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👉 Join WhatsApp Channel🌊 WAVES (Mechanical Waves)
Stage 3 – Page 1 : Standing Waves & Superposition Principle
1️⃣ Superposition Principle (Foundation)
When two or more waves overlap in a medium, the resultant displacement at any point is the algebraic sum of individual displacements.
✔ Superposition is valid only for linear waves.
Mathematically:
y = y₁ + y₂
2️⃣ Formation of Standing Waves
Standing waves are formed due to superposition of two identical waves traveling in opposite directions.
- Same frequency
- Same wavelength
- Same amplitude
- Opposite directions
⚠ If amplitudes are not equal → pure standing waves are NOT formed.
3️⃣ Mathematical Expression
Two waves:
y₁ = A sin(ωt − kx)
y₂ = A sin(ωt + kx)
Resultant:
y = 2A sin(ωt) cos(kx)
This equation clearly shows:
- Time-dependent term → sin(ωt)
- Position-dependent term → cos(kx)
4️⃣ Nodes and Antinodes
📌 Nodes
- Points of zero displacement
- cos(kx) = 0
- x = (2n + 1)λ/4
📌 Antinodes
- Points of maximum displacement
- cos(kx) = ±1
- x = nλ/2
5️⃣ Key Distances (Very Important)
- Distance between two consecutive nodes = λ/2
- Distance between node and antinode = λ/4
- Distance between two antinodes = λ/2
6️⃣ Energy in Standing Waves
Unlike travelling waves:
- No net energy transfer
- Energy oscillates locally
- Nodes have zero energy
- Antinodes have maximum energy
✔ This concept is frequently tested conceptually in JEE Advanced.
7️⃣ Standing Waves vs Travelling Waves
| Standing Wave | Travelling Wave |
|---|---|
| No energy transfer | Energy propagates |
| Nodes & antinodes present | No fixed nodes |
| Amplitude varies with position | Amplitude same everywhere |
| Formed by interference | Single wave motion |
8️⃣ IIT-JEE Concept Traps
- Assuming energy flow in standing waves
- Forgetting phase difference at nodes
- Wrong node–antinode spacing
- Confusing time variation with motion
🎯 Final Takeaway
✔ Standing waves = interference phenomenon ✔ No net energy flow ✔ Nodes and antinodes are fixed in space
Stage 3 – Page 1 Completed Successfully ✅
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