Notes on Relativity and Cosmology - Physics Department, UCSB

84 CHAPTER 3. EINSTEIN AND INERTIAL FRAMES
Receiver
Source
Light Ray
Light
Pulses or
Crests
For comparison, the Newtonian formula is
( ) c + vS
τ R = τ S (3.7)
c + v R
where c is the speed of the wave relative to the medium that carries it (i.e.,
sound in air, waves on water), v S is the velocity of the source relative to
the medium, and v R is the velocity of the receiver relative to the medium.
[For fun, you might try deriving that one, too ...]
3-14. By expanding equations (3.6) and (3.7) in a Taylor series, show that the
Newtonian and relativistic effects agree to first order in v c = vS−vR
c
. This
means that engineers who design radar devices do not in fact need to understand
relativity. You can do this using the expansions from problem
(2-2). Note that “to first order” means that we ignore any terms proportional
to v 2 , vR 2 , v2 S or higher powers of v, v R, v S .