Longitudinal frequency shift from speeds

General relativistic Doppler effect that is classical Doppler effect with relativistic coefficient. This law is not used for actual calculations because relativistic effects are not visible for acoustic waves. And for electromagnetic waves it is hard to define velocity relative to medium. This law is used to show the connection between classical and relativistic Doppler laws.

Notes:

1. The speed of light is used in this law, so replace it with the correct value of the speed of light in the medium.

2. When wave velocity is getting close to speed of light, we are no longer having observer and source velocities, but relativistic relative velocity.

Conditions:

1. Motion is in 1D space.

2. Coordinate system is at rest with respect to the medium of the wave, or any coordinate system for electromagnetic wave in vacuum.

3. Medium is not fixed in space.

4. Source and observer velocities are no greater than wave velocity.

5. The source and observer velocities are collinear.

observer_frequency

Observer temporal_frequency.

Symbol:

f_o

Latex:

\(f_\text{o}\)

Dimension:

frequency

source_frequency

Source temporal_frequency.

Symbol:

f_s

Latex:

\(f_\text{s}\)

Dimension:

frequency

source_speed

Source speed, positive when moving away from observer and negative otherwise.

Symbol:

v_s

Latex:

\(v_\text{s}\)

Dimension:

velocity

observer_speed

Observer speed, positive when moving away from source and negative otherwise.

Symbol:

v_o

Latex:

\(v_\text{o}\)

Dimension:

velocity

wave_speed

phase_speed of the wave.

Symbol:

v

Latex:

\(v\)

Dimension:

velocity

law

f_o = f_s * (1 - v_o / v) / (1 + v_s / v) * sqrt((1 - (v_s / c)^2) / (1 - (v_o / c)^2))

Latex:

\[f_\text{o} = \frac{f_\text{s} \left(1 - \frac{v_\text{o}}{v}\right)}{1 + \frac{v_\text{s}}{v}} \sqrt{\frac{1 - \left(\frac{v_\text{s}}{c}\right)^{2}}{1 - \left(\frac{v_\text{o}}{c}\right)^{2}}}\]