Doppler Shift Theory

Introduction

In this document formulas are derived for the Doppler shift in various cases.

Stationary Transmitter, Moving Receiver

Assume we have a medium in which waves are transmitted with a frequency of f₀ and propagate with velocity v_w relative to the medium. If the wave transmitter is at rest relative to the medium, the wavelength everywhere in the medium will be:

Equation 1:

If the receiver is moving with velocity v_Rx relative to the medium and in the direction of the transmitter, then the receiver's velocity relative to the waves will be:

Equation 2:

Thus, the received waves will have the apparent frequency:

Equation 3:

By substituting Equation 1 and Equation 2 into Equation 3, we obtain the following expression for the apparent frequency:

Equation 4:

This can be rewritten to express the relative Doppler shift:

Equation 5:

Keep in mind that the velocities v_w and v_Rx are relative to the medium.

Moving Transmitter, Stationary Receiver

Now let's assume that the transmitter is moving with velocity v_Tx relative to the medium and towards the receiver, which is at rest relative to the medium. The wavelength in front of the transmitter will be reduced by the distance it travels during one wave period T₀ (=1/f₀), so the wavelength becomes:

Equation 6:

The received waves will therefore have the apparent frequency:

Equation 7:

By substituting Equation 6 into Equation 7, we obtain the following expression for the apparent frequency:

Equation 8:

And the relative Doppler shift becomes:

Equation 9:

When v_Tx is small compared to v_w, Equation 8 and Equation 9 become similar to the stationary transmitter case (Equation 4 and Equation 5) as shown below:

Equation 10:

Equation 11:

Moving Transmitter, Moving Receiver

If both the transmitter and the receiver are moving relative to the medium and towards each other with the velocities v_Tx and v_Rx, respectively, relative to the medium, then the wavelength λ will be described by Equation 6, and the receiver's velocity v relative to the waves will be described by Equation 2. For the receiver the waves will therefore have the apparent frequency:

Equation 12:

and the relative Doppler shift:

Equation 13:

When v_Rx and v_Tx are small compared to v_w, these equations can be simplified to:

Equation 14:

Equation 15:

Note that v_Tx and v_Rx should have the same sign when they are pointed in opposite directions. v_w should always be positive.

Electromagnetic Waves

For electromagnetic waves it is not necessary to relate the motion of the transmitter, receiver, and waves to a medium. Only the velocities relative to the receiver are important. If the velocity v_Tx of the transmitter relative to the receiver (and towards it) is much smaller than the velocity v_w of the waves (which is 299,792,458 m/s in a vacuum), we can reuse Equation 10 and Equation 11, regardless of whether the receiver is moving or not:

Equation 16:

Equation 17:

Remember that in this case v_Tx is the transmitter's velocity relative to the receiver - not relative to the medium. Equation 16 and Equation 17 could of course also be obtained by setting v_Rx=0 in Equation 14 and Equation 15, respectively.

If the transmitter velocity v_Tx is not small compared to the electromagnetic wave velocity v_w, relativistic effects (according to Einstein's Theory of Relativity) will be non-negligible and need to be taken into account. In this case, time dilation will reduce the transmitted frequency f₀ in Equation 8 by the Lorentz factor γ when observed from the receiver's frame of reference (before accounting for the Doppler shift). The Lorentz factor is determined by the transmitter's velocity v_Tx relative to the receiver as follows:

Equation 18:

Replacing f₀ in Equation 8 with the time dilated value f₀/γ, we get the combined result of time dilation and Doppler shift:

Equation 19:

Equation 20: