Why Do Equations for Doppler Effect Differ for Moving Source and Observer?

[Terocamo]

I have just read about the principle of Doppler effect.
However there is a point which seemed a bit tricky.
According to the book, when the source of wave is moving the apparent frequency to a stationary observer is given by the equation:
f'=true frequency*speed of wave/(speed of wave + speed of source)

On the other hand, if an observer is moving towards a stationary source, the frequency is given by:
f'=original frequency * (speed of wave - speed of observer)/(speed of wave)

But referring to the basic principle of displacement, the velocity of the observer and that of source is a relative value. So theoretically there is no difference whether is the source moving or the observer moving, because its only relative motion.
However, with respect to the equation, they are not exactly equal (very close though).

What I want to know is the reason behind this or is there something that I missed?

 

[VanOosten]

the speed at which the two are approaching each other is the same in both reference frames, therefor the equation is the same in both cases. no?

 

[billslugg]

In the first equation, if you consider the speed of the source to be a negative value it will work out just fine.
In the second equation, if you also consider the speed of the observer to be a negative value (which is then multiplied by -1) it too will also work out fine.

 

[Terocamo]

Fact is the equation is not the same.
The apparent frequency of a moving source differs from the that of a moving observer. This is what I am puzzled by the two equation.

To be more detail, the two equation is based on two different approach.
For a moving source, the apparent wavelength is first considered. But for a moving observer, the apparent velocity of the traveling wave is first considered.
These give rise to two different equation as stated in the first post. This two equation give different values.

 

[Terocamo]

In the first equation, if you consider the speed of the source to be a negative value it will work out just fine.
In the second equation, if you also consider the speed of the observer to be a negative value (which is then multiplied by -1) it too will also work out fine.

My true confusion is not about the direction of the velocity vector. It is about the phenomenon that the two equation give two very close but not the identical (which I think should be) regardless that there is no difference whether the source is moving or the observer is moving.

 

[Doc Al]

But referring to the basic principle of displacement, the velocity of the observer and that of source is a relative value. So theoretically there is no difference whether is the source moving or the observer moving, because its only relative motion.

That would be true if there was no medium, as is the case for light in a vacuum. But here the speed of the wave is with respect to the air (presumed stationary). The speed of the wave with respect to the observer depends on whether the observer moves with respect to the air.