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🌊 WAVES (Mechanical Waves)

Stage 3 – Page 3 : Standing Waves in Air Columns (Pipes)

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1️⃣ Basics of Sound in Pipes

• Sound waves in air are longitudinal waves
• Standing waves form due to reflection at pipe ends
• Air particles oscillate, not travel

✔ Nodes & antinodes are formed in air displacement, not pressure.

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2️⃣ Boundary Conditions in Pipes

📌 Open End

• Displacement → Antinode
• Pressure → Node

📌 Closed End

• Displacement → Node
• Pressure → Antinode

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3️⃣ Open Pipe (Open at Both Ends)

Let length of pipe = L

Allowed Wavelengths

λₙ = 2L / n   (n = 1, 2, 3, …)

Allowed Frequencies

fₙ = n (v / 2L)

• Fundamental → n = 1
• All harmonics present

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4️⃣ Closed Pipe (One End Closed)

Length of pipe = L

Allowed Wavelengths

λₙ = 4L / (2n − 1)

Allowed Frequencies

fₙ = (2n − 1) (v / 4L)

• Only odd harmonics present
• Even harmonics are absent

⚠ Closed pipe behaves like fixed–free string.

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5️⃣ Comparison Table (High-Scoring Area)

Feature	Open Pipe	Closed Pipe
Ends	Antinode – Antinode	Node – Antinode
Harmonics	All	Only Odd
Fundamental λ	2L	4L
Fundamental f	v / 2L	v / 4L

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6️⃣ End Correction (JEE Advanced)

Due to air bulging at open ends:

• Effective length > Actual length
• End correction per open end ≈ 0.6r

L_eff = L + 0.6r (per open end)

• Open pipe → correction at both ends
• Closed pipe → correction at open end only

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7️⃣ Effect of Temperature

Speed of sound:

v = 331 + 0.6T (T in °C)

• Increase temperature → frequency increases
• Wavelength remains constant for fixed pipe

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8️⃣ JEE Advanced Traps

• Mixing displacement & pressure nodes
• Wrong harmonic counting in closed pipes
• Ignoring end correction
• Forgetting temperature dependence

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🎯 Final Takeaway

✔ Pipes behave like strings with different boundary conditions ✔ Closed pipe → only odd harmonics ✔ End correction is crucial in numerical problems

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Stage 3 – Page 3 Completed Successfully ✅