Standing waves — formation, nodes, antinodes: Question bank

1.6.18 · D5 · Physics › Oscillations & Waves › Standing waves — formation, nodes, antinodes

Shuru karne se pehle, woh ek equation yaad karo jis par sab kuch tika hai:

True ya false — justify karo

Do identical waves ek hi direction mein jaake standing wave bana sakti hain.

False — same direction sirf ek bada travelling wave deta hai (phir bhi

f (x - v t)

, phir bhi move karta hai); tumhe opposite directions chahiye taaki shape factor mein

ω t

cancel ho jaye. Dekho Superposition principle.

Ek standing wave string ke along net energy transport karta hai.

False — yeh do Travelling waves se bana hai jo opposite directions mein equal energy carry karte hain, isliye dono flows cancel ho jaate hain; energy local rehti hai, kinetic↔potential ke beech swap karti rehti hai.

Standing wave ka har point same amplitude se oscillate karta hai.

False — amplitude

2 A sin (k x)

hai, jo position par depend karta hai: nodes par zero, antinodes par

2 A

. Position-dependent amplitude hi "standing" ki definition hai.

Adjacent nodes ek wavelength apart hote hain.

False —

sin (k x)

ek wavelength mein do baar zero cross karta hai, isliye consecutive nodes

λ /2

apart hote hain.

Do adjacent nodes ke beech ke saare points ek hi instant par apni peak par pahunchte hain.

True — sab

cos (ω t)

factor share karte hain, isliye maximum saath mein hit karte hain; sirf unka maximum size

2 A sin (k x)

differ karta hai, timing nahi.

Ek node ke opposite sides ke points in phase move karte hain.

False — ek node ke across

sin (k x)

sign flip karta hai, isliye woh do regions bilkul out of phase move karte hain (ek upar jab doosra neeche hota hai).

Node par physically kuch nahi ho raha.

False — displacement zero hai lekin slope/strain wahan maximum hai, isliye restoring tension peak par hai; ek node mechanically bahut active hota hai.

Standing wave ki ek definite propagation speed hoti hai.

False — iska shape travel nahi karta, isliye pattern ki koi propagation speed nahi hoti; sirf underlying counter-propagating waves ki speed

v = ω / k

hoti hai.

Standing wave ki frequency us har travelling wave ki frequency ke barabar hoti hai jisne use banaya.

True — time factor

cos (ω t)

hai same

ω

ke saath; superposition waves ko add karta hai, unki frequency nahi badalta.

Error dhundo

"Standing wave ki amplitude

A

hai, same as har incoming wave."

Galat — ek antinode par amplitude

2 A

hoti hai (dono waves peak times par fully add hoti hain); constant "

A

" sirf travelling ingredients ko describe karta hai.

"Nodes aur antinodes

λ /2

apart hain, toh ek node apne antinode ke

λ /2

par baitha hai."

λ /2

spacing node↔node ya antinode↔antinode hai. Ek node se uske nearest antinode tak

λ /4

hai — do nodes ke beech ka midpoint.

"Nodes dhundhne ke liye main

cos (ω t) = 0

set karta hoon."

Galat factor — nodes fixed positions hain, isliye tum space part

sin (k x) = 0

set karte ho.

cos (ω t) = 0

set karne se woh instants milte hain jab poori string momentarily flat hoti hai, nodes nahi.

x = 0

hamesha node hai kyunki

y = 2 A sin (k x) cos (ω t)

mein

sin (0) = 0

aata hai."

Yeh sirf is particular phase choice ke liye true hai. Agar shape factor

cos (k x)

hota (different boundary/reflection phase), toh

x = 0

antinode hota. End condition, jo Reflection of waves at boundaries set karta hai, decide karta hai.

"Node par energy zero hai kyunki displacement zero hai."

Displacement zero hai lekin node wahan phir bhi potential energy maximum stretch/strain mein store karta hai, aur tension mechanical work karta hai; energy absent nahi hai, yeh kinetic ki jagah elastic hai.

"Har incoming wave ki amplitude double karne se nodes ek doosre ke paas aa jaate hain."

Nahi — node positions sirf

sin (k x) = 0

par depend karti hain, yaani

k

par (isliye

λ

par),

A

par nahi. Amplitude change karne se antinodes kitna swing karte hain yeh change hota hai, still points kahan hain yeh nahi.

Why questions

x

aur

t

parts

f (x) g (t)

mein separate kyun ho jaate hain?

Sum-to-product identity ek factor mein

ω t

cancel karti hai aur doosre mein

k x

, jo

2 A sin (k x) \cdot cos (ω t)

chodti hai — space aur time ab ek bracket mein tangle nahi hain.

Ek fixed shape travel kyun nahi kar sakti?

Travel ka matlab hai ki baad ke time par shape same shape shifted hai,

f (x - v t)

; lekin

sin (k x)

mein koi

t

nahi hai, isliye shape fixed

x

-positions par nailed hai aur sirf uska overall size breathe karta hai.

Ek node apne next node se

λ /2

kyun space karta hai lekin antinode se sirf

λ /4

sin (k x)

half wavelength mein zero → peak → zero jaata hai; peak (antinode) midpoint par baitha hai, isliye node→antinode

λ /2

ka half hai

= λ /4

Fixed end par reflection doosri wave kyun create karta hai?

Ek fixed end move nahi kar sakta, isliye returning reflected wave ko wahan incoming wave ko cancel karna hoga — woh inverted, opposite-direction-travelling wave exactly woh left-mover hai jo standing pattern banane ke liye chahiye.

Fixed end node kyun hona chahiye, antinode kyun nahi?

Ek fixed end har time zero displacement par clamped hai, aur sirf ek node ki

R (x) = 0

hoti hai; antinode swing karta hai, jo clamp forbid karta hai.

Sirf certain wavelengths hi bounded string par clean standing waves kyun banate hain?

Pattern ko boundary conditions fit karne chahiye (har end par node/antinode), isliye ek integer number of half-wavelengths length span karni chahiye — yahi normal modes aur harmonics ki origin hai.

Node par zero displacement ke bawajood maximum restoring force kyun hoti hai?

Force string ki curvature/slope se aati hai, displacement se nahi; string ek node se most steeply bend karti hai, isliye transverse tension component (restoring force) wahan peak karta hai.

Edge cases

Amplitude

x = 0

par kya hai jab shape factor

sin (k x)

hai?

Zero —

sin (0) = 0

x = 0

ko node banata hai; yeh origin par fixed end se reflect hone wali wave ka natural pattern hai.

Standing wave

cos (ω t) = 0

ke instants par kaisi dikhti hai?

Momentarily har jagah flat —

y = 0

har

x

par — kyunki poori shape zero se scale ho rahi hai; us instant par saari energy kinetic hai.

Agar do counter-propagating waves ki unequal amplitudes hain (

A_{1} \neq = A_{2}

), toh kya true nodes exist karte hain?

Perfect nodes nahi hote — choti wave badi wave ko fully cancel nahi kar sakti, isliye minima ki nonzero amplitude

∣ A_{1} - A_{2} ∣

hoti hai; tumhe ek partial standing wave milti hai (ek standing part plus ek leftover travelling part).

Zero frequency (

ω = 0

) par "standing wave" ka kya hota hai?

cos (0) = 1

shape ko ek static

2 A sin (k x)

curve ke roop mein freeze kar deta hai — koi breathing nahi, koi oscillation nahi; yeh ek fixed displacement mein degenerate ho jaata hai, wave nahi.

Kya ek akeli travelling wave kisi special amplitude choice ke liye standing wave hai?

Nahi — ek wave hamesha

x

aur

t

sin (k x - ω t)

ke roop mein tangle rakhti hai; sirf apna opposite-direction twin add karne se shape factor mein

ω t

cancel hota hai. Koi bhi amplitude yeh fix nahi karta.

Antinode par string ki velocity zero hai ya maximum?

Yeh instant par depend karta hai: maximum displacement par velocity zero hai, aur equilibrium se guzarte waqt velocity maximum hai (

2 A ω

). Antinode sabse bada swing batata hai, fixed speed nahi.

Recall Har trap ki one-line summary

Nodes/antinodes positions hain ( $sin (k x)$ se set hote hain, yaani $k$ se); amplitude position-dependent hai ( $2 A sin (k x)$ ); net energy transport zero hai; shape travel nahi kar sakti kyunki $t$ iske andar nahi hai; aur boundaries decide karti hain kaunse wavelengths survive karte hain.

Connections

Standing waves — formation, nodes, antinodes — parent jise yeh bank interrogate karta hai.
Superposition principle — kyun do waves simply add ho jaati hain.
Travelling waves — moving ingredients.
Reflection of waves at boundaries — returning wave banata hai; end nodes force karta hai.
Resonance and normal modes — kyun sirf kuch $λ$ fit hote hain.
Wave number k and wavelength — $k = 2 π / λ$ har spacing ke peechhe.