Visual walkthrough — Mechanical waves — transverse and longitudinal

1.6.13 · D2 · Physics › Oscillations & Waves › Mechanical waves — transverse and longitudinal

Pehli line se pehle, ek promise: neeche har ek symbol ko pehle simple words mein introduce kiya jaayega aur ek picture se pin kiya jaayega, usse kisi equation mein use karne se pehle.

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Step 1 — Ek dot jo bas upar-neeche karta hai (source)

KYA. Socho ek single particle ek lambi rope ke bilkul shuru mein baitha hai, us jagah jo hum $x=0$ kehte hain. Hum use pakad ke smoothly upar aur neeche move karte hain — wahi gentle back-and-forth jo spring pe mass karta hai. Yeh special motion hai Simple Harmonic Motion (dekho Simple Harmonic Motion).

KYU yeh aur koi random jiggle nahi. Hum SHM choose karte hain kyunki yeh simplest repeating motion hai, aur kyunki almost koi bhi real vibration (ek plucked string, ek tuning fork) SHM pieces se bani hoti hai. Agar hum ek SHM dot ko describe kar sakte hain, toh puri wave ko describe kar sakte hain usse copy karke.

PICTURE. Figure mein, amber dot ek vertical dashed line pe upar neeche jaata hai. Middle line se uski height ko hum $y$ kehte hain.

Yahan do words apne symbols earn karte hain:

• $y$ = dot ki displacement — yeh middle se kitna upar (positive) ya neeche (negative) hai.
• $t$ = time, aage badhta hua.

Waqt ke saath height ek sine curve trace karti hai, isliye hum likhte hain:

$y(0,t)=A\sin (\omega t)$

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Step 2 — Padosi copy karta hai, lekin late

KYA. $x=0$ wala dot rope se apne right wale dot se bandra hua hai. Jab hamara dot upar jaata hai, toh yeh apne padosi ko bhi upar kheenchta hai — lekin padosi ko respond karne mein thoda waqt lagta hai. Toh padosi bilkul wahi dance karta hai, thodi der baad shuru karke. Agla dot padosi ko copy karta hai, aur bhi der se, aur aise hi rope ke aage tak.

KYU. Yeh "copy-but-delayed" idea ek travelling wave ka poora raaz hai. Rope ke saath kuch bhi move nahi karta siwaaye dance ki timing ke. Har dot apni jagah pe hi rehta hai (matter wahi rehta hai); sirf kaun-abhi-upar-hai ka pattern aage badhta hai.

PICTURE. Figure mein dots ki ek row hai. Sabse baayi wali apne bounce ke top pe hai; har ek usse right wali apne cycle mein thodi "peeche" hai, toh tops-and-bottoms space mein ek frozen sine shape banaate hain.

Ab hume batana hai ki har dot kitna der baad shuru karta hai. Do naye plain-word symbols:

• $x$ = rope ke saath position, source se right ki taraf ki doori.
• $v$ = wave speed, "start signal" rope ke saath kitni fast travel karta hai (metres per second). Yeh rope khud set karta hai — Step 5 dekho.

Disturbance $x$ doori $v$ speed pe travel karta hai, isliye yeh itne time baad pahunchta hai

$t_{\text{delay}}=\frac{x}{v}$

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Step 3 — Delay ko formula mein likhna

KYA. Position $x$ wala dot abhi wahi karta hai jo source ne $x/v$ waqt pehle kiya tha. Toh jahan bhi source formula mein $t$ tha, woh door wala dot $t-x/v$ use karta hai (ek chhota, pehle ka waqt).

KYU. Delay subtract karna exactly yahi hai jo "baad mein wahi kaam karna" maths mein matlab rakhta hai. Agar tum door wale dot ki ghadi ko $x/v$ se rewind karo, toh woh source se match karta hai.

PICTURE. Do ghariyan: source ki ghadi $t$ dikhati hai, dur wali ghadi $x/v$ se peeche set hai. Wahi sine mein rewound ghadi daalo aur tumhe door wale dot ki height milti hai.

$y(x,t)=A\sin \left(\omega \right(t-\frac{x}{v}\left)\right)$

Andar $\omega$ multiply karo:

$y(x,t)=A\sin (\omega t-\frac{\omega }{v}x)$

Woh messy $\frac{\omega }{v}$ apna khud ka naam deserve karta hai. Ise $k$ kaho:

Substitute karne par milta hai show ka star:

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Step 4 — Minus sign wave ko right kyun bhejna hai (aur $+$ left ko)

KYA. Hum sign test karte hain. Ek "crest pe sawaar" rehne ke liye, tumhe phase $(\omega t-kx)$ ko ek fixed value pe rakhna hoga (maan lo woh value jahan sine peak karta hai). Jab time $t$ badhta hai, toh $x$ ko phase constant rakhne ke liye kya karna chahiye?

KYU. Ek crest bas "ek aisi jagah hai jahan phase ek peak value ke barabar hai." Us jagah ko follow karna hume batata hai kis taraf — aur kitni fast — pattern move karta hai.

PICTURE. Thode waqt ke antar se do snapshots. Amber crest right ki taraf khisal gayi hai; dashed vertical line use track karti hai.

Phase fixed rakho: $\omega t-kx=\text{constant}$ Thodi der $dt$ baad, phase nahi badalna chahiye, toh $\omega dt-kdx=0$. Crest ki speed ke liye solve karo: $\frac{dx}{dt}=\frac{\omega }{k}>0\Rightarrow \text{crest }+x\text{ mein move karti hai.}$

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Step 5 — $\omega$ aur $k$ aate kahan se hain? (Medium aur source ko naam dena)

KYA. Hum $\omega$ aur $k$ ko un cheezon se jodenge jo tum ruler aur stopwatch se measure kar sakte ho.

KYU. Symbols tabhi useful hain jab tum unhe real world mein point kar sako. $\omega$ aur $k$ do everyday quantities ke "per second" aur "per metre" versions hain.

PICTURE. Space mein rope ka ek frozen snapshot (upar): do crests ke beech ki repeat-distance $\lambda$ hai, wavelength. Ek single dot waqt ke saath dekha gaya (neeche): uske bounce ka repeat-time $T$ hai, period.

• Time mein ek poora cycle $2\pi$ radians of phase hota hai aur $T$ waqt leta hai: $\omega =\frac{2\pi }{T}=2\pi f,f=\frac{1}{T}(\text{frequency, cycles per second}).$
• Space mein ek poora cycle $2\pi$ radians of phase hota hai aur $\lambda$ doori leta hai: $k=\frac{2\pi }{\lambda }.$

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Step 6 — Universal payoff: $v=f\lambda$

KYA. Dono definitions combine karo. Kyunki $k=\omega /v$, $\omega =2\pi f$ aur $k=2\pi /\lambda$ daalo: $v=\frac{\omega }{k}=\frac{2\pi f}{2\pi /\lambda }=f\lambda .$

KYU yeh "ek wavelength per period" wala hai. Exactly ek period $T$ mein, pattern exactly ek poori repeat-distance $\lambda$ aage khisak jaata hai. Speed = distance ÷ time = $\lambda /T=f\lambda$. Yeh do tareekon se nikalti hai kyunki dono same geometric truth keh rahe hain.

PICTURE. $T$ ki ek tick mein, puri sine pattern exactly ek $\lambda$ right ki taraf shift hoti dikhti hai — amber crest exactly wahin land karti hai jahan agla crest pehle tha.

Yahi $v=f\lambda$ sab kuch aage drive karta hai: Sound waves, Standing waves & resonance, Doppler effect, aur yeh poore Wave equation ke andar ka engine hai.

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Step 7 — Edge aur degenerate cases (reader ko kabhi adhura mat chodo)

KYA & KYU. Ek formula jis par tum trust karte ho use apne extreme settings mein survive karna chahiye. Hum knobs ko unki limits tak push karte hain aur check karte hain ki picture abhi bhi sahi lagti hai.

PICTURE. Char mini-panels, ek per case, har ek rope ki shape dikhaata hai.

• $A=0$ (koi shake nahi). $y=0$ har jagah har waqt — flat, dead rope. Sahi: koi disturbance nahi, koi wave nahi.
• $f=0$ (source hamesha ke liye still rakha). $\omega =0$, toh $y=A\sin (-kx)$ waqt mein frozen — ek static shape, travelling wave nahi. Wave ke liye source ka actually move karna zaroori hai.
• $k=0$ yaani $\lambda \to \infty$. Puri rope saath mein, step mein upar neeche move karti hai — bilkul SHM, koi spatial pattern nahi. Yeh Simple Harmonic Motion ki taraf wapsi ka bridge hai: infinite wavelength ki wave bas synchronised bobbing hai.
• Left-mover, $y=A\sin (\omega t+kx)$. Same shape, crest doosri taraf khisak jaati hai (Step 4 se). Dono signs legitimate waves hain; ek left-mover aur right-mover ko add karna hi Standing waves & resonance aur Superposition and Interference ka janam hai.

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Ek-picture summary

Upar sab kuch, ek canvas pe: ek source dot SHM karta hua (left), har padosi use delay $x/v$ ke saath copy karta hua, space mein ek sine mein freeze hota hua jiska repeat-distance $\lambda$ hai; pattern right ki taraf $v=\omega /k=f\lambda$ pe march karta hai, jabki har dot sirf height $A$ ke saath bobs karta hai.

Recall Feynman retelling — poora walkthrough simple words mein

Line ke saamne ek akeli bacchi smoothly upar neeche hilti hai, spring jaisi (yahi sine hai, $A$ se kitna upar aur $\omega$ se kitni fast). Uske paas wali bacchi instantly move nahi kar sakti — woh wiggle ko ek heartbeat baad copy karti hai, kyunki tug $x/v$ waqt mein pahunchta hai. Line ke baaki sabhi log apne padosi ko ek heartbeat baad copy karte hain. Picture freeze karo aur "kaun-abhi-upar-hai" pattern line ke saath drawn ek sine jaisa lagta hai, har $\lambda$ metres pe repeat karta hua. Waqt chalne do aur woh poora pattern aage khisak jaata hai — ek poori repeat-distance $\lambda$ har period $T$ mein — toh yeh $v=f\lambda$ pe travel karta hai. Bachchiyan apni jagah nahi chalti (matter wahi rehta hai); sirf wiggle-pattern travel karta hai (energy move karti hai). Zyada fast hilao aur crests paas aa jaate hain, kyunki rope sirf ek speed pe pattern travel karne deti hai. Source band karo aur pattern freeze ho jaata hai; wavelength infinite kar do aur poori line saath bobbing karne lagti hai — yahi wahin se tha jahan hum shuru kiye the, plain SHM.

Recall Fast self-check

$-kx$ term physically kya karta hai? ::: Yeh har point ka phase delay karta hai us hisaab se ki wave ne kitni doori travel ki, aur uska minus sign wave ko $+x$ mein bhejna hai. Sine ka use kyun hota hai? ::: Kyunki har particle SHM karta hai, jiska displacement-vs-time ek sine hai. Ek fixed medium mein agar tum $f$ ko triple karo, toh $\lambda$ ka kya hoga? ::: Yeh ek tehai ho jaata hai, kyunki $v=f\lambda$ aur $v$ fixed hai. $k\to 0$ limit kya hai? ::: Infinite wavelength — har point unison mein oscillate karta hai, yaani pure SHM.