Example

Question:

A pipe, \(30.0\,\mathrm{cm}\) long, is open at both ends. Which harmonic mode of the pipe resonates a \(1.1\,\mathrm{kHz}\) source? Will resonance with the same source be observed if one end of the pipe is closed? Take the speed of sound in air as \(330\,\mathrm{m\,s^{-1}}\).

Solution:

Open at both ends:
For an open pipe, first harmonic frequency: \[ v_1 = \frac{v}{2L} \] nth harmonic: \[ v_n = \frac{n v}{2L}, \quad n = 1, 2, 3, \dots \] Here \(L = 0.30\,\mathrm{m}\), \(v = 330\,\mathrm{m/s}\): \[ v_n = \frac{n \times 330}{2 \times 0.30} = \frac{n \times 330}{0.6} = 550n \ \mathrm{Hz} \] For \(v_2 = 2 \times 550 = 1100\,\mathrm{Hz} = 1.1\,\mathrm{kHz}\): resonant with the source at the second harmonic.

One end closed:
For a pipe closed at one end, fundamental frequency: \[ v_1 = \frac{v}{4L} \] Only odd harmonics: \[ v_3 = \frac{3v}{4L}, \quad v_5 = \frac{5v}{4L}, \dots \] Here \(v_1 = \frac{330}{4 \times 0.30} = \frac{330}{1.2} = 275\,\mathrm{Hz}\)
\(1.1\,\mathrm{kHz}\) would correspond to the fourth harmonic if all were allowed: \(275 \times 4 = 1100\,\mathrm{Hz}\). But only odd harmonics are possible, so no resonance at this frequency with one end closed.