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🌊 WAVES (Mechanical Waves)

Stage 1 – Page 10 : Standing Waves in Air Columns (JEE / Intermediate)

1️⃣ Sound Waves in Air Columns

Sound waves in air are longitudinal waves. When sound reflects inside pipes, standing waves are formed.

Important Rule:
• Displacement antinode ↔ Pressure node
• Displacement node ↔ Pressure antinode

2️⃣ Open Pipe (Both Ends Open)

In an open pipe:

Both ends are open → Displacement antinodes
Pressure nodes at both ends

3️⃣ Allowed Wavelengths (Open Pipe)

L = n(λ / 2)

⇒

λₙ = 2L / n

where n = 1, 2, 3, ...

4️⃣ Frequencies of Open Pipe

fₙ = n(v / 2L)

Fundamental Frequency:
f₁ = v / 2L

5️⃣ Harmonics in Open Pipe

All harmonics are present:

1st harmonic → f₁
2nd harmonic → 2f₁
3rd harmonic → 3f₁

6️⃣ Closed Pipe (One End Closed)

In a closed pipe:

Closed end → Displacement node
Open end → Displacement antinode

Key Point:
Only odd harmonics are present

7️⃣ Allowed Wavelengths (Closed Pipe)

L = (2n − 1)(λ / 4)

⇒

λₙ = 4L / (2n − 1)

n = 1, 2, 3, ...

8️⃣ Frequencies of Closed Pipe

fₙ = (2n − 1)(v / 4L)

Fundamental Frequency:
f₁ = v / 4L

9️⃣ Comparison Table (Open vs Closed Pipe)

Property	Open Pipe	Closed Pipe
Ends	Antinode–Antinode	Node–Antinode
Harmonics	All	Odd only
Fundamental f	v / 2L	v / 4L

🔟 JEE / Advanced Exam Tips

Check end condition first (open or closed)
Odd harmonics → immediately think closed pipe
Temperature change affects v → affects frequency

One-Line Memory Trick

Open pipe = string fixed-fixed
Closed pipe = string fixed-free

Stage 1 – Page 10 Completed ✅