Doppler Effect with Arbitrary Motion of Source
and Observer
Setup and Notation
Assume:
• A source emitting a wave with frequency fs .
• An observer moving relative to the source in a general direction.
• Let vs and vo denote the velocities of the source and observer, respectively.
• θs and θo are the angles that the source and observer’s velocities make
with the line connecting the source and observer.
Relative Velocity Components
The Doppler shift depends on the relative velocity between the source and ob-
server along the line connecting them, the radial component vrad , and the
transverse component vtrans .
Radial Component
The radial component of the relative velocity vrad between the source and ob-
server is given by:
vrad = (vo − vs ) · r̂
where r̂ is the unit vector pointing from the source to the observer.
Expanding this in terms of the angles θo and θs :
vrad = vo cos θo − vs cos θs
where vo = |vo | and vs = |vs |.
1
Transverse Component
The transverse component vtrans is given by:
vtrans = vo sin θo − vs sin θs
However, for most types of waves (e.g., sound), this transverse component does
not directly affect the Doppler shift. It only affects the Doppler shift for light
waves in the special theory of relativity (transverse Doppler shift).
General Doppler Shift Equation
The observed frequency fo can be found by adjusting the emitted frequency fs
according to the relative radial velocity vrad between the source and observer:
c ± vrad
fo = fs
c
where:
• c is the speed of the wave in the medium (e.g., the speed of sound in air
or the speed of light in vacuum).
• The sign depends on the relative motion: use + if the source and observer
are moving closer, and − if they are moving apart.
Substituting for vrad :
c + (vo cos θo − vs cos θs )
fo = fs
c
This is the general Doppler shift equation for a moving source and moving
observer in a plane with arbitrary velocities.
Special Case: Light Waves and Transverse Doppler
Effect
For light waves, relativistic effects come into play, especially due to the trans-
verse component of motion. The general relativistic Doppler effect equation
is: √
1 − vrel
c
fo = fs
1 + vrel
c
For transverse motion, vrad = 0, and the transverse Doppler shift for light is:
√
v2
fo = fs 1 − 2
c
This results in a redshift purely due to the transverse component of relative
motion.
