Huygens' principle — wavefront propagation: Foundations

2.5.10 · D1 · Physics › Optics › Huygens' principle — wavefront propagation

Is page par koi assumption nahi hai. Agar parent note Huygens' Principle — Wavefront Propagation mein koi symbol use hua hai, toh hum use yahan se bilkul scratch se build karenge.

1. Ek wave = ek wobble jo travel karta hai

Sabse basic cheez se shuru karte hain. Ek lambi rope imagine karo. Tum ek end ko upar-neeche jhatkate ho. Ek hump rope ke neeche travel karta hai, lekin rope ka har chhota tukda sirf upar-neeche move karta hai — woh hump ke saath travel nahi karta. Yeh travelling pattern hi ek wave hai.

Figure — Huygens' principle — wavefront propagation

Yeh pehle kyun chahiye: Huygens' principle ek rule hai is baare mein ki travelling pattern kaise aage badhta hai. Agar tum "material ruka rehta hai, pattern move karta hai" ka picture nahi bana paate, toh baad ke har statement ("wave ne distance $v Δ t$ cover ki") ka koi matlab nahi banega.

2. Oscillation aur phase — "main wobble mein kitna aage hun?"

Rope ke ek akele tukde ko pick karo aur sirf use watch karo. Woh upar jaata hai → middle mein aata hai → neeche jaata hai → middle mein aata hai → upar… baar baar. Ek poora trip (upar, neeche, wapas start) ko ek cycle kehte hain.

Phase is sawaal ka jawaab deta hai: is waqt, yeh point apne cycle mein kitna aage hai?

Figure s02 dekho: yellow dots sab apne wobble ke top par hain is waqt — same phase. Red dot neeche hai — ek alag phase.

Phase kyun chahiye: parent note define karta hai wavefront ko equal phase ke points ke roop mein, aur kehta hai phase "interference control karta hai". Phase is poore topic par sabse important word hai — baaki sab kuch phase ki bookkeeping hai.

Phase differences kaisa path differences bante hain, yeh dekhne ke liye Phase and Path Difference dekho.

3. Wavefront — equal phase ki surface

Ab ek point ko watch karna band karo aur poori rope (ya poore pond) ko ek frozen instant mein dekho. Har woh point mark karo jo abhi in phase hai — maano, har woh point jo bilkul ek crest par baithe hain.

Ek giraye gaye stone se pond par, ek instant mein saare crest-points ek circle (ek ring) banate hain.
Bahut door se, ya ek wide flat source se, woh equal-phase points flat line up hote hain.

Yeh kyun chahiye: Huygens' construction ek wavefront abhi leti hai aur agli wavefront banati hai. Yahi woh object hai jis par poora principle operate karta hai.

4. Ray — wavefront ke perpendicular arrow

Wavefront tumhe batata hai kahan equal phase hai; ray tumhe batati hai energy kis direction mein ja rahi hai.

5. Speed $v$ , time $Δ t$ , aur ek wavelet kitni door travel karta hai

Do symbols jo parent note mein baar baar aate hain: $v$ aur $Δ t$ .

Yeh kyun matter karta hai: Huygens' recipe ka step 2 har point ke around $ρ = v Δ t$ radius ka ek circle draw karta hai. Woh radius yahi formula hai. Same $v$ aur same $Δ t$ ⇒ har wavelet same size ka hota hai, aur yahi wajah hai ki ek plane wave plane hi rehti hai.

6. Secondary wavelet aur envelope (common tangent)

Figure s04 mein, purani wavefront par har blue point se $ρ = v Δ t$ radius ka ek green circle bada hua. Yellow line jo har green circle ko graze karti hai woh nayi wavefront hai. Hum forward envelope rakhte hain aur backward waali chodh dete hain (obliquity factor backward amplitude ko zero force kar deta hai — parent note yeh explain karta hai).

Yeh pair of words kyun chahiye: "secondary wavelet" aur "envelope" literally woh do nouns hain jo Huygens' principle ke statement mein hain. Inke bina principle bola bhi nahi ja sakta.

7. Ek angle ka sine — do lengths ka ratio

Isse pehle ki hum measure kar sakein ki ek wavefront ek surface par kaise tilt karta hai, hume trigonometry ka ek chhota sa piece chahiye. Hum ise abhi introduce karte hain, use karne se pehle.

Sine kyun, kuch aur kyun nahi? Jab ek slanted wavefront kisi surface se milti hai, toh woh piece jo "upar stick karta hai" (opposite side) slanted length (hypotenuse) ke compare mein exactly woh hai jo tilt angle encode karta hai. Sine woh tool hai jo "ek edge ne kitni door travel ki" ko "wavefront konsa angle banata hai" mein convert karta hai — yahi kaam hai Snell's law mein neecha.

8. Normal se angle — $i$ aur $r$

Jab ek wavefront kisi surface se milti hai (ek mirror ya glass), hum angles measure karte hain. Lekin kis line se measure karte hain?

Normal se kyun measure karte hain, surface se kyun nahi? Kyunki physics laws clean aati hain: mirror ke liye $∠ i = ∠ r$ , aur refraction ke liye $\frac{sin i}{sin r} = \frac{v _{1}}{v _{2}}$ . Surface se measure karne par sirf clutter badhta (tum $9 0^{\circ} - i$ har jagah likhte).

Yeh dekhne ke liye ki sines kyun aate hain, figure s06 mein labelled triangle dekho. Ek slanted wavefront $A B$ surface se milti hai: end $A$ pehle surface point $A$ par touch karta hai, jabki end $B$ ko abhi surface point $C$ tak neeche jaana hai.

Time $t$ mein, corner $B$ , medium 1 mein $B C = v_{1} t$ cover karta hai — yeh length incidence angle $i$ ke opposite hai.
Usi time mein $A$ se wavelet medium 2 mein $A D = v_{2} t$ distance tak pahunchti hai — yeh refraction angle $r$ ke opposite hai.
Dono right triangles hypotenuse ke roop mein same slanted line $A C$ share karte hain.

Section 7 ki sine definition ko har triangle par apply karte hain: $sin i = \frac{B C}{A C} = \frac{v _{1} t}{A C}, sin r = \frac{A D}{A C} = \frac{v _{2} t}{A C}$ Divide karo, aur shared $A C$ (aur $t$ ) cancel ho jaate hain: $\frac{s i n i}{s i n r} = \frac{v _{1}}{v _{2}} .$

Yeh angles kahan jaate hain, yeh dekhne ke liye Laws of Reflection aur Snell's Law and Refractive Index dekho.

9. Refractive index $n$

$n = c / v$ ko upar ke ratio mein feed karna Snell's law ke speed form ko familiar index form $n_{1} sin i = n_{2} sin r$ mein badal deta hai (parent note mein work out kiya gaya hai). Snell's Law and Refractive Index dekho.

Foundations topic ko kaise feed karte hain

Equipment checklist

Khud test karo — right side cover karo aur out loud answer do.

Ek wave carry karta hai

energy (aur pattern), lekin material nahi — har point sirf wobble karta hai apni jagah.

Phase ka matlab hai

ek point apne upar-neeche cycle mein kitna aage hai, ek angle ke roop mein measure kiya gaya jahan ek poora cycle

36 0^{\circ}

hai.

Do points "in phase" hote hain jab

woh exactly same kaam ek hi instant mein kar rahe hon (jaise dono ek crest par).

Ek wavefront hai

woh surface jo un saare points ko join karti hai jo ek instant mein same phase mein hain.

Point source ke paas wavefront ki shape hoti hai

spherical (2D mein circle).

Source se door wavefront

plane (flat) dikhti hai.

Ek ray hai

wavefront ke perpendicular ek line, us direction mein point karti hui jisme energy travel karti hai.

Time

Δ t

mein ek wavelet jo distance travel karta hai woh hai

wavelet radius

ρ = v Δ t

Wavelet radius ko

ρ

kyun likhte hain,

r

kyun nahi?

Kyunki

r

ka matlab pehle se refraction ka angle hai —

ρ

use karne se letter clash avoid hota hai.

Ek secondary wavelet hai

woh tiny spherical wave jo wavefront ka har point speed

v

par emit karta hai.

Envelope hai

woh surface jo saare wavelet circles ko touch (tangent) karti hai — agli wavefront.

sin (angle)

hai

right triangle mein (opposite side) / (hypotenuse) ka ratio,

0

1

tak.

Normal hai

contact point par surface ke perpendicular drawn line.

i

aur

r

normal se measure karte hain, surface se nahi.

Refractive index

n

barabar hai

c / v

; bada

n

matlab us medium mein slower light.

Connections

Huygens' Principle — Wavefront Propagation (woh parent jiske liye yeh page prepare karta hai)
Phase and Path Difference (phase jo measurable path lengths mein badal jaata hai)
Wave Optics — Interference (equal-phase surfaces overlap karti hain)
Young's Double Slit Experiment (har slit ek secondary source ke roop mein)
Diffraction (wavelets edges ke around bend karte hain)
Snell's Law and Refractive Index
Laws of Reflection
Fresnel–Kirchhoff Diffraction