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12. Expression for apparent frequency due to doppler effect

Let $S$ and $L$ represent the source and listener, respectively, moving at velocity

V_{S}

and

V_{L}

. Consider the case of the source and the listener approaching each other as shown in the below diagram.

Source and the listener

The apparent frequency will be greater than the actual source frequency as the distance between them decreases.

Let $n$ and $n'$ represent the frequency of the sound produced by the source and the sound heard by the listener. The apparent frequency n' is then expressed as

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 the below 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) $V_{S}$ and $V_{L}$ become opposite 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 velocity of the source and the listener. b) $V_{S}$ 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. The listener follows the source.	a) Apparent frequency depends on the velocity of the source and the listener. b) $V_{S}$ and $V_{L}$ 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 decreases. b) Apparent frequency is more than actual frequency. c) $V_{S}$ $=$ $0$ 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 increases. b) Apparent frequency is less than actual frequency. c) $V_{S}$ $=$ $0$ 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) $V_{L}$ $=$ $0$ 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) $V_{L}$ $=$ $0$ in $case-2$.	$n^{"} = (\frac{V}{V + V_{S}}) n$

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12. Expression for apparent frequency due to doppler effect

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