What is Doppler Effect? Obtain an expression for the apparent frequency of sound heard, when the source is in motion with respect to an observer at rest. Mention its applications.

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> [!TLDR]
> The Doppler Effect is the apparent shift in frequency due to relative motion between a wave source and observer; for a source moving toward a stationary observer, n' = nv/(v − vs).
>
> **Doppler Effect:**
> The Doppler effect is the phenomenon in which the frequency of a wave (sound, light, etc.) as detected by an observer appears to change (from the frequency actually emitted by the source) whenever there is relative motion between the source and the observer. The apparent frequency increases when the source and observer approach each other, and decreases when they recede from each other.
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> **Derivation — source moving, observer at rest:**
> Let the actual frequency of the source be $n$, wavelength $\lambda = v/n$ (where $v$ is the speed of sound in the medium), and let the source move with speed $v_s$ toward a stationary observer.
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> Consider the source emitting successive wave crests. In the time interval of one period $T = 1/n$ (the time between the emission of two successive crests), the source moves a distance $v_s T$ toward the observer. Because the source has moved closer by this amount before emitting the next crest, the distance between two successive crests (as measured toward the observer) is compressed:
> $$\lambda' = \lambda - v_s T = \dfrac{v}{n} - \dfrac{v_s}{n} = \dfrac{v - v_s}{n}$$
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> Since the observer is at rest, the crests reach the observer with speed $v$ (the normal wave speed in the medium), so the apparent frequency heard by the observer is:
> $$n' = \dfrac{v}{\lambda'} = \dfrac{v}{\dfrac{v - v_s}{n}} = \dfrac{n v}{v - v_s}$$
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> This shows $n' > n$: the apparent frequency (pitch) increases as the source approaches — hence the higher pitch heard as a car or train approaches.
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> If instead the source recedes from the stationary observer with speed $v_s$, a similar argument (with $\lambda$ effectively stretched) gives:
> $$n' = \dfrac{nv}{v + v_s}$$
> so $n' < n$: the apparent frequency decreases as the source moves away

Situation	Numerator ( $v \pm v_o$ )	Denominator ( $v \mp v_s$ )	Effect on $f'$
Observer moves toward source	$v + v_o$	$v$ (source stationary)	$f' > f$
Observer moves away from source	$v - v_o$	$v$	$f' < f$
Source moves toward observer	$v$	$v - v_s$	$f' > f$
Source moves away from observer	$v$	$v + v_s$	$f' < f$

Q.What is Doppler Effect? Obtain an expression for the apparent frequency of sound heard, when the source is in motion with respect to an observer at rest. Mention its applications.

Concept understanding — Doppler Effect in Sound

The Doppler Effect in Sound – From Intuition to Formula

Why does the pitch change? The intuition

The precise statement

The sign convention — the only tricky part

Worked example

What the formula does NOT cover

Q.What is Doppler Effect? Obtain an expression for the apparent frequency of sound heard, when the source is in motion with respect to an observer at rest. Mention its applications.

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