Doppler effect of light Doppler effect for light is created for relative motion . But shouldn't light always approach us at the same velocity 'c' regardless of the relative speed ?Then how does Doppler effect work ?
 A: Special relativity is a generally accepted and experimentally confirmed physical theory that in its original form has two postulates:


*

*the laws of physics are invariant in all inertial frames of reference

*speed of light in vacuum is the same for all observers, regardless of the motion of the light source or observer
You are correct, light approaches us at the same c speed regardless of our relative speed.
The relativistic Doppler effect is the change in frequency (and wavelength) of light, caused by the relative motion of the source and the observer. It includes time dilation and Lorenz symmetry.

Light ahead of the observer is blueshifted, and light behind the observer is redshifted.
It is very important that you need to use time dilation in relativistic Doppler shift. 
This is because clocks on the receiver are time dilated relative to clocks at the source.
$$\frac{f_s}{f_r} = \sqrt{\frac{1 + \beta}{1 - \beta}}$$
Is called the Doppler factor of the source relative to the receiver.
https://en.wikipedia.org/wiki/Special_relativity
https://en.wikipedia.org/wiki/Relativistic_Doppler_effect
A: A wave is travelling oscillation having frequency from oscillation that depends upon medium, source and relative speed, while wavelength is feature of a wave dependent upon source, similarly amplitude is dependent on source. These three are characterstics of a wave and energy of a wave is dependent on them. Any change in energy of a wave is indication of change in any one of these parameters but travelling wave is independent of source, so there is only possibility of change in frequency. This change in frequency of a wave is known as doppler shift or effect, and that depends upon change in relative speed or change in medium.
Now we come to important claim that according to theory of relativity, doppler shift can’t be possible or theory doesn’t show any shift in frequency. Reason is obvious that theory of relativity claims that speed of light is relative invariant, so there is no possible shift due to relative motion between source and observer. Theory of relativity claims transverse doppler shift also but that is not possible because there is always inclination as in stellar abberation. I assume that concept of time dilation came from doppler shift, but this shift is change in duration or interval due to change in speed not the flow of time itself.
Suppose a light wave is travelling with speed, $c$ having angular frequency $\omega$ and wavevector $k$. As phase $\phi=kx-\omega t$ of a wave is determining its position in space and time, which remains constant for any value of time and corresponding value of distance. For any two instances of either time or distance, there must be no phase difference,
$\phi_2-\phi_1=k(x_2-x_1)-\omega(t_2-t_1)=0\tag*{}$
$\implies \frac{\Delta x}{\Delta t}=\frac{\omega}{k}=c\tag1$
From classical relativity of addition of velocities, speed of given wave emitting from a source moving with relative speed $v$ in the direction of wave is, then from $(1)$
$k(\Delta x+v\Delta t)=k(c+v)\Delta t\Rightarrow \frac{\Delta x}{\Delta t}=c\tag2$
where $\ k(c+v)=\omega'$
If the wave from source moving with relative speed $v$, approaches to observer then from $(1)$,
$k(\Delta x)=k(c+v)\Delta t\Rightarrow \omega'=\omega+\Delta{\omega}\tag3$
where $\ k\frac{\Delta x}{\Delta t}=\omega'$ and $kv=\Delta{\omega}$
Now from theory of relativity and using (1), the result for condition (2) and (3) is given in (4) and (5) as following,
$k\gamma(\Delta x+v\Delta t)=\omega\gamma(\Delta t+\frac{v\Delta x}{c^2})\Rightarrow\frac{\Delta x}{\Delta t}=c\tag4$
$k(\Delta x)=\omega(\Delta t)\Rightarrow \frac{\Delta x}{\Delta t}=c\tag5$
From (3), it is clear that classical relativity inherently consists of doppler effect while it’s equivalent (5) for theory of relativity shows no doppler shift. Reason is obvious, shift occur when there is change in speed of wave, if theory insists on constancy of speed in relative motion there is no doppler effect.
$\quad$Addendum
This is added to show that how doppler shift's expression used in theory of relativity is incorrect.
Many authors use time dilation for transverse direction in expression for longitudnal doppler shift which is incorrect for three reasons. First is that meaurement is not for relative frame, so there is no relative motion observed by observer thus no use of lorentz transformation. Second as said above, there is no transverse shift because there is no relative effect in transverse direction, so all apparently transverse are inclination from longitudnal direction. As in stellar abberation, there is no same angular inclination of a star for apparent and actual position, star above head is behind from exact overhead position.
Third reason is that one can't use motion of source because one is not measuring how it appears in relative frame of source. Also this assumed speed of relative speed prior to observer doppler shift which is used for measuring relative speed of source. Some authors equate component by component of phase of a wave and relativistic phase, where we already assumed that phase of a wave and its relative part is constant. If a vector remains invariant by transformation then their components are not equal if there is transformation.
