Noesis
1.6.21Oscillations & Waves

Doppler effect — all cases - source moving, observer moving, both

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🇮🇳 Yeh note Hinglish mein padho →

WHY does the frequency change at all?

A sound source emits a crest, then waits one period TT before emitting the next. In that time the first crest has travelled outward. The frequency you hear is just:

f′=number of crests passing youtime=speed of crests relative to youdistance between crests (the wavelength you meet)f' = \frac{\text{number of crests passing you}}{\text{time}} = \frac{\text{speed of crests relative to you}}{\text{distance between crests (the wavelength you meet)}}

So we only ever need to track two things:

  1. The wavelength in the medium (changed if the source moves — it chases its own waves).
  2. The speed at which crests sweep past you (changed if the observer moves).

Everything below is bookkeeping of these two effects. vv = speed of wave in medium (e.g. sound ≈340\approx 340 m/s). The medium is the referee — all speeds are measured relative to it.

Deriving each case from scratch

Case 1 — Observer moves, source still

The wavelength in the medium is unchanged: λ=v/f\lambda = v/f (the source isn't chasing anything). But if the observer moves toward the source at vov_o, the crests sweep past at speed v+vov+v_o (you run into them faster).

f′=relative speed of crestsλ=v+vov/f=f v+vovf' = \frac{\text{relative speed of crests}}{\lambda} = \frac{v+v_o}{v/f} = f\,\frac{v+v_o}{v}

Why this step? Frequency = (how fast crests come at you) ÷ (spacing between them). The spacing λ\lambda is fixed; only the approach speed changed.

Moving away: replace +vo→−vo+v_o \to -v_o.

Case 2 — Source moves, observer still

Now the source chases its own waves. In one period TT it moves vsTv_s T toward you, so the next crest is emitted closer — the wavelength in front is compressed:

λ′=λ−vsT=vf−vsf=v−vsf\lambda' = \lambda - v_s T = \frac{v}{f} - \frac{v_s}{f} = \frac{v-v_s}{f}

Why this step? Each crest is born vsTv_s T nearer than the last, so the gap shrinks by exactly that amount.

The crests still travel at vv in the medium (the medium sets wave speed, not the source). So:

f′=vλ′=f vv−vsf' = \frac{v}{\lambda'} = f\,\frac{v}{v-v_s}

Source receding: −vs→+vs-v_s \to +v_s.

Case 3 — BOTH move (the master formula)

Combine: observer changes the approach speed of crests; source changes the wavelength. Stack them:

f′=f v±vov∓vs\boxed{f' = f\,\frac{v \pm v_o}{v \mp v_s}}

Figure — Doppler effect — all cases - source moving, observer moving, both

Worked Examples

Common Mistakes (steel-manned)

Recall Feynman: explain to a 12-year-old

Imagine throwing a ball every second to a friend. If your friend runs toward you, they catch balls more often → "higher frequency." If you walk toward your friend while throwing, each ball leaves from a spot closer to them, so the balls bunch up — they still catch them more often. Sound crests are the balls. Moving closer in any way = faster catching = higher pitch. Moving apart = slower = lower pitch (the ambulance dropping in tone as it passes).

Flashcards

Master Doppler formula for sound
f′=f v±vov∓vsf' = f\,\dfrac{v\pm v_o}{v\mp v_s}, signs chosen so approach raises pitch.
Why does a moving SOURCE change frequency?
It chases its own waves, compressing/stretching the wavelength: λ′=(v∓vs)/f\lambda'=(v\mp v_s)/f.
Why does a moving OBSERVER change frequency?
It changes the speed crests sweep past: relative crest speed =v±vo=v\pm v_o, while λ\lambda stays fixed.
Which speed goes in the numerator?
The observer's speed vov_o (always top).
Which speed goes in the denominator?
The source's speed vsv_s (always bottom).
Sign when observer moves toward source
+vo+v_o in numerator (raises f′f').
Sign when source moves toward observer
−vs-v_s in denominator (raises f′f').
What is the reference frame for all speeds?
The medium (e.g. still air); the medium sets the wave speed vv.
What happens when vs→vv_s \to v?
Denominator →0\to 0, f′→∞f'\to\infty: wavefronts pile up into a shock wave / sonic boom.
Observer still, source receding at vsv_s: f′f'?
f′=f vv+vsf' = f\,\dfrac{v}{v+v_s} (lower pitch).
Is source-toward equal to observer-toward at same speed?
No — different mechanisms; they only approximately agree at low speeds.
Effect of wind speed ww toward observer?
Replace v→v+wv \to v+w in numerator and denominator (medium drifts).

Connections

  • Wave speed in a medium — sets vv; source can't change it.
  • Wavelength and frequency relation — λ=v/f\lambda = v/f underpins the whole derivation.
  • Sonic boom and shock waves — the vs≥vv_s \ge v limit.
  • Relative velocity — observer-frame crest speeds.
  • Doppler effect of light — relativistic version, no medium, fully symmetric.
  • Beats — what you hear when two Doppler-shifted tones combine.

Concept Map

defined by

needs

needs

compresses

changes

gives

gives

combined into

combined into

applied via

Frequency heard f prime

f prime = crest speed / wavelength met

Wavelength in medium

Speed crests sweep past you

Source moves

Observer moves

Case 1 observer only

Case 2 source only

Master formula both

Sign rule approach raises f prime

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Doppler effect ka core idea simple hai: jo pitch (frequency) tum sunte ho, wo depend karti hai ki wavefronts (crests) tumhare kaan tak kitni jaldi pahunch rahe hain. Agar tum source ke paas ja rahe ho, ya source tumhare paas aa raha hai — dono case me crests jaldi-jaldi aate hain, to pitch high ho jaati hai. Door jaane par opposite — pitch low. Yahi reason hai ambulance pass karte waqt uski siren ki tone pehle high lagti hai, phir achanak low ho jaati hai.

Do alag-alag mechanism hain, isko yaad rakhna. Jab observer move karta hai, wavelength to wahi rehti hai, par crests tum tak aane ki speed badal jaati hai — isiliye observer ka speed vov_o upar (numerator) aata hai: v±vov \pm v_o. Jab source move karta hai, wo apni hi waves ko chase karta hai, isse wavelength compress/stretch ho jaati hai — isiliye source ka speed vsv_s neeche (denominator) aata hai: v∓vsv \mp v_s. Isi liye master formula hai f′=f v±vov∓vsf' = f\,\frac{v\pm v_o}{v\mp v_s}.

Sign yaad rakhne ka shortcut: paas aana matlab pitch high. To sign aise chuno ki paas aane par f′f' bada ho jaaye. Observer toward → +vo+v_o top me; source toward → −vs-v_s bottom me. Bas itna. Char alag formula ratne ki zarurat nahi — ek formula aur ye logic kaafi hai.

Ek important baat: saari speeds medium (hawa) ke respect me measure hoti hain, kyunki medium hi wave ki speed vv decide karta hai. Aur agar source ki speed vsv_s sound ki speed vv ke barabar ho jaaye, to denominator zero ho jaata hai aur f′f' infinite — yahi sonic boom (shock wave) banta hai. Exam me ye limit aksar pucha jaata hai!

Go deeper — visual, from zero

D1FoundationsEvery symbol from zero · 1,686 wordsD2Visual walkthroughThe derivation in pictures · 1,932 wordsD3Worked examplesEvery scenario covered · 4,263 wordsD4ExercisesGraded, full solutions · 2,891 wordsD5Question bankConcept traps, quick reveals · 2,637 words

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Connections

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