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_r 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_r are relative to the medium.

Moving Transmitter, Stationary Receiver

Now let's assume that the transmitter is moving with velocity v_t 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_t 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_t and v_r, 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_r and v_t are small compared to v_w, these equations can be simplified to:

Equation 14:

Equation 15:

Note that v_t and v_r 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. The velocity v_t of the transmitter relative to the receiver (and towards it) will usually be very much smaller than the velocity v_w of the waves (which is approximately 300000 km/s in a vacuum), so 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_t 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_r=0 in Equation 14 and Equation 15, respectively.

For electromagnetic waves we will not consider the general case when v_t is not small compared to v_w. Relativistic effects (according to Einstein's Theory of Relativity) would be non-negligible in the general case.