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Stage 3 – Page 4 : Energy, Power & Intensity of Waves
1️⃣ Energy Transport in Waves
- Waves transfer energy without transporting matter
- Each particle oscillates about its mean position
- Energy flows in the direction of wave propagation
✔ Wave motion = Energy motion, not particle motion
2️⃣ Energy in a Stretched String Wave
For a sinusoidal wave:
y = A sin(kx − ωt)
Average Energy per Unit Length
U = ½ μ ω² A²
- μ = mass per unit length
- ω = angular frequency
- A = amplitude
3️⃣ Power Transmitted by a Wave
Average Power
P = ½ μ ω² A² v
- Power ∝ square of amplitude
- Power ∝ square of frequency
- Power ∝ wave speed
⚠ Doubling amplitude → Power becomes 4 times
4️⃣ Intensity of a Wave
Intensity = Power transmitted per unit area
I = P / A
- SI unit → W m⁻²
- Applicable mainly for sound waves
5️⃣ Intensity of Sound Waves
For plane progressive sound wave:
I = ½ ρ v ω² s₀²
- ρ = density of medium
- s₀ = displacement amplitude
- v = speed of sound
6️⃣ Intensity Level (Decibel Scale)
Human ear responds logarithmically.
β = 10 log₁₀ (I / I₀)
- I₀ = 10⁻¹² W m⁻² (threshold of hearing)
- Unit → decibel (dB)
| Sound | Intensity Level |
|---|---|
| Whisper | 30 dB |
| Conversation | 60 dB |
| Traffic | 90 dB |
| Pain Threshold | 120 dB |
7️⃣ Inverse Square Law (Sound)
For a point source:
I ∝ 1 / r²
- Doubling distance → intensity becomes 1/4
- Sound spreads spherically
8️⃣ JEE Advanced Traps
- Confusing amplitude with intensity
- Using linear instead of logarithmic scale
- Forgetting ω = 2πf
- Mixing sound intensity and loudness
🎯 Final Takeaway
✔ Energy & power depend on A² ✔ Intensity decreases with distance ✔ Decibel scale is logarithmic
Stage 3 – Page 4 Completed Successfully ✅
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