Derivation of Doppler Effect Equations

[Pseudo Statistic]

Hey,
I need a bit of help with the derivation of Doppler's equations as shown in the attached image. (From Fundamentals of Physics)
The paragraph where it states "Now let us again consider the situation..."
I don't understand why they use vt + vDt
I mean, the sound wave would initially have to travel a distance vt to where the observer was, but then the observer D would be moving to the left towards it at velocity vD...
And in a time t, the person would have already traveled a distance vDt to the left, so wouldn't that have to make it -vDt which would indicate that the sound wave has less distance to travel? Or am I totally missing the point?
And the same goes for the second derivation, wouldn't it be + in that case?

I don't know, I totally lack the understanding of what's going on...
I'd really appreciate it if someone could explain this a bit... as I'm not really accustomed to just remembering the formula.

Thanks for any help.

 

[Pseudo Statistic]

...bump...

 

[Doc Al]

Think this way: How far away is the wavefront if the observer meets the wavefront in time t? The total distance is the sum of how far the wavefront traveled (vt) plus how far the observer traveled (v_D t). (True, the wavefront doesn't have to travel the entire distance to reach the observer. If it did, it would take longer than just t.)

Just like if two people run towards each other, they cover more distance between them faster.