Superposition principle: Visual walkthrough — Hinglish deep dive

1.6.16 · D2 · Physics › Oscillations & Waves › Superposition principle

Hum is ek boxed fact ke peeche ja rahe hain: $A_{res} = 2 A cos (\frac{ϕ}{2})$

Neeche sab kuch — har symbol use karne se pehle usse earn karta hai.

Step 1 — Ek wave kya hoti hai, ek moving shape ki tarah draw ki hui?

KYA. String par ek wave ek aisi shape hai jo side mein slide karti hai bina form change kiye. Hum ise aise likhte hain: $y_{1} = A sin (k x - ω t)$

Har symbol ko picture par padhte hain:

$y_{1}$ — ek jagah aur ek waqt par string ki upar/neeche height. Positive = string upar uthee, negative = neeche dabee.
$x$ — string ke saath-saath position (hum kitni door tak dekhte hain).
$t$ — time.
$A$ — amplitude: sabse badi height jo wave kabhi reach karti hai. Yeh flat line se crest tak ki doori hai.
$k$ — wavenumber: string ke har metre mein sine wave ke kitne radians fit hote hain. Bada $k$ = tightly packed wiggles.
$ω$ — angular frequency: wiggle kitni tez upar-neeche bob karti hai time mein.

Sine kyun? Kyunki sine sabse simple repeating shape hai — ek smooth hump upar, ek neeche, hamesha. Combination $(k x - ω t)$ wave ki "clock reading" hai: jaise-jaise $t$ badhta hai, clock reading same rakhne ke liye aapko $x$ ko aage badhana padega, toh poori shape right ki taraf travel karti hai.

PICTURE.

Step 2 — Doosri wave, ek kadam peeche

KYA. Ab usi string par ek doosri identical wave bhejo — same $A$ , same $k$ , same $ω$ — lekin thodi der baad launch ki gayi. Yeh delay phase difference $ϕ$ ke roop mein likha jata hai: $y_{2} = A sin (k x - ω t + ϕ)$

Naye symbol ko padhna:

$ϕ$ — phase difference, radians mein measured. Yeh ek head start hai: $+ ϕ$ wave ko shift karta hai taaki uske crests wave 1 ke crests se thoda aage (ya peeche) baith jaayein.

Phase kyun, doori kyun nahi? Kyunki ek wave apne clock ke $2 π$ radians mein repeat karti hai. Phase mein baat karne ka matlab hai "wave 2 repeating cycle mein wave 1 se kitni door hai?" — woh akela number wavelength ki parwah kiye bina offset capture karta hai.

PICTURE. Red wave, wave 2 hai, $ϕ$ se shifted.

Step 3 — Superposition: bas heights add karo

KYA. Superposition principle kehta hai ki medium dono instructions ek saath karta hai. Har jagah actual height hai: $y = y_{1} + y_{2} = A sin (k x - ω t) + A sin (k x - ω t + ϕ)$

Hum bas add kyun kar sakte hain? Kyunki Wave equation linear hai — parent note ne prove kiya ki agar $y_{1}$ aur $y_{2}$ dono ise solve karte hain, toh $y_{1} + y_{2}$ bhi karta hai. Koi cross-terms nahi, koi fighting nahi. Pushes add karo, truth milo.

PICTURE. Black + red, point-by-point add kiya, blue resultant deta hai.

Step 4 — Humein jo tool chahiye: sines ke sum ko product mein convert karna

KYA. Do sines add karna messy lagta hai. Hum ek exact identity ka sahara lete hain: $sin P + sin Q = 2 sin (\frac{P + Q}{2}) cos (\frac{P - Q}{2})$

Yeh tool kyun, doosra kyun nahi? Hum har sine ko angle-addition formulas se expand kar sakte the, lekin yeh "sum-to-product" identity purpose-built hai: yeh hume ek akela sine (ek wave) deta hai jo constant cosine (ek amplitude) se multiply hai. Woh split exactly woh cheez hai jo hum read off karna chahte hain — "wave part" aur "size part" alag-alag nikl aate hain.

Do clock readings naam dete hain taaki cleanly plug in kar sakein:

$P = k x - ω t$ (wave 1 ka phase)
$Q = k x - ω t + ϕ$ (wave 2 ka phase)

Tab identity ke do ingredients hain: $\frac{P + Q}{2} = \frac{( k x - ω t ) + ( k x - ω t + ϕ )}{2} = k x - ω t + \frac{ϕ}{2}$ $\frac{P - Q}{2} = \frac{( k x - ω t ) - ( k x - ω t + ϕ )}{2} = - \frac{ϕ}{2}$

PICTURE. Yeh halves geometrically kahan se aate hain — average clock, aur half the gap.

Step 5 — Resultant wave assemble karo

KYA. Un do pieces ko identity mein feed karo: $y = 2 A cos (- \frac{ϕ}{2}) sin (k x - ω t + \frac{ϕ}{2})$

Cosine even hai — $cos (- θ) = cos (θ)$ — toh minus sign harmlessly gayab ho jaata hai: $y = amplitude 2 A cos (\frac{ϕ}{2}) ek travelling wave sin (k x - ω t + \frac{ϕ}{2})$

Do boxed groups ko padhna:

$sin (k x - ω t + \frac{ϕ}{2})$ — ek wave jiska $k$ aur $ω$ same hai pehle jaisa. Toh result abhi bhi same frequency ki wave hai, bas thoda nudge hoke dono originals ke beech mein baithne ke liye.
$2 A cos (\frac{ϕ}{2})$ — ek constant hai (isme koi $x$ ya $t$ nahi hai). Yeh wave ko multiply karta hai, isliye yeh naya amplitude hai: $A_{res} = 2 A cos (\frac{ϕ}{2})$

Aise split kyun? Kyunki ab phase $ϕ$ exactly ek jagah mein rehta hai — amplitude mein. Phase knob ghuma lo aur sirf height badlegi; wave bina ruke chalti rahegi.

PICTURE. Blue resultant apne envelope amplitude $2 A cos (ϕ /2)$ ke saath marked.

Step 6 — Case $ϕ = 0$ : constructive (woh double ho jaati hain)

KYA. $ϕ = 0$ set karo: $A_{res} = 2 A cos (0) = 2 A \times 1 = 2 A$

KYUN. Zero phase gap ke saath crests exactly ek doosre ke upar baithte hain. Har upar-push doosre upar-push se add hota hai — string do guna zyada oonchi swing karti hai. Yeh constructive interference hai.

PICTURE. Do crests aligned, resultant $2 A$ tak pahunchta hai.

Step 7 — Case $ϕ = π$ : destructive (woh gayab ho jaati hain)

KYA. $ϕ = π$ set karo: $A_{res} = 2 A cos (\frac{π}{2}) = 2 A \times 0 = 0$

KYUN. Aadha cycle apart hone ka matlab hai wave 2 ka trough exactly wave 1 ke crest ke upar baitha hai. Har upar-push ek equal neeche-push se milta hai; woh har jagah aur hamesha cancel karte hain. String flat rehti hai. Yeh destructive interference hai.

PICTURE. Crest over trough, resultant flat line hai.

Step 8 — Beech wala aur cosine ka sign

KYA. $0$ aur $π$ ke beech general $ϕ$ ke liye, amplitude smoothly $2 A$ se $0$ tak slide karti hai. Lekin $π$ se aage badhte raho aur kuch subtle hota hai: $cos (ϕ /2)$ negative ho jaata hai ( $π < ϕ < 3 π$ ke liye, yaani $ϕ /2$ $\frac{π}{2}$ aur $\frac{3 π}{2}$ ke beech).

Negative amplitude kyun sense karta hai? Sine ke aage negative sign use usse ulta kar deta hai — jo aadha cycle shifted sine jaisi hi cheez hai. Toh ek "negative amplitude" ek extra flip ke saath positive amplitude hai. Physically measured amplitude (sabse badi height jo aap actually dekhoge) magnitude hai: $A_{res} = 2 A cos (\frac{ϕ}{2})$

Yeh SAB cases cover karta hai:

$ϕ = 0$ → $∣ cos 0∣ = 1$ → $2 A$ (max).
$ϕ = π$ → $∣ cos \frac{π}{2} ∣ = 0$ → $0$ (min).
$ϕ = 2 π$ → $∣ cos π ∣ = 1$ → $2 A$ phir se — kyunki $2 π$ ek full cycle hai, $ϕ = 0$ se alag nahi.
Koi bhi $ϕ$ → smoothly $0$ aur $2 A$ ke beech, har $2 π$ par repeat karta hua.

PICTURE. Amplitude curve $2 A ∣ cos (ϕ /2) ∣$ phase ki full range mein, uske peaks aur zeros marked.

Recall

$∣ \cdot ∣$ kyun, raw cosine kyun nahi? Amplitude ka matlab "sabse badi size" hai, jo kabhi negative nahi hoti ::: Raw $2 A cos (ϕ /2)$ negative ho sakta hai; woh sign ek half-cycle flip hai jo wave mein absorb ho jaata hai, toh measured amplitude uska absolute value hai.

Ek-picture summary

Left se right padho: $ϕ$ se offset do identical waves → superpose (heights add karo) → sum-to-product identity → ek akeli travelling wave jiska amplitude $2 A cos (ϕ /2)$ hai. Phase knob ghuma lo aur amplitude $2 A$ (constructive) aur $0$ (destructive) ke beech slide karti hai.

Recall Feynman retelling — poora walkthrough plain words mein

Do identical ripples ek string par ek doosre ka peecha karte hain, ek thoda peeche — woh "thoda peeche" phase $ϕ$ hai. Superposition kehta hai: har jagah, bas do heights add karo. Jab main same shape ke do sines add karta hoon, ek neat trick hai jo jawab ko ek sine (abhi bhi ek rolling wave, dono ke beech mein aadha baitha hua) times ek fixed number aage ki taraf split karta hai. Woh fixed number naya amplitude hai, aur woh $2 A cos (ϕ /2)$ nikalta hai. Agar ripples step mein march karte hain ( $ϕ = 0$ ) toh cosine $1$ hai aur wave $2 A$ tak double ho jaati hai. Agar ek aadha cycle peeche hai ( $ϕ = π$ ) toh cosine $0$ hai aur woh ek doosre ko flat kar dete hain. Beech mein, cosine amplitude ko un extremes ke beech smoothly slide karta hai — aur kyunki ek wave har full cycle mein repeat karti hai, pattern $ϕ = 2 π$ par wapas $2 A$ par aa jaata hai. Yahi poori kahani hai: heights add karo, trick use karo, knob read off karo.

Connections

Superposition principle — parent law jiska yeh page picture-proof derive karta hai.
Interference of waves — constructive/destructive cases (Steps 6–7) interference hi hain.
Beats — kya hota hai jab do waves ka $ω$ sirf phase gap ki jagah slightly alag hota hai.
Standing waves — same addition, lekin doosri wave opposite direction mein travel karti hai.
Phasor method — isi amplitude formula ke liye geometric shortcut.
Simple Harmonic Motion — string ka har point SHM execute karta hai; superposition do SHMs add karta hai.
Wave equation — uski linearity wajah hai ki Step 3 (bas add karo) legal hai.