Doppler effect — all cases - source moving, observer moving, both: Visual walkthrough

1.6.21 · D2 · Physics › Oscillations & Waves › Doppler effect — all cases - source moving, observer moving

Hume sirf do ideas chahiye, aur dono ko hum earn karenge:

Wavelength — do paas-paas crests ke beech ki doori.
Crest-arrival speed — wo crests kitni tezi se tumhare kaan tak pahunchti hain.

Frequency jo suno $=$ (arrival speed) $\div$ (wavelength jo tum meet karo). Baaki sab bookkeeping hai.

Step 1 — "Wavefront" hota kya hai

KYA HAI. Ek sound source (horn, chidiya, baja hua ghanta) hawa ka pressure baar baar upar-neeche karta hai. Jab bhi pressure peak karta hai, woh peak bahar ki taraf ek expanding sphere ke roop mein phailti hai — ek crest. Ise ek circle ki tarah draw karo. Har period $T$ mein ek naya circle paida hota hai.

YE YAHAN SE KYUN SHURU KAREIN. Koi bhi symbol aane se pehle hume us cheez ko picture karna hoga jo count ho rahi hai. Jo pitch tum sunte ho woh literally har second mein kitne circles tumhare kaan se takrate hain — wahi asli frequency $f$ hai, aur $T = 1/ f$ janmon ke beech ka intezaar hai.

PICTURE. Neeche, ek still source (magenta dot) ne $0, T, 2 T, 3 T$ times par circles release kiye hain. Kyunki source nahi hila, har circle ka center same hai — woh evenly spaced hain jaise ek still pond mein ripples.

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

Paas-paas circles ke beech ki spacing hi wavelength $λ$ hai. Ek period $T$ mein ek crest $v T$ doori taय karta hai, isliye:

$crests ke beech gap λ = crest speed v \times janmon ke beech waqt T = \frac{v}{f}$

Yeh bilkul wahi hai jo Wavelength and frequency relation mein hai. Abhi koi nahi hil raha, isliye yeh $λ$ hamara baseline hai.

Step 2 — Counting rule (sab kuch ka dil)

KYA HAI. Apna kaan ek fixed point par rakho. Count karo har second mein kitne crests guzarte hain.

KYUN. Woh count hi woh frequency hai jo tum perceive karte ho. Yeh barabar hai kitni tezi se crests tumhare paas aati hain, divided by unke beech ki doori:

$f^{'} = \frac{crests ki speed jo tumhare kaan se guzre}{crests ke beech ki doori jo tum meet karo}$

PICTURE. Source se kaan tak line ke along ek ruler rakha gaya. Agar crests $λ$ dooron par hain aur speed $u$ se guzar rahi hain, toh ek second mein "crest-train" ki length $u$ guzar jaati hai, jisme $u / λ$ crests hain.

Neeche har case sirf sahi $u$ aur sahi $λ_{met}$ is ek rule mein plug karta hai.

Step 3 — Observer move kare, source still

KYA HAI. Source still hai, isliye circles evenly spaced rehte hain: $λ_{met} = λ = v / f$ (unchanged). Ab observer source ki taraf $v_{o}$ speed se chalta hai.

WAVELENGTH KYUN NAHI BADLTI. Source kisi cheez ka peecha nahi kar raha — Step 1 ki picture hawa mein frozen hai. Sirf observer ki motion matter karti hai, aur woh badlti hai kitni tezi se crests kaan tak pahunchti hain, unki spacing nahi.

PICTURE. Crests abhi bhi $v$ speed se bahar ja rahi hain. Lekin kaan apni speed $v_{o}$ se unke paas daud raha hai. Toh closing speed — dekho Relative velocity — $v + v_{o}$ hai.

$f^{'} = \frac{u}{λ _{met}} = \frac{v + v _{o} crests tumhare paas tezi se aati hain}{wahi purani spacing v / f} = f \frac{v + v _{o}}{v}$

Term by term: $v$ crest ki apni speed hai; $+ v_{o}$ tumhari extra speed hai unki taraf; neeche $v / f$ unchanged wavelength hai. Bada top $\Rightarrow$ zyada pitch.

Away case. Agar tum door bhago, tum crests se bhagte ho, closing speed $v - v_{o}$ hai: $f^{'} = f \frac{v - v _{o}}{v} (pitch neeche jaati hai) .$

Degenerate check. Agar $v_{o} = 0$ toh $f^{'} = f$ milta hai — koi motion nahi, koi shift nahi. Theek hai. Agar tum $v_{o} = v$ pe door bhaago (sound ke liye impossible, lekin algebraically), $f^{'} = 0$ : tum ek crest ke saath pace mein ho aur agli se kabhi nahi milte.

Step 4 — Source move kare, observer still

KYA HAI. Ab TUM still ho aur source tumhari taraf $v_{s}$ speed se aata hai. Yahan crests ki speed jo tumhare kaan tak pahunche woh abhi bhi sirf $v$ hai (medium yeh set karta hai, source nahi) — lekin spacing chhoti ho jaati hai.

SPACING KYUN CHHOTI HOTI HAI. Ek crest banane aur agli ke beech, source khud $v_{s} T$ aage badh jaata hai. Toh agli crest pehli waali ke zyada paas paida hoti hai. Har circle ka center aage nudge ho jaata hai. Circles saamne bunch up hote hain, peeche phail jaate hain.

PICTURE. Offset centers dekho. Moving source ke saamne crest gap squeeze ho jaata hai; peeche, stretch hota hai. Yeh classic "ambulance" picture hai.

$λ^{'} = v / f λ - source ka forward step v_{s} T = \frac{v}{f} - \frac{v _{s}}{f} = \frac{v - v _{s}}{f}$

Term by term: $λ = v / f$ woh spacing hai jo hoti; $v_{s} T = v_{s} / f$ woh doori hai jitni source janmon ke beech creep karta hai; subtract karne par squeezed front gap milta hai. Ab isse counting rule mein feed karo crest speed $u = v$ ke saath:

$f^{'} = \frac{u}{λ ^{'}} = \frac{v}{( v - v _{s} ) / f} = f \frac{v}{v - v _{s}}$

Term by term: top $v$ = crests abhi bhi $v$ pe chalti hain; bottom $v - v_{s}$ = compressed spacing. Chhota bottom $\Rightarrow$ zyada pitch.

Receding case. Source bhage $\Rightarrow$ crests stretch hote hain, $λ^{'} = (v + v_{s}) / f$ : $f^{'} = f \frac{v}{v + v _{s}} (pitch neeche — tone guzarte waqt drop karta hai) .$

Step 5 — Dono effects alag lines par kyun hain

KYA HAI. Steps 3 aur 4 compare karo. Observer motion ne top badla (arrival speed). Source motion ne bottom badla (wavelength). Yeh ek hi lever nahi hain.

KYUN MATTER KARTA HAI. Log expect karte hain ki "speed $x$ par toward" ek hi answer dega chahe kaun move kare. Aisa nahi hota, kyunki dono physically alag mechanisms hain — ek speed par additive hai, ek distance par squeeze hai.

PICTURE. Ek split panel: left, kaan fixed circles mein speed karta hua (top-line effect); right, source circles ko zyada paas kheeenchta hua (bottom-line effect).

Step 6 — Dono move karein: dono levers stack karo

KYA HAI. Observer move karo (top) aur source move karo (bottom) ek saath. Bas dono independent changes combine karo.

KYUN SIRF MULTIPLY/STACK KAR SAKTE HO. Observer ne sirf $u$ badla; source ne sirf $λ_{met}$ badla. Woh counting rule $f^{'} = u / λ_{met}$ ke alag parts touch karte hain, isliye interfere nahi karte — dono plug in karo.

PICTURE. Source aur kaan ek line par, dono ek doosre ki taraf move kar rahe hain: front-crests $(v - v_{s}) / f$ tak squeeze hue, aur kaan $v + v_{o}$ par unme daud raha hai.

$f^{'} = \frac{u}{λ _{met}} = \frac{v + v _{o}}{( v - v _{s} ) / f} = f \frac{v \pm v _{o}}{v \mp v _{s}}$

Har combination, koi gap nahi:

Observer	Source	Top sign	Bottom sign	Pitch
toward	toward	$+ v_{o}$	$- v_{s}$	sabse zyada ↑↑
toward	away	$+ v_{o}$	$+ v_{s}$	mixed
away	toward	$- v_{o}$	$- v_{s}$	mixed
away	away	$- v_{o}$	$+ v_{s}$	sabse kam ↓↓

Step 7 — Formula ki wall: $v_{s} \to v$

KYA HAI. Source ki speed badhao jab tak $v_{s} = v$ na ho jaaye. Denominator $v - v_{s} \to 0$ , isliye $f^{'} \to \infty$ .

YE ASLI INFINITY KYUN NAHI HAI. Saare front crests ek hi aage-daudne wale point par paida hote hain — woh ek single surface par pile ho jaate hain. Formula ne assume kiya tha saaf alag crests ki line; woh assumption yahan khatam ho jaati hai.

PICTURE. Step 4 ke offset circles, push karte jaao jab tak sab ek cone ke along touch na ho jayein. Woh stacked wall hi shock front hai.

Worked examples (har ek step se traced)

Ek-picture summary

Ek line, do knobs: observer knob top tweak karta hai (arrival speed $v \pm v_{o}$ ); source knob bottom tweak karta hai (met wavelength $\to v \mp v_{s}$ ). Dono ko toward-each-other ghoomao aur dono pitch upar push karte hain.

Recall Feynman retelling — ek 12-saal ke bachche ko samjhao

Ek drummer har second drum bajata hai; har bang ek ring hai jo hawa mein phailti hai. Agar tum drummer ki taraf bhaago, tum rings se tezi se takraoge — tum bangs zyada baar sunoge (fraction ka top upar jaata hai). Agar iski jagah drummer tumhari taraf chalta hua bajaye, toh har nayi ring pichhli se zyada paas se shuru hoti hai, isliye rings bunch up ho jaati hain aur tum unhe zyada baar pakad lete ho (fraction ka bottom chhota ho jaata hai). Door bhaagna, ya drummer ka door chala jaana, rings ko spread out karta hai aur bangs slow karta hai. Bas itna hai: kisi bhi tarah paas aana = tezi se bangs = zyada pitch. Agar drummer kabhi itna tez bhaage jitna rings khud, saari rings ek wall mein stack ho jaati hain — yahi sonic boom hai, aur woh neat little formula kaam karna band kar deta hai.

Medium-less, fully symmetric relativistic version ke liye, dekho Doppler effect of light.