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I'm reading "Introduction to Electrodynamics" by David J. Girffiths and the following assumption is confusing me:

We have an EM wave inciding on a surface,

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Then, when applying the boundary conditions, the following is done:

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What confuses me, regarding the highlighted part, is, why should those exponentials be equal? Isn't it sufficient for them to be constant at $z = 0$?

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  • $\begingroup$ If they are equal for all the times, is it not a constant when z = 0? $\endgroup$ Jan 25 '19 at 19:06
  • $\begingroup$ What have you tried? Suppose $\alpha e^{iax} + \beta e^{ibx} = \gamma e^{icx}$ for all $x$ and for nonzero $\alpha, \beta, \gamma, a, b, c$. What can we say about $a, b, c$? $\endgroup$ Jan 25 '19 at 19:14
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I see that "$\exp{(ikx)}$" gave the idea in commentary while I was typing. Simply to complete (with a poor english) :

With $z = 0$ and taking $y = 0$ you have a linear combination $A{{e}^{i{{k}_{1x}}x}}+B{{e}^{i{{k}_{2x}}x}}=C{{e}^{i{{k}_{3x}}x}}$ equality valid for all $x$, with $A$, $B$ and $C$ different from $0$.

If you agree that $\left\{ {{e}^{i\alpha x}} \right\}_{\alpha}$ form a linearly independent set of functions, the only solution is that the argument of the exponential must be equal.

For the proof of linear independence, I think mathematical arguments can easily found on the net. Hope he can help!

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