Foundations — Wave equation — derivation for string

1.6.15 · D1 · Physics › Oscillations & Waves › Wave equation — derivation for string

Is page par assume kiya gaya hai ki aapne kuch bhi nahi dekha. Hum har letter, har squiggle, aur har picture build karte hain jinpar parent note Wave Equation derivation silently rely karta hai. Upar se neeche padhein — har item agle ke liye ladder ki ek rung hai.

1. String aur uski picture: $x$, $y$, aur $y(x,t)$

Ek lambi horizontal string imagine karein, jaise ek stretched skipping rope. Hum us par ek ruler rakhte hain.

• $x$ = string ke saath aap kitni door hain, horizontally measure kiya hua. Ye batata hai ki hum string ke kis point ki baat kar rahe hain.
• $y$ = wo point apni resting straight line se kitna upar ya neeche push hua hai. Upar positive hai, neeche negative.

Ab key notation: $y(x,t)$.

Topic ko iske kyon zaroorat hai. Ek wave ek shape hai jo move karti hai, isliye height depend karti hai dono par — kahan dekhte ho aur kab dekhte ho. Ek variable kaafi nahi hai — isliye do inputs, $x$ aur $t$. Derivation mein baaki sab is ek function ke behaviour ke baare mein hai.

2. Tension $T$ — string ke andar ka pull

Do haath rope pakad ke alag-alag kheench rahe hain — rope ab tense hai. Ise pluck karo aur pluck tezi se race karta hai; grip dhiloo karo aur wahi pluck dhire chalega.

Topic ko iske kyon zaroorat hai. Tension restoring pull hai — ye hi ek displaced piece of string ko straight ki taraf wapas kheenchta hai. Tension nahi, wave nahi. Dekho Wave speed on a string — v = sqrt(T over mu).

3. Linear mass density $\mu$ — har metre mein kitna bhaari

String ko 1-metre ke tukdon mein kaatne aur ek tukda weighing karne ki imagine karo — wo weight (kg mein) $\mu$ hai.

Isse ek tiny piece ki mass aati hai:

Topic ko iske kyon zaroorat hai. Newton ka law hai $F=ma$, aur yahan mass hai $dm=\mu dx$. Bhaari string ($\mu$ bada) zyada sluggish hoti hai → slower wave.

4. Tiny piece: $dx$, $dm$, aur "$d$-kuch bhi"

Derivation mein har jagah aapko chezon ke aage ek chota $d$ milega: $dx$, $dm$. Ye koi naya letter multiply nahi kar raha — ye ek signal hai.

Topic ko iske kyon zaroorat hai. Hum $F=ma$ poori floppy string par ek saath apply nahi kar sakte kyunki alag-alag hisson mein alag-alag cheezein hoti hain. Toh hum ek tiny piece par zoom in karte hain, uska law nikalte hain, aur tininess ($dx$) hume straight-line geometry aur derivatives use karne deti hai. Baad mein magic: $dx$ dono taraf se cancel ho jaata hai, isliye final law har piece ke liye hold karta hai — isliye poori string ke liye.

5. Angle $\theta$ aur slope — string kitni tilted hai

Hamare tiny piece ke siron par string thodi tilted hai. Us tilt angle ko $\theta$ (Greek "theta") kehte hain, horizontal se measure kiya hua.

Do trig tools saamne aate hain. Yahan har ek kyun hai, plain words mein:

Aur $\tan \theta$:

Topic ko iske kyon zaroorat hai. Ye geometry (ek tilt angle) ko calculus (ek slope = ek derivative) mein convert karte hain, jo poori derivation ka bridge hai.

6. Partial derivatives: $\frac{\partial y}{\partial x}$ aur $\frac{\partial y}{\partial t}$

Kyunki $y$ do chezon par depend karta hai ($x$ aur $t$), jab hum rate of change lete hain toh hume batana padta hai kis ko wiggle kar rahe hain aur doosre ko hold karo. Wahi curly $\partial$ ka matlab hai.

Topic ko iske kyon zaroorat hai. Slope (space) aur transverse velocity (time) usi $y$ ki alag-alag rates hain. Ordinary $\frac{d}{dx}$ ambiguous hoga; curly $\partial$ unhe alag rakhta hai. Dekho Partial derivatives and curvature.

7. Second derivatives: acceleration aur curvature

Ek rate ka rate lo. Do important hain.

Topic ko iske kyon zaroorat hai. Poora result yeh padhta hai: acceleration ($t$-curvature) spatial curvature ($x$-curvature) ke proportional hai. Dono second derivatives hain — inke bina wave equation nahi padh sakte.

8. Wave speed $v$ aur shape-mover $f(x-vt)$

Topic ko iske kyon zaroorat hai. Ye solutions hain — actual waves. Special sine wave $y=A\sin (kx-\omega t)$ (dekho Travelling wave function y = A sin(kx - omega t)) aur uska rule $v=\omega /k$ (Dispersion relation omega = vk) is shape-mover ke particular cases hain.

9. Scaffolding law: $F=ma$

Ye poori derivation ka engine hai — dekho Newton's Second Law. Upar ki saari cheezein bas yeh batati hain ki $F$ (tension geometry), $m$ ($dm=\mu dx$), aur $a$ (${\partial }^{2}y/\partial t^2$) ko kaise likhein.

Prerequisite map

Equipment checklist

Right side chhupaao aur parent derivation padhne se pehle khud test karo.

Connections

• Wave equation — derivation for string
• Newton's Second Law
• Partial derivatives and curvature
• Wave speed on a string — v = sqrt(T over mu)
• Travelling wave function y = A sin(kx - omega t)
• Dispersion relation omega = vk
• Transverse vs Longitudinal Waves
• Standing waves on a string
• Energy carried by a wave