Why do we assume harmonic time dependence in wave equation solutions?

[Enjolras1789]

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?

 

[jasonRF]

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?

This is commonly done with linear wave equations of various sorts. If you prefer, you can think of it as taking the Fourier transform of the equation with respect to time. Of course Fourier analysis then allows you to synthesize any temporal function as a linear combination of complex exponentials.

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?

What 'gives them the right' is that there exist solutions of the form $e^{-i \omega t + i \mathbf{k \cdot r}}$, which they show by presenting the analysis that you are reading. These solutions are particularly interesting since functions of the form $f(\omega t - \mathbf{k\cdot r})$ are traveling waves, which are physically interesting. Furthermore, assuming $e^{-i \omega t + i \mathbf{k \cdot r}}$ allows you derive the dispersion relation $\omega = g(\mathbf{k})$ which gives physical insight into the phenomena you are studying. For example, the group velocity is given by $\mathbf{v}_g = \nabla_\mathbf{k} \omega$, where the gradient is with respect to $\mathbf{k}$.
And then, since these wave equations are linear, waves of arbitrary shape can be synthesized via Fourier analysis.

The main limitation of this kind of 'assume harmonic dependence' analysis is that it doesn't include transient effects. Including these effects gets complicated in a hurry. In Jackson's book there is some of this in the section on precursors.

Remember these are physics books, written to teach physics. The techniques they use are commonplace in their field, and they quickly lead to results that provide physical insight. Furthermore, these time-harmonic cases are often practical as well. These texts are not written to present a rigorous analysis of well-known differential equations, so don't mathematically justify every step. If they did, they would have no space left for the physics!

 

[vanhees71]

Another argument is that you can use harmonic-time-dependence solutions to build more general solutions via Fourier representations.