There are at least three significant errors in your attachment. Before getting into the details, note the graph you present is clearly incorrect, which provides a strong clue that there are errors. Note that it predicts that it's impossible to hear an ambulance siren approaching you. You say that the frequency of the sound as the source approaches you goes "beyond the hearing range" as the angle
. Actually the graph says it goes all the way to infinity. And that this happens even if V_s is very small. Consequently - your graph says that it's impossible to hear a car passing you at, say, 30 MPH. Obviously this is not correct.
So given the clearly incorrect results let's see where the errors in your math are. There are three that I see:
1. The relative velocity of the the source as seen by the observer is
. Your formula goes to infinity at
when clearly it should go to zero (as a source passes you for a brief instant it is neither approaching nor receeding - so it's relative velocity is 0). hence it's clear that the secant term is incorrect.
2. The velocity of the wave front is set by the properties of the medium (air, in the case of sound), NOT by the speed of the source. So your equation
is not right on two counts - first because the secant tem is incorrect and second because of your incorrect assumption that the velocity of the wave front is dependent on the velocity of the source.
3. Third error is in coming up with the function
You don't indicate why you think it appropriate to add together what you claim are the relative velocity and relative frequency.
Back in post #6 of this thread I had directed you to a web site that demonstrates the Doppler effect pretty clearly - did you look at at that, and if so do you have any questions about it? I had also posted a formula in post #8 for frequency as measured by the observer as determined by the source frequency and relative velocity - again, have you read it?