Superposition principle

1.6.16 · Physics › Oscillations & Waves

Superposition Principle HAI KYA?

Iske andar packed key consequences:

Waves ek doosre se scatter nahi hoti; cross karne ke baad woh unchanged shape, speed aur direction mein nikl jaati hain.
Yahi cheez interference, beats, aur standing waves ko possible banati hai.

YEH SACH KYUN HAI? (First principles se derivation)

Superposition principle koi alag law nahi hai — yeh wave equation ke linear hone ka consequence hai.

String par ek wave is equation ko follow karti hai: $\frac{\partial ^{2} y}{\partial t ^{2}} = v^{2} \frac{\partial ^{2} y}{\partial x ^{2}}$

Step 1 — Maano ki $y_{1}$ aur $y_{2}$ dono solutions hain. $\frac{\partial ^{2} y _{1}}{\partial t ^{2}} = v^{2} \frac{\partial ^{2} y _{1}}{\partial x ^{2}}, \frac{\partial ^{2} y _{2}}{\partial t ^{2}} = v^{2} \frac{\partial ^{2} y _{2}}{\partial x ^{2}}$ Yeh step kyun? Hum shuru karte hain yeh maanke ki har wave akele ek valid solution hai — iska matlab hi yahi hai ki "ek wave hona".

Step 2 — Sum $y = y_{1} + y_{2}$ try karo. Left side mein plug karo: $\frac{\partial ^{2} ( y _{1} + y _{2} )}{\partial t ^{2}} = \frac{\partial ^{2} y _{1}}{\partial t ^{2}} + \frac{\partial ^{2} y _{2}}{\partial t ^{2}}$ Yeh step kyun? Differentiation addition par distribute hoti hai — yahi crucial property hai jo waves ko "ek doosre ki physics mein interfere" nahi karne deti.

Step 3 — Jane-maane results substitute karo: $= v^{2} \frac{\partial ^{2} y _{1}}{\partial x ^{2}} + v^{2} \frac{\partial ^{2} y _{2}}{\partial x ^{2}} = v^{2} \frac{\partial ^{2} ( y _{1} + y _{2} )}{\partial x ^{2}}$ Yeh step kyun? $v^{2}$ factor out karo aur dobara combine karo — hume exactly wave equation wapas milti hai, lekin ab $y_{1} + y_{2}$ ke liye.

Conclusion: $y_{1} + y_{2}$ wave equation ko bhi satisfy karta hai ⇒ sum khud ek valid wave hai. Principle isliye hold karti hai kyunki governing equation linear hai.

ISKI USE KAISE KAREIN — do waves add karna

Do waves lo jo same direction mein travel kar rahi hain, equal amplitude aur frequency ke saath, phase difference $ϕ$ ke saath: $y_{1} = A sin (k x - ω t), y_{2} = A sin (k x - ω t + ϕ)$

Inhe add karo $sin P + sin Q = 2 sin \frac{P + Q}{2} cos \frac{P - Q}{2}$ use karke:

$y = 2 A cos (\frac{ϕ}{2}) sin (k x - ω t + \frac{ϕ}{2})$

Yeh kyun matter karta hai: Resultant phir bhi usi frequency ki wave hai, lekin amplitude ke saath $A_{res} = 2 A cos (\frac{ϕ}{2})$

$ϕ = 0$ → $A_{res} = 2 A$ → constructive (in phase).
$ϕ = π$ → $A_{res} = 0$ → destructive (out of phase).

Worked Examples

Forecast-then-Verify

Recall Forecast: Point P par teen pulses, har ek

$+ 2$ cm, plus ek $- 2$ cm. Net displacement predict karo, PHIR check karo. Forecast: unhe add karo. Verify: $2 + 2 + 2 - 2 = + 4$ cm. Superposition sirf signed addition hai — kisi bhi number of waves ke liye kaam karta hai.

Common Mistakes (Steel-manned)

Active Recall

Superposition principle batao.

Kisi bhi point par resultant displacement un sabhi displacements ka vector (1-D mein algebraic) sum hota hai jo har wave akele produce karti.

Superposition wave equation ki kis mathematical property ka consequence hai?

Uski linearity (

y

sirf first power mein aata hai).

Phase difference φ ke saath amplitude A ki do equal waves ka resultant amplitude kya hoga?

A_{res} = 2 A cos (ϕ /2)

Fully constructive interference ke liye phase difference?

φ = 0 (ya 2nπ), jisse

A_{res} = 2 A

milta hai.

Fully destructive interference ke liye phase difference?

φ = π (ya odd multiples), jisse

A_{res} = 0

milta hai.

Kya do pulses ek doosre ko cross karne ke baad shape change karti hain?

Nahi — har ek unchanged emerge hoti hai; overlap temporary aur reversible hai.

Do waves ke liye general phasor amplitude formula kya hai?

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

Superposition fail kab hoti hai?

Large-amplitude / non-linear media ke liye (shock waves, breaking waves), jahan wave equation ab linear nahi rehti.

Destructive interference mein energy kahan jaati hai?

Yeh constructive interference ke regions mein redistribute hoti hai; total energy conserved rehti hai.

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho tum aur tumhara dost dono talab mein ek-ek patthar phenkte ho. Ripples ki do sets failti hain aur ek doosre ko cross karti hain. Jahan woh milti hain, paani ek saath dono amounts se upar jaata hai — agar dono upar push karna chahein, toh woh extra upar jaata hai; agar ek upar aur doosra neeche equally push kare, toh paani us pal ke liye flat rehta hai. Lekin phir ripples chalti rehti hain, bilkul unbothered, jaise do ghosts ek doosre se guzar rahe hon. Woh "pushes add karo" wali rule hi superposition principle hai.

Connections

Interference of waves — superposition ka direct application do coherent sources par.
Beats — thodi alag frequencies ki do waves ka superposition.
Standing waves — opposite directions mein travel karti do identical waves ka superposition.
Wave equation — iski linearity hi woh reason hai ki superposition hold karti hai.
Phasor method — phase differences wali waves add karne ka geometric tool.
Simple Harmonic Motion — same/perpendicular directions mein SHMs ka superposition.