Resonance ColumnJEE Main

Concept

An air column closed at the bottom (by water) resonates with a tuning fork when its length fits an odd number of quarter wavelengths — a node at the water, an antinode near the open top. Raising the water and hearing the loudness peak twice gives two lengths whose difference is exactly $\lambda/2$, eliminating the messy end effects.

Key formula

Derivation

Closed pipe modes: $L = (2n-1)\lambda/4$. The antinode actually forms slightly ABOVE the tube mouth — the end correction e.

Subtracting the two resonance conditions kills e: $l_2 - l_1 = \lambda/2$. Then $v = f\lambda = 2f(l_2-l_1)$ — a clean lab measurement. Back-substituting gives $e = (l_2 - 3l_1)/2$.

Scenarios to explore

• Resonance Column — Measure the speed of sound with a tuning fork & water column.

Real-world applications

• Standard JEE/NEET lab experiment for v_sound.
• Organ pipe & wind-instrument tuning.
• Acoustic impedance tubes in materials testing.

JEE exam tips

• v = 2f(l₂ − l₁) needs no end correction — that's the whole point of two readings.
• Third resonance sits at l₃ = l₁ + λ (≈ 5λ/4 − e).
• Temperature: v ∝ √T — quote v at the lab temperature.

Common mistakes

• Using l₂ − l₁ = λ (it's λ/2).
• Forgetting e when computing λ from a SINGLE resonance length.
• Expecting a resonance at l₂ = 2l₁ — closed pipes have only odd multiples.

Exam traps to avoid

• The water surface is always a displacement NODE (pressure antinode).
• e = (l₂ − 3l₁)/2 comes straight from the two equations — derivable, no memorising.