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🌊 WAVES (Mechanical Waves)

Stage 2 – Page 5 : Standing Waves in Air Columns

1️⃣ What Are Air Columns?

When sound waves travel inside a pipe, they reflect at the ends and form standing waves.

The pattern depends on whether the pipe ends are:

Open
Closed

2️⃣ Open Pipe (Both Ends Open)

At an open end:

Displacement → Antinode
Pressure → Node

Rule: Open end always behaves like a displacement antinode.

3️⃣ Allowed Wavelengths (Open Pipe)

λₙ = 2L / n

n = 1, 2, 3, ...

All harmonics are allowed.

4️⃣ Allowed Frequencies (Open Pipe)

fₙ = n × (v / 2L)

Fundamental → n = 1
Overtones → n = 2, 3, 4...

5️⃣ Closed Pipe (One End Closed)

At a closed end:

Displacement → Node
Pressure → Antinode

Rule: Closed end always behaves like a displacement node.

6️⃣ Allowed Wavelengths (Closed Pipe)

λₙ = 4L / (2n − 1)

n = 1, 2, 3, ...

Only odd harmonics are present.

7️⃣ Allowed Frequencies (Closed Pipe)

fₙ = (2n − 1) × (v / 4L)

Fundamental → n = 1
1st overtone → 3rd harmonic

8️⃣ Comparison Table

Property	Open Pipe	Closed Pipe
Displacement at ends	Antinode	One node, one antinode
Harmonics	All	Only odd
Fundamental λ	2L	4L
Fundamental f	v / 2L	v / 4L

9️⃣ JEE Important Tricks

Open pipe → frequency ratio: 1 : 2 : 3
Closed pipe → frequency ratio: 1 : 3 : 5
Length correction applies only for air columns

🔟 Next Page Preview

End correction in air columns
Real pipe behavior

Stage 2 – Page 5 Completed Successfully ✅