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My question is, whether this definition $E=E_0e^{-i\omega t}$ includes that it is a plane wave, since I am confused by the fact that we do not have any dependence on the position. So about what kind of electromagnetic waves are we talking if we use $E=E_0e^{-i\omega t}$?

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You're right that says nothing about position dependence. It only says that the (angular) frequency is $\omega$. The position dependence is hidden in $E_0$ and there is no way to guess without context what it might be, except that it has to satisfy the wave equation. – Michael Brown Aug 1 '13 at 9:30
up vote 1 down vote accepted

If you assume that $E_0$ depends on position, Michael Brown is right and there is no information about the wave's nature, except that we know its frequency. However, if you assume $E_0$ to be constant (independent of time and position) and $\omega$ to be nonzero, the expression is not a solution to the wave equation.

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thanks to both of you – Xin Wang Aug 1 '13 at 10:04

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