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🌊 WAVES (Mechanical Waves)

Stage 3 – Page 6 : Superposition & Beats (JEE Advanced)

1️⃣ Principle of Superposition

When two or more waves travel simultaneously in a medium, the resultant displacement at any point is the algebraic sum of the individual displacements.

y = y₁ + y₂ + y₃ + ...

✔ Valid only in linear media
✔ Waves do not permanently affect each other

2️⃣ Superposition of Two Harmonic Waves

Consider two waves of same direction:

y₁ = a sin(ωt)
y₂ = a sin(ωt + φ)

Resultant displacement:

y = 2a cos(φ/2) sin(ωt + φ/2)

👉 Resultant amplitude depends on phase difference

3️⃣ Constructive & Destructive Interference

📌 Constructive Interference

Phase difference = 0, 2π, 4π…
Maximum amplitude = 2a

📌 Destructive Interference

Phase difference = π, 3π…
Minimum amplitude = 0

4️⃣ Beats – Concept

Beats are periodic variations in loudness of sound due to interference of two sound waves of slightly different frequencies.

✔ Frequency difference must be small
✔ Amplitudes should be nearly equal

5️⃣ Mathematical Derivation of Beats

Let:

y₁ = a sin(2πf₁t)
y₂ = a sin(2πf₂t)

Resultant wave:

y = 2a cos[π(f₁ − f₂)t] sin[2π(f₁ + f₂)t / 2]

6️⃣ Beat Frequency

Beat Frequency = | f₁ − f₂ |

Number of beats per second
Independent of amplitude

7️⃣ Conditions for Hearing Beats

Same medium
Frequency difference ≤ 10 Hz
Comparable amplitudes
Same direction of propagation

8️⃣ JEE Advanced Traps

Confusing beats with interference pattern
Assuming beats with large frequency difference
Ignoring medium condition
Wrong interpretation of loudness vs intensity

9️⃣ Applications of Beats

Tuning musical instruments
Finding unknown frequency
Detecting defects in machines

🎯 Final Takeaway

✔ Superposition explains interference & beats
✔ Beats depend on frequency difference only
✔ Extremely important for JEE Advanced MCQs

Stage 3 – Page 6 Completed Successfully ✅