Doppler Shift Formula

When the speed of the source and receiver is slower than the velocity of the waves in the medium, the relationship between the observed frequency (f) and the emitted frequency (f0) is described by the Doppler shift or Doppler effect formula.

f = [(c ± v_r)/ (c ± v_s)] f₀

where,
“c” is the velocity of waves in the medium,
“v_r” is the velocity of the receiver relative to the medium (if the receiver is moving towards the source, then “v_r” is added to c, or if the receiver is moving away from the source, then “v_r” is subtracted from c.)
“v_s” is the velocity of the source relative to the medium (if the source is moving away from the receiver, then “v_s” is added to c, or if the source is moving towards the receiver, then “v_s” is subtracted from c.) 
“f” is the observed frequency, and
“f₀” is the emitted frequency.

Case 1: When the Source is Moving Toward the Observer at Rest 

If the source is moving toward the observer at rest (v_r = 0) relative to the medium and the source is emitting waves with an actual frequency of f₀, then the frequency at which the observer detects waves is given as follows:

f = f₀

where,
“c” is the velocity of waves in the medium,
“v_r” is the velocity of the receiver,
“f” is the observed frequency, and
“f₀” is the emitted frequency.

Case 2: When the Source is Moving Away From the Observer at Rest 

If the source is moving away from the observer at rest (v_r = 0) relative to the medium and the source is emitting waves with an actual frequency of f₀, then the frequency at which the observer detects waves is given as follows:

f = f₀

where,
“c” is the velocity of waves in the medium,
“v_r” is the velocity of the receiver,
“f” is the observed frequency, and
“f₀” is the emitted frequency.

Case 3: When the Observer is Moving Toward the Source at Rest

If the observer is moving toward the source at rest (v_s = 0), then the equation for observed frequency is given as follows:

f = [(c + v_r)/c] f₀

where,
“c” is the velocity of waves in the medium,
“v_r” is the velocity of the receiver,
“f” is the observed frequency, and
“f₀” is the emitted frequency.

Case 4: When the Observer is Moving Away From the Source at Rest

If the observer is moving away from the source at rest (v_s = 0), then the equation for observed frequency is given as follows:

f = [(c − v_r)/c] f₀

where,
“c” is the velocity of waves in the medium,
“v_r” is the velocity of the receiver,
“f” is the observed frequency, and
“f₀” is the emitted frequency.

Doppler effect or Doppler shift phenomenon was described in 1842 by an Austrian physicist, Christian Doppler, and it is named after him. The Doppler effect or Doppler shift is a change (increase or decrease) in the frequency of a wave as the source and the observer move (towards or away from) each other relative to the medium. Based on the direction of the source and the observer and the magnitudes of their velocities, the observed frequency can be less or more than the source frequency. For example, the pitch of the sound of an ambulance siren changes as it passes us. It happens because of the relative velocity between the source and the observer. When the ambulance is approaching us, the relative velocity is negative, and the relative velocity is positive when it is moving away. So, that is the reason behind the difference in the pitch of the sound of the ambulance siren while it is approaching and while it is moving away. The Doppler effect applies to all types of waves, including sound and light. The Doppler effect is responsible for the Blue Shift or Red Shift phenomenon observed in light waves. The Doppler effect is used in various fields such as radar, astronomy, satellite communication and navigation, medical imaging, etc.