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Stage 1 – Page 8 : Standing Waves (JEE / Intermediate Level)
1️⃣ What is a Standing Wave?
A standing wave is formed due to the superposition of two waves having the same frequency, same amplitude and same speed but travelling in opposite directions.
Key Feature:
Standing waves do not transfer energy from one point to another.
2️⃣ Formation of Standing Waves
Standing waves are formed when:
- An incident wave reflects from a boundary
- The reflected wave overlaps with the incident wave
- Both waves have same frequency and amplitude
Example: Waves on a stretched string fixed at one end
3️⃣ Mathematical Representation
Incident wave:
y₁ = A sin (ωt − kx)
Reflected wave:
y₂ = A sin (ωt + kx)
Resultant wave:
y = 2A sin (ωt) cos (kx)
This equation represents a Standing Wave
4️⃣ Nodes
Node is a point on the standing wave where the displacement is always zero.
- Amplitude = 0
- Particles never vibrate
- Energy is minimum
Condition for node:
cos (kx) = 0
⇒ kx = (2n + 1)π/2
5️⃣ Antinodes
Antinode is a point where the displacement is maximum.
- Amplitude = maximum (2A)
- Particles vibrate with maximum energy
Condition for antinode:
cos (kx) = ±1
⇒ kx = nπ
6️⃣ Distance Relations (Very Important)
| Points | Distance |
|---|---|
| Two consecutive nodes | λ / 2 |
| Two consecutive antinodes | λ / 2 |
| Node to adjacent antinode | λ / 4 |
7️⃣ Energy Distribution
- Energy is localized
- No net energy flow along the medium
- Energy oscillates between KE and PE
8️⃣ Comparison: Progressive vs Standing Waves
| Progressive Wave | Standing Wave |
|---|---|
| Transfers energy | No energy transfer |
| All particles vibrate | Nodes don’t vibrate |
| Same amplitude everywhere | Amplitude varies |
9️⃣ Common JEE Mistakes
- Assuming energy flow in standing waves ❌
- Confusing node distance with wavelength ❌
- Forgetting λ/4 relation ❌
🔟 One-Line Memory Tricks
Node = No motion
Antinode = All motion
Stage 1 – Page 8 Completed ✅
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