Doppler Effect Intuition and Equation

This video introduces the Doppler effect, provides some applications and intuition behind it, and finally introduces the governing equation. Below is a loose script:

Time to discuss the Doppler Effect
This is a fairly well known effect you may have heard of and an effect you have certainly witnessed whether or not you realized it
It's very important in applications from the radar gun the police use to discofer your speed, but perhaps the most profound discovery based on the Doppler effect was that the universe had a beginning. The 'red shift' of distant objects in the universe, due to the Doppler effect, is the most commonly understood reason scientists believe the universe had a beginning, the big bang, which is kind of a big deal.
But speaking of such, the TV show big bang theory had this little clip on the doppler effect, which is worth checking out:
So perhaps you 'get it' and that explanation was sufficient for you, but if not, let's take a closer look at what sheldon was talking about, by first considering what that sound was.
Car:
So here is car honking its horn whilst driving by an observer
Notice the sound changed as the car approached and passed right?
It was higher pitched on the approach, then lower as it left
Let's look at why that's the case with a little cartoon below it
So the car pulls up and begins honking its horn
The sound waves travel down
And they reach the observer on the side of the road, denoted by the ear.
Now as the car approaches, you can see the sound waves get bunched up, which makes the observed frequency higher, hence the higher pitch
And as the car leaves, the sound waves get more spread out, leading to a lower perceived frequency, hence the lower pitch.
So that's the basic concept of how motion of a source can affect the perceived frequency
Let's look at this in 2D now
So say we have a source here of electromagnetic radiation
This source emit equally in all directions
Observers at any place around this will perceive the same emitted frequency
Now, say this source is moving
In this case you can see the waves bunch up on the side it's moving toward
Wavelength gets smaller
Frequency gets higher
And spread out on the side its moving away from
Wavelength gets larger
Frequency gets lower
Same concept as the car earlier.