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In D.K.Cheng's Field and wave electromagnetics he states the following:

The phasor electric field intensity for a uniform plane wave propagating in the +$z$-direction is $$\mathbf{E}(z)=E_0e^{-jkz} $$ where $E_0$ is a constant, $j$ is the imaginary unit and $k$ is the wavenumber.

My question is, why is this a wave? Isn't a wave supposed to be time-dependent? I tried finding the real-notation of the field. $$\Re(\mathbf{E}(z))=E_0\cos(kz) $$

I get that when $z$ changes the electromagnetic field becomes a cosine function, but can the wavelike behavior also be coordinate dependent. I thought it should be time-dependent?

Can someone clarify?

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    $\begingroup$ Have you read section 7-7.2 (in my edition), "Time-Harmonic Electromagnetics"? - "Field vectors that vary with space coordinates and are sinusoidal functions of time can similarly be represented by vector phasors that depend on space coordinates but not on time." - The key word here is phasor. $\endgroup$ Nov 17, 2020 at 13:18
  • $\begingroup$ See here en.wikipedia.org/wiki/Phasor#Definition $\endgroup$ Nov 17, 2020 at 13:26

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The time dependent factor, $e^{j\omega t}$ has been omitted, to save clutter. Readers are supposed to supply it for themselves. The omission is usually made only when dealing with waves with the same $\omega$, or with the same wave at different points $z$ in space. In such cases, $e^{j\omega t}$ can be factored out of derivations and put in again at the end, if leaving it out causes worry!

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  • $\begingroup$ Can one say: "The electromagnetic field is a wave, because the field magnitude changes like a consine-wave as a function of the $z$-coordinate"? $\endgroup$
    – Carl
    Nov 17, 2020 at 13:24
  • $\begingroup$ I wouldn't say that. There's no indication of a time variation. It's an understood convention to leave out the time factor in the mathematical representation of the wave, but there is no such convention for a verbal explanation. $\endgroup$ Nov 18, 2020 at 16:56
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We hid the time dependence when we said we were using a phasor representation of the field.

A phasor is a complex number $A$ representing a time dependent quantity $|A|\cos(\omega t+\angle{A})$, where $\omega$ is the previously defined angular frequency at which the instantaneous value of the quantity varies.

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