Name: Wave Motion:Transverse and Longitudinal Waves
Uploaded: 2019-07-19
Duration: 3 min 47 s
Description: Waves transfer energy from one place to another. There are two types of wave motions depending upon the movement of the particles about their axes. Watch the...

Question

Explain the formation of stationary wave in an air column enclosed in an open pipe. Derive the equation for the frequencies of the harmonics produced.

Accepted Answer

When a source of sound is kept at the open end of a pipe of an open pipe, compression and rarefactions are se up in the pipe. At the end a rarefaction is formed and returns as a compression. Thus, incident and reflected waves superimpose and stationary wave is formed. An open pipe has anti nodes at both ends. The wavelength of the wave is given by $λ = \frac{2 L}{n}$

Where $L$ = length of pipe $n$ = integer for mode no.

First normal mode of vibration:

In this mode there will be two anti nodes and one node corresponding to $n = 1$

$λ_{1} = \frac{2 L}{1} or L = \frac{λ_{1}}{2}$ and frequency $ν = \frac{v}{λ_{1}} = \frac{v}{2 L}$

This is the lowest (fundamental frequency) or first harmonic

Second normal mode of vibration:

In this case there will be three antinode and two nodes and the wavelength is corresponding to $n$ $= 2$ , the wave length is given by:

$λ_{2} = \frac{2 L}{2} = L frequency = ν_{2} = \frac{2 v}{2 L} = 2 ν_{1}$

This is double of the fundamental and called second harmonic or first overtone.

Third normal mode of vibration:

In this case there will be $4$ anti nodes and $3$ nodes. The wavelength is given by

$λ_{3} = \frac{2 L}{3} or L = \frac{3 λ_{3}}{2} and frequency = ν_{3} = \frac{v}{λ_{3}} = \frac{3 v}{2 L} or 3 ν_{1}$

This is third harmonic or second overtone.

Thus, it is clear that both even and odd harmonics are produced in open organ pipe.

Explain the formation of stationary wave in an air column enclosed in an open pipe. Derive the equation for the frequencies of the harmonics produced.

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