I would be very eager to have someone explain to me why it is justified to assume harmonic time dependence when seeking solutions to a wave equation. This is done many times in Jackson or Kittel. Isn't assuming harmonic time dependence in solving the wave equation not using part of the solution in solving the equation? I could appreciate if one is saying that one tries a separation of variables to solve the wave equation, and that one can show that a harmonic time dependence exists under separation of variables, but then the assumption made is that the resultant solution of separation variables has a completeness theorem to justify use of the procedure. This seems like a very convoluted way of saying "assuming there is a completeness theorem for the result of separation of variables" is = "assuming harmonic time dependence." Also, in Kittel, he seeks solutions for the wave equation in a non-magnetic isotropic medium at one point, and assumes a solution that has harmonic time dependence as well as harmonic dependence on the quantity of ( i K r) for i is the imaginary number, K is the wave vector dotted into r. I don't understand what gives the right to assume a solution involving the dot product of the wave vector and radial distance. How does this not limit the solution set obtained from solving the wave equation in this manner? If so, why would all other solutions be unimportant?