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Stage 3 – Page 9 : Superposition, Beats & Resonance (JEE Advanced)
1️⃣ Principle of Superposition
When two or more waves overlap at a point, the resultant displacement is the algebraic sum of individual displacements.
y = y₁ + y₂
✔ Valid only for linear media
✔ Independent of the nature of waves
2️⃣ Constructive & Destructive Interference
🔹 Constructive Interference
- Waves meet in phase
- Resultant amplitude increases
Δφ = 2nπ
🔹 Destructive Interference
- Waves meet out of phase
- Resultant amplitude decreases
Δφ = (2n + 1)π
3️⃣ Mathematical Superposition (Same Frequency)
For two waves of equal amplitude and frequency:
y = A sin(ωt) + A sin(ωt + φ)
Resultant amplitude:
R = 2A cos(φ/2)
👉 Phase difference controls resultant amplitude
4️⃣ Beats (Important JEE Topic)
Beats occur when two waves of slightly different frequencies superpose.
fbeat = |f₁ − f₂|
- Amplitude varies periodically
- Sound becomes loud and faint alternately
✔ Beat frequency ≠ average frequency
✔ Beats give information about frequency difference
5️⃣ Energy Variation in Beats
- Energy ∝ Amplitude²
- Maximum energy during loud sound
- Minimum energy during faint sound
⚠ Energy oscillates but is conserved
6️⃣ Resonance
Resonance occurs when the frequency of external force equals the natural frequency of the system.
f = f₀
- Maximum amplitude
- Maximum energy absorption
👉 Resonance can be useful or destructive
7️⃣ Sharpness of Resonance
- Low damping → sharp resonance
- High damping → broad resonance
Quality factor:
Q = f₀ / Δf
8️⃣ Forced Oscillations
- System oscillates with driving frequency
- Amplitude depends on frequency
- Phase changes with frequency
✔ Below resonance → displacement in phase
✔ Above resonance → displacement out of phase
9️⃣ JEE Advanced Traps
- Confusing beat frequency with average frequency
- Ignoring phase relation in superposition
- Assuming resonance only increases frequency
- Forgetting energy ∝ A²
🔟 Final Takeaway
✔ Superposition is linear addition
✔ Beats depend on frequency difference
✔ Resonance gives maximum amplitude & energy
Stage 3 – Page 9 Completed Successfully ✅
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