Huygens' principle — wavefront propagation

2.5.10 · Physics › Optics

Wavefront KYA hota hai?

Aise kyun define karte hain? Phase hi interference ko control karta hai. Agar hum equal phase ki surfaces track karein, toh hum bina poori wave equation har jagah solve kiye yeh predict kar sakte hain ki bright/dark kahan honge aur bending kahan hogi.

Figure — Huygens' principle — wavefront propagation

Huygens' Principle — statement

Sirf forward envelope kyun? Ek pure Huygens construction mein ek flaw hai — wavelets backward bhi jaani chahiye. Huygens ne simply backward part ko haath se ignore kar diya. Baad mein (Fresnel/Kirchhoff) ne dikhaya ki backward wave ek obliquity factor $\propto \frac{1}{2} (1 + cos θ)$ se khatam ho jaati hai, jo forward mein $1$ aur backward mein $0$ hota hai. Toh forward-only rule physically justified hai, koi fudge nahi hai.

Agla wavefront KAISE construct karein (recipe)

DERIVATION 1 — Plane wave, plane hi rehti hai (aur seedhi propagation ka law)

Ek plane wavefront $W_{1}$ lo. Uske upar points $A, B, C$ chuno, sab same phase mein hain.

Har ek $r = v Δ t$ radius ki wavelet emit karta hai — same radius, kyunki medium speed $v$ same hai aur sab ne same phase mein start kiya. Same radius kyun? Same $v$ , same start time.
Un circles ka tangent jiske centres ek straight line par hain, khud bhi ek straight line hoti hai jo $W_{1}$ ke parallel hoti hai, $v Δ t$ door.

$distance moved = v Δ t, W_{2} ∥ W_{1}$

Toh ek plane wave ek parallel plane ki tarah aage badhti hai → light uniform medium mein straight lines mein travel karti hai. ✔

DERIVATION 2 — Huygens se Reflection ka Law

Ek plane wavefront $A B$ ek mirror $M M^{'}$ se takraati hai. End $A$ pehle touch karta hai; doosra end $B$ ko abhi surface tak point $C$ par pahunchna baaki hai.

Maano $B C = v t$ (time jo $B$ ko surface tak pahunchne mein lagta hai $C$ par). Usi time mein, $A$ se nikli wavelet same side par medium mein $A E = v t$ radius ki ek hemisphere tak phailti hai.

Yeh kyun kaam karta hai: reflected wavefront $C E$ reflected wavelets ki tangent hai; congruent triangles incidence aur reflection ki geometry ko match karne par force karte hain.

DERIVATION 3 — Huygens se Snell's Law (Refraction)

Plane wavefront $A B$ surface $X Y$ se takraati hai jo medium 1 (speed $v_{1}$ ) aur medium 2 (speed $v_{2}$ ) ke beech mein hai. End $A$ pehle enter karta hai.

$B$ ko surface tak $C$ par pahunchne mein time: $B C = v_{1} t$ .
Usi time mein $A$ se nikli wavelet medium 2 mein travel kar chuki hai: $A D = v_{2} t$ .

Common line $A C$ par right triangles mein: $sin i = \frac{B C}{A C} = \frac{v _{1} t}{A C}, sin r = \frac{A D}{A C} = \frac{v _{2} t}{A C}$

Divide karo: $\frac{s i n i}{s i n r} = \frac{v _{1}}{v _{2}} = constant =_{1} n_{2}$

Frequency kyun constant rehti hai lekin wavelength kyun change hoti hai: wavefronts boundary ke across continuous hote hain, isliye crests per second ki sankhya (frequency) change nahi ho sakti. Lekin speed change hoti hai, toh $λ = v / f$ change hoti hai: $λ_{2} / λ_{1} = v_{2} / v_{1}$ .

Common mistake Common errors ko steel-man karna

Mistake A: "Wavelets sabhi directions mein jaati hain, toh ek flat wave backward aur sideways bhi phailni chahiye ek blur mein." Yeh sahi kyun lagta hai: har point sach mein ek poora sphere emit karta hai. Fix: obliquity factor $\frac{1}{2} (1 + cos θ)$ amplitude ko forward mein maximum aur backward mein zero banata hai, aur envelope ke saath sideways wavelets out of phase hoti hain aur destructively cancel ho jaati hain. Sirf forward tangent bachta hai.

Mistake B: "Refraction mein frequency change ho jaati hai kyunki colour/speed change hoti hai." Yeh sahi kyun lagta hai: speed aur wavelength dono change hote hain, toh kuch toh dena padega. Fix: wavefronts boundary par continuously join hote hain → crests pile up ya vanish nahi ho sakte → frequency conserved rehti hai. Sirf $λ$ aur $v$ change hote hain.

Mistake C: "Light glass mein slow hoti hai matlab uski energy kam ho jaati hai." Yeh sahi kyun lagta hai: "slower = weaker" ka intuition. Fix: speed change medium mein phase propagation ke baare mein hai; intensity amplitude ke baare mein hai. Slow hona ≠ energy ka kho jaana.

Mistake D: ray aur wavefront direction mein confuse hona. Yeh dono perpendicular hain, parallel nahi.

Recall Feynman: ek 12-saal ke bachche ko explain karo

Ek line mein khade doston ki kalpana karo, sab ek saath taali baja rahe hain. Ab socho ki har taali ki awaaz ek circle ki tarah phailti hai. Thodi der baad, un sabhi circles ko milaane par ek nayi line banti hai awaaz ki, ek kadam aage. Woh nayi line wavefront hai. Light bhi yahi karti hai: chashani wali line ka har ek jagah ek tiny naya lamp ban jaata hai, aur saare tiny lamps milke agli chashni wali line banate hain. Jab ripples ki yeh line ek tilted surface se milti hai, toh surface ke sabse karib wale dost pehle "pahunchte" hain, isliye poori line tilt ho jaati hai — wahi tilt bilkul woh reason hai jis se light bend aur reflect karti hai.

Flashcards

Wavefront kya hota hai?

Kisi diye hue instant par wave ke un sabhi points ka locus jo same phase mein hote hain.

Ray aur wavefront ke beech kya relation hai?

Ray wavefront ke perpendicular hoti hai aur energy propagation ki direction mein point karti hai.

Huygens' principle ke 2 parts kya hain?

(1) Wavefront ke har point se secondary spherical wavelets nikalti hain jo speed v se travel karti hain; (2) naya wavefront in wavelets ka forward envelope (common tangent) hota hai.

Huygens' construction mein backward wavefront kyun ignore kiya jaata hai?

Obliquity factor ½(1+cosθ) wavelet amplitude ko backward direction mein zero bana deta hai (θ=180°) aur forward mein maximum.

Point source se bahut door wavefronts ki shape kaisi hoti hai?

Plane (flat) wavefronts.

Huygens' reflection derivation mein kaunsa fact ∠i = ∠r deta hai?

BC = AE = vt, jisse triangles ABC aur AEC congruent ho jaate hain.

Huygens se sin i / sin r ka ratio derive karo.

BC = v₁t, AD = v₂t, common hypotenuse AC ⇒ sin i/sin r = v₁/v₂ = n₂/n₁.

Refraction mein kaunsi quantity constant rehti hai: frequency ya wavelength?

Frequency (wavefronts boundary ke across continuous hote hain); wavelength aur speed change hoti hain.

Agar light rarer se denser medium mein jaaye, toh normal ki taraf bend hogi ya door? Kyun?

Normal ki taraf, kyunki v kam ho jaati hai (n badhta hai), isliye sin r < sin i.

Time t ke baad har secondary wavelet ki radius kya hoti hai?

r = v·t, jahan v medium mein wave speed hai.

Connections

Wave Optics — Interference (wavefronts overlap → constructive/destructive)
Young's Double Slit Experiment (do slits par secondary sources)
Diffraction (Huygens edges ke around bending explain karta hai)
Snell's Law and Refractive Index
Laws of Reflection
Fresnel–Kirchhoff Diffraction (obliquity factor ke saath rigorous fix)
Phase and Path Difference