Superposition principle: Question bank — Hinglish deep dive

1.6.16 · D5 · Physics › Oscillations & Waves › Superposition principle

Shuru karne se pehle, ek shared picture dhyan mein rakho: kisi wave ka displacement matlab hai medium ka ek point apni rest position se kitna door move kiya hai kisi given instant par — socho ek rope par ek single dot upar-neeche bob kar raha hai. Superposition kehta hai: jab kayi waves ek saath visit karti hain, us dot ki position nikaalane ke liye, tum har wave ki instruction ko add karo (har ek signed displacement ko) aur dot ko sum ko follow karne do. Isme $+ 3$ aur $- 5$ add karne se zyada koi mystery nahi hai.

True ya false — justify karo

Do identical waves hamesha ek doosre ko reinforce karti hain.

False. Do identical-amplitude waves reinforce tabhi karti hain jab wo in phase hon (

ϕ = 0

); ek ko aadhi wavelength (

ϕ = π

) shift karo aur wo cancel ho jaati hain. "Identical amplitude" phase ke baare mein kuch nahi kehta.

Agar do waves kisi point par completely cancel ho jaati hain, toh us point par energy destroy ho gayi.

False. Energy redistribute hoti hai, destroy nahi — wo un points par flow karti hai jahan waves reinforce karti hain. Poore medium par total phir bhi energy conserve karta hai.

Superposition principle nature ka ek fundamental law hai, wave equation se independent.

False. Ye wave equation ke linear hone ka consequence hai (

y

sirf first power mein appear karta hai). Linearity toot jaaye toh superposition bhi toot jaata hai.

Do pulses ek doosre se guzarne ke baad permanently distort ho jaate hain.

False. Overlap ke dauran wo ek temporary combined shape mein add ho jaate hain, lekin har pulse apni original shape, speed aur direction ke saath bahar nikalti hai — process reversible hai kyunki equation linear hai.

Superposition sirf do nahi, kisi bhi number of waves ke liye kaam karta hai.

True. Tum bas saare individual displacements add kar do:

y = y_{1} + y_{2} + \dots + y_{n}

. Linearity guarantee karti hai ki sum abhi bhi ek valid wave hai.

Do waves ka resultant hamesha kisi bhi ek se zyada badi amplitude rakhta hai.

False. Resultant amplitude

2 A cos (ϕ /2)

2 A

0

tak range karti hai. Out-of-phase waves chhoti amplitude produce kar sakti hain, zero bhi.

Superposition bahut tez sound aur bade breaking ocean waves ke liye utni hi achhi tarah kaam karti hai jitni chhoti ripples ke liye.

False. Large-amplitude waves medium ko non-linearly respond karwati hain (

y^{2}

-type terms appear hote hain), isliye waves ek doosre ko affect karti hain — superposition wahan fail ho jaata hai.

Alag-alag frequencies ki do waves ko phir bhi superposition se add kiya ja sakta hai.

True. Superposition ne kabhi equal frequencies ki demand nahi ki; tum bas displacements add karte ho. Haan, "

2 A cos (ϕ /2)

" wala neat formula sirf equal frequency aur amplitude ke liye apply hota hai — alag frequencies se Beats milte hain.

Agar do equal waves ka resultant amplitude zero hai har jagah aur hamesha, toh phase difference exactly

π

hona chahiye.

True (equal amplitudes sath sath travel kar rahe hain ke liye).

2 A cos (ϕ /2) = 0

ke liye

cos (ϕ /2) = 0

chahiye, yaani

ϕ = π

(ya odd multiples). Tabhi wo har point aur instant par cancel karte hain.

Error dhundho

A_{1} = 3

A_{2} = 4

, toh resultant amplitude

3 + 4 = 7

hai, hamesha."

Error: amplitudes directly tabhi add hoti hain jab $ϕ = 0$ . Generally wo phasors ki tarah add hoti hain:

A_{res} = A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} cos ϕ

, jo

∣ A_{1} - A_{2} ∣ = 1

A_{1} + A_{2} = 7

tak range karti hai.

"Do waves

9 0^{\circ}

out of phase hain, dono amplitude

A

ki, toh resultant amplitude

2 A

hai."

Error:

2 A

sirf

ϕ = 0

ke liye hai. Yahan use karo

2 A cos (ϕ /2) = 2 A cos 4 5^{\circ} = 2 A

2 A

nahi. Equivalently, perpendicular phasors dete hain

A^{2} + A^{2} = 2 A

"Destructive interference ka matlab hai waves wahan exist karna band ho gayi hain."

Error: waves abhi bhi poori tarah present hain aur pass through kar rahi hain — unki instructions sirf us instant aur jagah par cancel hoti hain. Ek moment baad, ya thoda aage, wo reinforce karti hain. Kuch bhi switch off nahi hua hai.

"Kyunki overlap ke dauran combined shape messy lagti hai, toh wave equation yahan non-linear honi chahiye."

Error: messy-looking sum bilkul wahi hai jo linear equation predict karta hai — wo bas do shapes ko add karta hai. Linearity equation ke algebra ke baare mein hai, picture tidy lagti hai ya nahi iske baare mein nahi.

y_{1} = A sin (k x - ω t)

aur

y_{2} = A sin (k x - ω t + ϕ)

ko superpose karne ke liye, main inhe multiply karta hoon."

Error: superposition addition hai, kabhi multiplication nahi.

y = y_{1} + y_{2}

. Multiply karne se

y^{2}

-type term introduce ho jaayega — bilkul wahi non-linear ingredient jo superposition forbid karta hai.

"Do overlapping waves ki energy har point par

E_{1} + E_{2}

hai."

Error: energy amplitude squared ke saath jaati hai, aur

(y_{1} + y_{2})^{2} \neq = y_{1}^{2} + y_{2}^{2}

generally — ek cross term hota hai. Energy spatially redistribute hoti hai; sirf poore medium par total

E_{1} + E_{2}

ke barabar hoti hai.

Why questions

Wave equation ke linear hone se superposition kyun follow karta hai?

Kyunki derivatives addition par distribute hote hain: agar

y_{1}

aur

y_{2}

dono equation satisfy karte hain, toh

y_{1} + y_{2}

plug karne par do copies mein split ho jaata hai jo dono hold karte hain, isliye sum bhi ise satisfy karta hai. Koi bhi

y^{2}

term us split ko tod deta.

Do crossing pulses unchanged kyun nikalte hain, jaise "ghosts walking through each other"?

Kyunki sum

y_{1} + y_{2}

reversible hai — linear equation sum ke andar har wave ki apni information intact rakhta hai, isliye alag hone par har ek apna original form exactly recover kar leta hai.

Resultant amplitude formula

2 A cos (ϕ /2)

hai aur

2 A cos ϕ

kyun nahi?

Sum-to-product identity

sin P + sin Q = 2 sin \frac{P + Q}{2} cos \frac{P - Q}{2}

half phase difference produce karta hai. Geometrically, resultant phasor donon ke beech ke angle ko bisect karta hai, isliye

ϕ

ka aadha appear karta hai.

Do waves zero displacement mein add ho sakti hain phir bhi energy kahin aur carry kar sakti hain, kyun?

Kisi node par displacement zero matlab wahan koi motion nahi hai, lekin wahi waves antinodes par medium ko double amplitude tak push karti hain. Energy sirf wahan concentrate hoti hai jahan wo reinforce karti hain — wo kabhi vanish nahi hoti.

Phasor method (waves ke liye arrows) kaam karta hi kyun hai?

Kyunki ek given frequency ki har sinusoidal wave ko ek rotating arrow se represent kiya ja sakta hai jiska vertical projection displacement hai; displacements add karna phir arrows ko tip-to-tail add karna ban jaata hai — vector addition — yehi phasors karte hain.

Superposition Standing waves ko possible kyun banata hai?

Standing wave opposite directions mein travel karti do identical waves ka sum hai. Superposition un donon ko coexist aur add karne deta hai; unka sum move karne ki jagah fixed nodes aur antinodes rakhta hai.

Edge cases

Kisi wave aur usse khud ke resultant ka kya hoga (

y_{1} = y_{2}

ϕ = 0

Perfectly constructive: amplitude double hokar

2 A

ho jaati hai, kyunki

2 A cos (0) = 2 A

. Har point simply do guna zyada door move karta hai.

Exactly

ϕ = π

par do equal waves ke saath kya hota hai?

Har jagah aur hamesha total cancellation:

2 A cos (π /2) = 0

. Medium flat rehta hai — lekin sirf isliye kyunki donon waves amplitude mein equal hain aur sath sath travel karti hain.

Agar donon waves ki unequal amplitudes hain aur

ϕ = π

Wo zero par cancel nahi hoti hain. Phasor formula deta hai

A_{res} = A_{1}^{2} + A_{2}^{2} - 2 A_{1} A_{2} = ∣ A_{1} - A_{2} ∣

. Sirf equal amplitudes complete destructive cancellation deti hain.

Agar do waves mein se ek ki amplitude zero ho toh resultant kya milega?

Bas doosri wave, unchanged. Har jagah zero displacement add karne se

y = y_{1}

rehta hai — ek sanity check ki superposition degenerate case mein sahi se reduce karta hai.

Ek non-linear medium mein superposition ka kya hota hai (ek supersonic jet se shock wave)?

Ye fail ho jaata hai: medium ke response mein

y^{2}

-type terms include hote hain, isliye waves distort hoti hain aur ek doosre ko influence karti hain, aur ek simple sum ab motion predict nahi karta.

Ek single instant par, agar wave 1 point P par

+ 3 cm

aur wave 2

- 3 cm

deti hai, lekin amplitudes

5 cm

hain, toh kya P permanently rest par hai?

Nahi. Us instant P

0 cm

par hai, lekin donon waves ke displacements time ke saath change hote hain; ek moment baad dono ek hi direction mein push kar sakti hain aur P move karta hai. Momentary cancellation permanent stillness nahi hai.

Frequencies

f

aur

f + Δ f

ki do waves superpose karti hain — kya result ek single amplitude wali simple wave hai?

Nahi. Phase difference time ke saath drift karta hai, isliye amplitude slowly

2 A

aur

0

ke beech pulse karta hai — yahi Beats phenomenon hai, koi single fixed-amplitude wave nahi.

Recall Yahan har trap ki one-line summary

Superposition displacements ka signed addition hai — linear, reversible, energy-conserving; direct amplitude addition aur "energy destroy ho gayi" — ye donon sabse seductive jhooth hain.

Connections

Superposition principle — parent idea jise yahan har question probe karta hai.
Interference of waves — constructive/destructive traps yahan rehte hain.
Beats — different-frequency edge case.
Standing waves — opposite-travelling waves ka superposition.
Wave equation — linearity isliye superposition hold karta hai.
Phasor method — "arrows kyun kaam karte hain" wala question.
Simple Harmonic Motion — wo building block jise add kiya ja raha hai.