What is the difference between the equations for Doppler effect and relativity?

[Xeniebop]

Homework Statement

Hello, I'm researching about relativity and Doppler effect, most text and book gives the equation for it as

(term with square root is on the numerator, and the term w/o is on the denominator)


while on some text, has the reverse of the fraction part (which is the term with the square root is on the numerator

Are they both equal to each other? I can't seem to express/derive the other one from the other one.
Can you explain what are the difference in both equations.

thanks

 

[Meir Achuz]

The two expressions differ in how the unit vector giving the direction of the EM wave is defined. Your first formula is for the direction in the source frame, and is used for stellar aberration. Your second formula defines the unit vector in the observer's frame, and is used for the Doppler effect. A good discussion of this is on page 456 of "Classical theory of Electric and Magnetic Fields" by Good and Nelson.