An organ pipe 40 cm long is open at both ends. The speed of sound in air is 360 ms⁻¹. The frequency of the second harmonic is ___________ Hz.

Solution

Physics sir · 3 steps

Open at both ends means every harmonic is allowed: find the fundamental, then double it.

1. Open pipe harmonics. Both ends of an open pipe are displacement antinodes. The pipe length must hold a whole number of half-wavelengths, so L = nλ/2 with n = 1, 2, 3, …

   fₙ = (n v)/2L

2. Put in numbers. Convert the length first: L = 40 cm = 0.4 m. The fundamental is f₁ = 360/(2 × 0.4) = 450 Hz.
3. So. The second harmonic is f₂ = 2 × 450 = 900 Hz.

The common slip: 450. Stopping at the fundamental v/2L gives 450 Hz. That is the first harmonic, and the second harmonic is twice it.

The idea to keep. An open pipe gives all harmonics, fₙ = nv/2L. A pipe closed at one end gives only odd ones, fₙ = nv/4L with n odd. Check the ends before you pick the formula.

Answer from the NTA final answer key. Solution by PrepWiser.