and confirm that it is a solution for appropriate values of a and b. This is a solution with frequency c, called an eigenfunction. From Fourier analysis, any solution can be constructed from a sum of such solutions.

Answer 3

Actually the solution is not always sinusoidal. It depends on the boundary conditions. In the situation where there are no boundaries then the solution is sinusoidal. You can see this by doing a Laplace transform in t and Fourier transform in x, solve for u and invert the transform.