Foundations — Superposition principle

1.6.16 · D1 · Physics › Oscillations & Waves › Superposition principle

Parent note Superposition principle apni pehli hi equations mein bahut saara notation throw karta hai: $y$, $x$, $t$, $A$, $k$, $\omega$, $\varphi$, aur woh curly $\partial$ symbols. Yeh page assume karta hai ki tumne inhe kabhi nahi dekha. Hum har ek ko banate hain, ek aisi order mein jahan har naya symbol sirf pehle se explain kiye gaye symbols use kare.

0 · Stage: ek string aur ek ruler

Kisi bhi symbol se pehle, physical cheez ko picture karo.

Ek lamba rope imagine karo jo flat pada hai. Hum ek axis rope ke saath pin karte hain (ise horizontal position kaho) aur ek axis across karte hain (ise rope ko kitna upar ya neeche push kiya gaya hai kaho). Ek wave ek bump hai — ya ek wiggle — jo rope ke saath travel karta hai jabki rope ka har chhota piece sirf upar aur neeche move karta hai.

Woh ek picture pehle teen symbols ka source hai.

1 · $x$ — medium mein position

• Picture: horizontal ruler par ek mark.
• Topic ko yeh kyun chahiye: wave ke alag-alag places par ek saath alag-alag heights hoti hain, isliye hum ko yeh kehna aana chahiye ki hum kaun sa place mean kar rahe hain.

2 · $t$ — time

• Picture: ek stopwatch jo rope dekhte waqt chal raha ho.
• Topic ko yeh kyun chahiye: superposition "kisi bhi point aur kisi bhi instant par" ke baare mein hai. "Instant" exactly $t$ hai.

3 · $y$ — displacement, aur function $y(x,t)$

Kyunki height dono par depend karti hai — kahan dekho aur kab dekho — hum $y(x,t)$ likhte hain, padha jaata hai "y as a function of x and t".

• Picture: figure mein ruler se rope tak vertical arrow.
• Sign cases jo tumhe handle karne padhenge: $y>0$ (upar), $y<0$ (neeche), $y=0$ (flat — yeh woh case hai jo destructive interference possible banata hai; do waves jo $+3$ aur $-3$ deti hain add hokar flat $0$ banaati hain).

4 · $A$ — amplitude

• Picture: figure mein amber bracket, flat line se crest tak measure kiya gaya.
• Topic ko yeh kyun chahiye: superposition ka poora payoff resultant amplitude $A_{\text{res}}=2A\cos (\varphi /2)$ ka formula hai. Jab tak $A$ ka matlab "peak height" nahi, tum woh read nahi kar sakte.
• Zero case: $A=0$ matlab koi wave nahi — flat rope.

5 · Ek repeating wiggle ko $\sin$ chahiye

Parent waves ko $A\sin (\dots )$ ke roop mein likhta hai. Sine kyun, aur koi bhi random bump kyun nahi?

• Picture: amplitude figure mein smooth curve — woh hai $\sin$, $A$ se scale kiya gaya.
• Kaunsa tool, kaunsa question: $\sin$ answer karta hai "main aisa shape kaise likhun jo ek single tidy formula se hamesha ke liye repeat kare?" Sharp-cornered zig-zags ko messy piecewise rules chahiye hote; sine ko ek symbol chahiye.

6 · Angles, aur $\pi$ kyun appear hota hai

$\sin$ ek angle khaata hai. Physicists woh angle radians mein measure karte hain, jahan ek full turn (360°) $2\pi$ ke barabar hai aur half turn (180°) $\pi$ ke barabar hai.

• Topic ko yeh kyun chahiye: constructive/destructive results $\varphi =0$ aur $\varphi =\pi$ ke roop mein state kiye jaate hain. Woh $\pi$ bas "sine ka aadha cycle" hai, yaani ulta hua.

7 · $k$ — wave space mein kitni tightly packed hai

• Picture: short, closely-packed ripples ka large $k$ hota hai; long lazy swells ka small $k$ hota hai (figure mein do rows dekho).
• Topic ko yeh kyun chahiye: wave argument $kx-\omega t$ hai. $kx$ part ek distance ko sine-angle mein convert karta hai taaki sine ko pata chale ki har point $x$ apne cycle mein kitna andar hai.

8 · $\omega$ — wiggle time mein kitni fast cycle karti hai

• Picture: rope par ek fixed point ko upar-neeche bob karte dekho; $\omega$ measure karta hai ki woh har bob kitni quickly complete karta hai.
• Topic ko yeh kyun chahiye: $-\omega t$ term poore pattern ko time ke saath slide along karaata hai — yahi ise ek travelling wave banata hai, frozen shape nahi. Dekho Wave equation ki $k$ aur $\omega$ kaise speed $v=\omega /k$ pin karte hain.

9 · $\varphi$ — phase difference

• Picture: do identical sine curves, ek doosre se amber gap ke barabar sideways nudged.
• Woh saare cases jo tumhe pata hone chahiye (yeh superposition ka poora point hain):
  • $\varphi =0$ — crests align hain, waves reinforce karti hain → constructive, $A_{\text{res}}=2A$.
  • $\varphi =\pi$ — ek ka crest doosre ke trough par baithta hai → destructive, $A_{\text{res}}=0$.
  • $0<\varphi <\pi$ — partial, beech mein.
  • $\varphi =2\pi$ — ek full lap head start = koi head start nahi → wapas constructive.
• Topic ko yeh kyun chahiye: $\varphi$ woh single dial hai jo control karta hai ki do overlapping waves add hoti hain ya cancel. Dekho Phasor method $\varphi$ ko arrows ke beech angle mein turn karne ke liye, aur Interference of waves jahan $\varphi$ physically aata hai.

10 · $\partial$ — partial derivative

Parent ki derivation $\frac{{\partial }^{2}y}{\partial t^2}$ use karti hai. Yeh sabse scary-looking symbol hai, isliye hum ise slowly banate hain.

• Kaunsa tool, kaunsa question: derivatives answer karte hain "ek quantity doosre mein small change par kaise respond karti hai?" Hume inki zaroorat hai kyunki wave entirely define hoti hai is se ki bending aur acceleration kaise relate karti hain.
• Topic ko yeh kyun chahiye — linearity link: parent prove karta hai superposition us fact se ki derivatives addition par distribute hote hain: $\frac{\partial (y_1+y_2)}{\partial t}=\frac{\partial y_1}{\partial t}+\frac{\partial y_2}{\partial t}$. Wahi ek property poori wajah hai ki waves cleanly add hoti hain.

11 · Phasors — arrows jo waves represent karte hain

Parent mein Examples 2 aur 3 waves ko phasors ke roop mein add karte hain bina unhe define kiye.

• Picture: length $4$ aur $3$ ke do arrows right angles par → unka tip-to-tail sum length $\sqrt{4^2+3^2}=5$ (parent ka Example 2).
• Topic ko yeh kyun chahiye: yeh "do wobbling sines add karo" ko "do static arrows add karo" mein turn karta hai, jo bahut aasaan hai. Full treatment Phasor method mein; underlying single-wave motion Simple Harmonic Motion hai.

Prerequisite map

Baayein taraf sab kuch woh symbols hain jo is page ne build kiye; woh sab right par parent topic mein flow karte hain.

Equipment checklist

Khud test karo — right side cover karo.

Connections

• Superposition principle — woh parent topic jisme har symbol yahan feed hota hai.
• Wave equation — jahan $k$, $\omega$ aur $\partial$-symbols rehte hain.
• Phasor method — §11 se arrow tool.
• Simple Harmonic Motion — rope ke ek point ki up-down motion.
• Interference of waves — $\varphi$ physically kahan se aata hai.
• Beats aur Standing waves — same symbols ke later applications.