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In reading about electrodynamics and quantum mechanics, I have come across the concepts of mode functions and mode solutions. For example, in Ballentine's Quantum Mechanics, the author discusses the mode functions of the EM field (link) when discussing the solution to the wave equation:

The solution of the wave equation (19.5) can be facilitated by representing the electric field as a sum of mode functions, $\textbf{u}_\textit{m}(\textbf{x})...$

Here are my questions and confusions:

  1. How are mode functions related to normal modes of vibration?
  2. How are mode functions related to the general solution of a differential equation?
  3. How would one find the mode functions (in general) for a specific wave equation?
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I think the explanation is right there in the text, but it might take a bit of translation. First, notice that the time dependence of a mode field is $e^{-i\omega t}$. A mode has a single frequency. Next notice that the "mode function" obeys the eigenvalue equation and subsidiary conditions which I take to mean boundary conditions. A mode function is a shape that satisfies both the eigenvalue equation and the boundary conditions. Multiply a mode function by it's associated time dependence, and you have a solution for the field.

The usefulness of mode functions is that any solution, with any time dependence and any allowable spatial distribution can be expressed as a sum of the mode fields.

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