Whenever there is a relative motion between a source and a listener, the frequency of the sound heard by the listener is different from the original frequency of sound emitted by the source. This is known as “Doppler effect".
For simplicity of calculation, it is assumed that the medium is at rest, That is the velocity of the medium is zero. Let S and L be the source and the listener moving with velocities v and v respectively. Consider the case of source and listener moving towards each other. As the distance between them decreases, the apparent frequency will be more than the actual source frequency.
Source and listener moving towards each other
Let n and n’ be the frequency of the sound produced by the source and the sound observed by the listener respectively. Then, the expression for the apparent frequency n’ is
\(n' = (\frac{V + V_L}{V- V_S})\,n\)
Here, v is the velocity of sound waves in the given medium. Let us consider different possibilities of motions of the source and the listener. In all such cases, the expression for the apparent frequency is given in table
| Case No. |
Position of source and listener |
Note |
Expression for apparent frequency |
| 1. |
- Both source and listener move
- They move towards each other
|
(a) Distance between source and listener decreases.
(b) Apparent frequency is more than actual frequency. |
\(n' = (\frac{V+V_L}{V-V_s}) n\) |
| 2. |
- Both source and listener move
- They move away from each other
|
(a) Distance between source and listener increases.
(b) Apparent frequency is less than actual frequency.
(c)Vs and VL become opposite to that in case - 1. |
\(n' = (\frac{V - V_L}{V + V_s}) n\) |
| 3. |
- Both source and listener move
- They move one behind the other
- Source follows the listener
|
(a) Apparent frequency depends on the velocities of the source and the listener.
(b)Vs becomes opposite to that in case-2. |
\(n' = (\frac{V - V_L}{V-V_s})\, n\) |
| 4. |
- Both source and listener move
- They move one behind the other
- Listener follows the source
|
(a) Apparent frequency depends on the velocities of the source and the listener.
(b) Vs and VL become opposite to that in case - 3. |
\(n' = (\frac{V -+V_L}{V + V_s})\, n\) |
| 5. |
- Source at rest
- Listener moves towards the source
|
(a) Distance between source and listener increases.
(b) Apparent frequency is less than actual frequency.
(c)Vs = 0 in case -1. |
\(n' = (\frac{V +V_L}{V})\, n\) |
| 6. |
- Source at rest
- Listener moves away from the source
|
(a) Distance between source and listener decreases.
(b) Apparent frequency is more than actual frequency.
(c) Vs = 0 in case - 2. |
\(n' = (\frac{V -V_L}{V})\, n\) |
| 7. |
- Listener at rest
- Source moves towards the listener
|
(a) Distance between source and listener decreases.
(b) Apparent frequency is more than actual frequency.
(c) VL = 0 in case - 1 |
\(n' = (\frac{V}{V - V_s}) n\) |
| 8. |
- Listener at rest
- Source moves away from the listener
|
(a) Distance between source and listener increases.
(b) Apparent frequency is less than actual frequency.
(c) VL = 0 in case - 2. |
\(n' = (\frac{V}{V + V_s}) n\) |
Suppose the medium (say wind) is moving with a velocity W in the direction of the propagation of sound. For this case, the velocity of sound, ‘v’ should be replaced with (v + W). If the medium moves in a direction opposite to the propagation of sound, then ‘v’ should be replaced with (v – W).
