Mechanical waves — transverse and longitudinal

1.6.13 · Physics › Oscillations & Waves

Intuition Badi picture

Ek wave ek disturbance hoti hai jo travel karti hai, energy aur momentum ko medium mein le jaati hai bina medium ke matter ko saath liye. Medium ka har particle apni hi resting jagah ke around thoda hilta-dulta hai — aage jo pattern move karta hai, woh sirf hilne ka pattern hai.

KYUN mechanical waves ko medium chahiye? Kyunki disturbance ek particle se doosre particle tak unke beech ki forces (elasticity) aur unki inertia (mass) ke zariye pass hoti hai. Particles nahi → springs nahi ki pass kar sake → wave nahi.
KYA cheez dono types ko alag karti hai? Particles jis direction mein vibrate karte hain woh wave ke travel karne ki direction ke relative.
KAISE shape move karti hai par matter ruka rehta hai? Har particle wahi karta hai jo uske neighbour ne ek tiny moment pehle kiya tha — jaise stadium ki "Mexican wave": log baithe rehte hain, wave poore stadium mein ghoomti hai.

1. Do types — core distinction

Figure — Mechanical waves — transverse and longitudinal

2. Wave equation ko scratch se banana (ek snapshot in time + motion)

Step 1 — Ek particle (source). Maano $x = 0$ wala particle SHM kare: $y (0, t) = A sin (ω t)$ Yeh step kyun? Hum source ki motion choose karte hain; baaki sab delay se follow karta hai.

Step 2 — Baaki ko delay karo. Speed $v$ se move karta hua disturbance position $x$ tak $t_{delay} = x / v$ time baad pahunchta hai. Toh $x$ wala particle abhi wahi karta hai jo source ne $x / v$ time pehle kiya tha: $y (x, t) = A sin (ω (t - \frac{x}{v}))$ Yeh step kyun? Yahi travelling wave ka dil hai: position $x$ time mein sirf peeche reh jaata hai.

Step 3 — Wave number se tidy karo. Define karo == $k = ω / v$ == (angular wave number). Tab:

Step 4 — Wave speed relation nikalo. Phase $(ω t - k x)$ constant rehna chahiye "crest par ride" karne ke liye. Differentiate karo: $ω d t - k d x = 0 \Rightarrow v = \frac{d x}{d t} = \frac{ω}{k} = \frac{2 π f}{2 π / λ} = f λ$

3. Speed medium par depend karti hai (dimensions se derive karo)

Derivation sketch (string), har factor kyun: length $d x$ ka ek chhota element mass $μ d x$ (inertia) ka hota hai aur tension $T$ ke net vertical component se wapas khicha jaata hai jo uske curved ends par act karta hai (restoring). Newton's law $F = ma$ ko element par apply karne se $T \frac{\partial ^{2} y}{\partial x ^{2}} = μ \frac{\partial ^{2} y}{\partial t ^{2}}$ milta hai, yaani wave equation $\frac{\partial ^{2} y}{\partial t ^{2}} = \frac{T}{μ} \frac{\partial ^{2} y}{\partial x ^{2}}$ . $v^{2} \frac{\partial ^{2} y}{\partial x ^{2}}$ se match karne par $v = T / μ$ milta hai.

4. Particle velocity vs wave velocity (inhe confuse mat karo!)

Ek neat link (snapshot ki slope): $\frac{\partial y}{\partial t} = - v \frac{\partial y}{\partial x} \Rightarrow v_{p} = - v \times (slope)$ Kyun: dono $y = A sin (ω t - k x)$ ko differentiate karne se aate hain; $\partial y / \partial t = A ω cos (\cdot)$ aur $\partial y / \partial x = - A k cos (\cdot)$ , aur $ω / k = v$ .

5. Worked examples

6. Common mistakes (Steel-manned)

7. Active recall

Recall Quick self-test (answers cover karo)

Transverse waves gas ke bulk mein kyun travel nahi kar sakti? → Koi shear rigidity nahi hoti.
Wave mein kya ruka rehta hai aur kya travel karta hai? → Matter ruka rehta hai (oscillate karta hai); energy/pattern travel karta hai.
$v = f λ$ mein, kaunsa source se set hota hai aur kaunsa medium se? → $f$ source se, $v$ medium se, toh $λ$ adjust hota hai.
Max particle speed? → $A ω$ .

Recall Feynman: 12-saal ke bachche ko samjhao

Socho haath pakde doston ki ek lambi line hai. Pehla bachcha hilta hai. Kyunki woh haath pakde hain, agla bachcha ek tiny moment baad tug feel karta hai aur hilta hai, phir agla, aur agla. Wiggle line ke neeche travel karta hai chahe har bachcha apni jagah ruka rahe. Agar bachche side-to-side hilte hain, woh transverse wave hai (jaise saanp). Agar woh aage-peeche push-and-pull karte hain, ek doosre ke paas aa kar phir door ho jaate hain, toh woh longitudinal wave hai (jaise sound). Bachche kahin nahi jaate — sirf wiggle jaati hai.

Connections

Simple Harmonic Motion — har particle ek SHM oscillator hai.
Wave equation — $\partial_{t}^{2} y = v^{2} \partial_{x}^{2} y$ .
Sound waves — prime longitudinal example.
Superposition and Interference — waves overlap honay par kya hota hai.
Standing waves & resonance — confined travelling waves.
Doppler effect — moving source/observer $f$ aur $λ$ ko kya karta hai.

Mechanical wave kya hota hai?

Ek travelling disturbance jo medium mein energy/momentum transfer karta hai bina matter ke net transport ke; elasticity + inertia chahiye.

Transverse wave — particle motion direction?

Wave propagation ke perpendicular (crests & troughs).

Longitudinal wave — particle motion direction?

Propagation ke parallel (compressions & rarefactions).

Fluids bulk transverse mechanical waves support kyun nahi kar sakte?

Unmein koi shear rigidity nahi hoti (sideways shape change resist nahi kar sakte).

Universal wave relation batao.

v = fλ (ek period mein wave ek wavelength aage badhti hai).

Wave speed — source set karta hai ya medium?

Medium; source frequency set karta hai, toh λ adjust hoti hai.

String par transverse wave ki speed?

v = √(T/μ), T = tension, μ = mass per unit length.

Fluid mein sound ki speed?

v = √(B/ρ), B = bulk modulus, ρ = density.

Wave speed ki general form?

v = √(elastic restoring property / inertial property).

Travelling wave equation (+x direction)?

y = A sin(ωt − kx), jahaan k = 2π/λ, ω = 2πf.

Angular wave number k define karo.

k = 2π/λ = ω/v (phase ke radians per metre).

Wave mein maximum particle speed?

A·ω (equilibrium position par hoti hai).

Particle velocity vs wave velocity ka fark?

Particle velocity = ∂y/∂t (vary karti hai, max Aω); wave velocity = ω/k = fλ (constant, medium se set).

Fixed medium mein agar frequency double ho, toh λ ka kya hoga?

λ aadhi ho jaayegi; v constant rahegi.

Particle velocity aur snapshot ki slope ka relation?

v_p = −v·(∂y/∂x).