Complex number representation of a wave

Question

There are some aspects to waves I am confused, for instance in Chapter 11. Fraunhofer Diffraction.

The incoming electric fields can be partially expressed as $e^{i(kr-\omega t)}$. I have two questions regarding this:

1. What does $\,(kr-\omega t)\,$ indicate, and how do we know it is these values, why not just use $\omega t?$

2. Why is there suddenly an introduction of an imaginary component of the oscillation? What is the significance of the imaginary component of the incoming oscillation, and why can we express them in this way?

I will appreciate some layman's term or down to earth explanation to 2. but some detailed explanation to 1.

Answer 1

We write $kr$ to show how the wave changes through space. For example you can fix $t$=constant, so the part $e^{i\omega t}$=constant, so you can see changes through space just shifting the "$x$" - space component.

"$kx-\omega t$" express the whole phase, so you take a real part if this exponent, and after multiplying the amplitude you get the value of vector if your field in this moment $t$ and this coordinate $x$.

When we write wave in complex form, we assume, that after all mathematical manipulations we take a real part of it. So you can write everywhere Re[...], and it would be the same