Standing waves — formation, nodes, antinodes

1.6.18 · Physics › Oscillations & Waves

Standing waves kyun exist karti hain?

Do opposite waves KAISE banti hain? Aam taur par ek wave kisi boundary se takraati hai (string ka fixed end, closed pipe ka end) aur wapas apne aap par reflect ho jaati hai.

KAISE: standing wave ko scratch se derive karna

Superpose karo (waves bas add hoti hain — yahi principle of superposition hai): $y = A sin (k x - ω t) + A sin (k x + ω t)$

$sin P + sin Q = 2 sin \frac{P + Q}{2} cos \frac{P - Q}{2}$ use karo.

$\frac{P + Q}{2} = \frac{( k x - ω t ) + ( k x + ω t )}{2} = k x$ — Kyun? $ω t$ cancel ho jaata hai.
$\frac{P - Q}{2} = \frac{( k x - ω t ) - ( k x + ω t )}{2} = - ω t$ — Kyun? $k x$ cancel ho jaata hai; aur $cos (- θ) = cos θ$ .

Nodes aur antinodes — seedha $R (x) = 2 A sin (k x)$ se padhna

Figure — Standing waves — formation, nodes, antinodes

Apni galtiyon ko steel-man karo

Active recall

Recall Quick self-test (answers chhupao)

Standing wave banane ke liye kya do conditions chahiye? → Equal amplitude/frequency, opposite directions.
$x$ ka function ke roop mein amplitude? → $2 A sin (k x)$ .
Node spacing? → $λ /2$ .
Node-to-antinode? → $λ /4$ .
Kya yeh energy transport karta hai? → Nahi (net zero).

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho do bachche ek lambi rope ko ek doosre ki taraf exactly same rhythm mein hila rahe hain. Wiggles ek doosre se takraati hain. Kuch jagahon par rope bilkul still baithti hai — jaise ek gaanth jo kabhi nahi uchhalti (ek node). Unke beech mein rope pagalon ki tarah upar-neeche fadfadaati hai (ek antinode). Pattern kabhi rope ke saath slide nahi karta; woh bas unhi jagahon par fadfadaata rehta hai. Yahi ek standing wave hai — yeh khadi hai kyunki still-spots wahan ki wahan rehte hain.

Standing wave kya hai?

Do identical waves jo opposite directions mein move karti hain unhe superpose karne se bana wave pattern; uski shape space mein fixed hoti hai jabki amplitude time mein oscillate karti hai, koi net energy transport nahi hota.

A sin (k x - ω t) + A sin (k x + ω t)

se standing wave equation?

y = 2 A sin (k x) cos (ω t)

Standing wave ki position-dependent amplitude?

R (x) = 2 A sin (k x)

Node ki condition?

sin (k x) = 0 \Rightarrow x = nλ /2

(displacement hamesha zero).

Antinode ki condition?

∣ sin (k x) ∣ = 1 \Rightarrow x = (2 n + 1) λ /4

(amplitude

2 A

Adjacent nodes ke beech distance?

λ /2

Node se nearest antinode tak ki distance?

λ /4

Kya standing wave net energy transport karta hai?

Nahi — equal energy dono taraf flow karti hai, net zero; energy trapped rehti hai KE↔PE swap karti hui.

x

aur

t

ke parts alag kyun hote hain?

Sum-to-product ek factor mein

ω t

cancel karta hai aur doosre mein

k x

, jo

f (x) g (t)

deta hai.

Node par mechanically kya khaas hai?

Maximum slope/strain aur maximum restoring force, jabki displacement zero hoti hai.

Connections

Superposition principle — woh additive law jo standing waves ko possible banata hai.
Travelling waves — opposite directions mein move karne wale building blocks.
Reflection of waves at boundaries — doosri (wapas aane wali) wave kaise banti hai.
Resonance and normal modes — sirf kuch khaas $λ$ hi boundary conditions mein fit hoti hain.
Waves on a string / Sound in pipes — applications (harmonics, fundamentals).
Wave number k and wavelength — $k = 2 π / λ$ poore mein use hota hai.

Standing waves — formation, nodes, antinodes

Standing waves kyun exist karti hain?

KAISE: standing wave ko scratch se derive karna

Nodes aur antinodes — seedha $R (x) = 2 A sin (k x)$ se padhna

Pehle Forecast, Phir Verify

Apni galtiyon ko steel-man karo

Active recall

Connections

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