Name: Closed Organ Pipe
Uploaded: 2019-07-19
Duration: 3 min 44 s
Description: An organ pipe is closed at one end and open at the other end. Bottle, whistle, etc., are examples of a closed organ pipe. Let's learn about its different mod...

Question

Explain the formation of standing waves in an organ pipe open at both ends and discuss the various mode of vibration.

Accepted Answer

Standing waves are formed in an open end organ pipe by incident of reflected waves.

Consider an organ pipe of length $l$ open of both ends lying along $x$ -axis with ends at $x = 0$ and $x = 1$ . As the pipe is open at both ends, there is an antinode at each end.

The incident and reflected waves are represented by

$y_{1} = a \sin \frac{2 π}{λ} (v t + x)$

$y_{2} = a \sin \frac{2 π}{λ} (v t - x)$

The reflection at open end introduces no change in phase.

On, superposition

$y = y_{1} + y_{2}$

$= a \sin \frac{2 π}{λ} (v t + x) + a \sin \frac{2 π}{λ} (v t - x) = 2 a \sin (\frac{2 πvt}{λ}) \cos (\frac{2 πx}{λ})$

At $x = l$ displacement will be maximum

$y = 2 a \sin (\frac{2 πvt}{λ}) \cos (\frac{2 π l}{λ})$

$y$ will always be maximum. If $\cos (\frac{2 π l}{λ}) = \pm 1$ or $\frac{2 πl}{λ} = m$ where $m = 1, 2, 3$

or $λ = \frac{2 l}{m} . . . . . . . . . . . . . . . . . . . . . . . . . . (1)$

Various modes of vibration

First three modes of vibration are shown in fig. $1$

First mode of vibration- Let $λ_{1}$ the wavelength of standing waves set up in the air column corresponding to $m = 1$ , then from eq. $1$

$λ_{1} = 2 l o r l = \frac{λ_{1}}{2}$

It is shown in fig. $1$ . It is the fundamental mode. Its frequency $n_{1} = \frac{ϑ}{λ} = \frac{ϑ}{2 l} . . . . . . . . . . . . . . . . . . . . (i i)$

It is also known as first harmonic.

Second mode of vibration- Let $λ_{2}$ is the wavelength of the standing wave corresponding to $m = 2$ .

These from, eq. $1$ $λ_{2} = \frac{2 l}{2}$ or $l = \frac{2 λ_{2}}{2}$

This mode of vibration is shown in fig. $1 b$ . Frequency of vibration of this mode is $n_{2} = \frac{v}{λ_{2}} = \frac{2 v}{2 l} = 2 h$

It is called first overtone or second harmonic of the open pipe.

Third mode of vibration- Let $λ_{3}$ , the positive wavelength for $m = 3$ . It is shown in fig. $1 c$

From eq. $1$ , $λ_{3} = \frac{2 l}{3}$ or $l = \frac{3 λ_{3}}{2}$

Frequency of vibration in this mode is given by

$n_{3} = \frac{v}{λ_{3}} = \frac{3 v}{2 l} = 3 (\frac{v}{2 l}) = 3 h$

The node produced in this mode is third harmonic or second overtone.

In an open pipe both the even and odd harmonics are produced.

Explain the formation of standing waves in an organ pipe open at both ends and discuss the various mode of vibration.

Important Questions on Superposition of Waves-2 : Stationary (Standing) Waves : Vibration of Air Columns

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