# Why is this electromagnetic field a wave?

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?

• 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. Nov 17, 2020 at 13:18
• Nov 17, 2020 at 13:26

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!
• 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"?
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.