MHT-CET Physics · Sound

The Doppler Effect

A listener hears f(v ± v_o)/(v ∓ v_s): approach raises the pitch and recession lowers it, the observer's speed adds to or subtracts from the numerator and the source's from the denominator, so the same speed shifts the pitch more when the source moves.

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Why this matters

19 PYQs, 3 of them HARD. Fourteen apply the formula once — observer or source moving, the ratio of approaching and receding pitches, the speed that halves or triples the pitch. Five combine two steps: a whistle heard before and after it passes, a siren echoed from a wall back to the driver, a car accelerating away from a siren. Two cards.

Concept 1 of 2: The Doppler Formula

Write f′ = f(v + v_o)/(v − v_s) with both speeds positive when they bring source and observer closer; flip a sign for each one moving apart. An observer approaching at v/5 hears 6/5 of the pitch, 20% more. A source approaching at v/10 gives 10/9. The same speed does more when the source moves, because it squeezes the waves: at 50 m/s with v = 330, a moving source gives 330/280 = 1.18f, a moving observer 380/330 = 1.15f. A source receding at v halves the pitch; an observer must approach at 2v to triple it. Frequency rises, so the wavelength the observer measures falls.

Definition

$f' = f\,\dfrac{v \pm v_o}{v \mp v_s}$ : upper signs when approaching.
Observer toward at $\tfrac{v}{5}$ ⇒ +20%; source toward at $\tfrac{v}{10}$ ⇒ $\tfrac{10}{9}f$ ; both toward at $\tfrac{v}{10}$ ⇒ $\tfrac{11}{9}f \approx 1.22f$ .
Observer toward and away at $V_1$ , ratio 2 ⇒ $V = 3V_1$ .
Half the pitch: source receding at v. Triple the pitch: observer approaching at 2v.
Same speed: moving source shifts more than moving observer.

Doppler effect

f' = f\,\frac{v \pm v_o}{v \mp v_s}

Worked example

A source approaches a stationary listener at v/3 while the listener moves away at v/5. Apparent frequency?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 16th May Shift 2 · Q39Moderate

Example 1 · Sound · Doppler Effect — Moving Source and Observer

A source and listener are both moving towards each other with speed

\frac{V}{10}

(where

V

is speed of sound). If the frequency of sound note emitted by the source is

n

, then the frequency heard by the listener would be nearly

Treating moving source and moving observer alike

The observer's speed sits in the numerator, the source's in the denominator. At 50 m/s the moving source gives 330/280; the moving observer only 380/330.

Getting the sign for recession

Moving apart LOWERS the pitch: observer v − v_o on top, source v + v_s underneath.

Drill 13 more on the doppler formula

Concept 2 of 2: Passing Sources, Echoes and Changing Speeds

A source that passes a listener gives v/(v − v_s) before and v/(v + v_s) after: the ratio (v + v_s)/(v − v_s) fixes its speed. A driver approaching a wall is a moving source for the wall, which then re-emits as a stationary source to the driver now moving as an observer: n(v + V₁)/(v − V₁). A vehicle accelerating away from a siren hears f(v − v_o)/v; once that ratio fixes v_o, the distance is v_o²/2a.

Definition

Passing source: before/after = $\dfrac{v + v_s}{v - v_s}$ (11 : 9 ⇒ $v_s = v/10$ ); a train at 20 m/s, 510 Hz, v = 320 ⇒ 544 and 480 Hz.
Pitch drops 30% as it recedes: $\dfrac{v}{v + v_s} = 0.7$ .
Echo from a wall to a driver at $V_1$ : $n\,\dfrac{v + V_1}{v - V_1}$ .
Accelerating away: $\dfrac{f'}{f} = \dfrac{v - v_o}{v}$ , then $s = \dfrac{v_o^2}{2a}$ (94% at 330 m/s, 2 m/s² ⇒ 98 m).

Echo from a wall

n' = n\,\frac{v + V_1}{v - V_1}

Worked example

A car at 20 m/s sounds a 500 Hz horn while approaching a wall (v = 340 m/s). Frequency of the echo heard by the driver?

Practice this conceptself-check · 1 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 4th May Shift 2 · Q43Moderate

Example 2 · Sound · Doppler Effect — Moving Source and Observer

A train sounding a whistle of frequency 510 Hz approaches a station at 72 km/hr. The frequency of the note heard by an observer on the platform as the train (1) approaches the station and then (2) recedes the station are respectively (in hertz) (velocity of sound in air = 320 m/s)

Using the source formula twice for an echo

The wall receives as a stationary observer and re-sends as a stationary source; the driver is a source on the way out and an OBSERVER on the way back. The two steps use different places in the formula.

Using one Doppler factor for before and after a pass

Approaching, the source term is v − v_s; receding, v + v_s. The before-to-after ratio is (v + v_s)/(v − v_s), not the square of either factor.

Drill 4 more on passing sources, echoes and changing speeds

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

The Doppler Formula
Doppler effect
$f' = f\,\frac{v \pm v_o}{v \mp v_s}$
Passing Sources, Echoes and Changing Speeds
Echo from a wall
$n' = n\,\frac{v + V_1}{v - V_1}$

Watch out for (4)

Treating moving source and moving observer alike
→ The Doppler Formula
Getting the sign for recession
→ The Doppler Formula
Using the source formula twice for an echo
→ Passing Sources, Echoes and Changing Speeds
Using one Doppler factor for before and after a pass
→ Passing Sources, Echoes and Changing Speeds

Test yourself on Sound

15 past MHT-CET questions from this chapter, timed at 14 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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Related notes

Atoms — the Doppler shift of spectral lines →