Name: Oscillations of a Stretched String
Uploaded: 2019-07-19
Duration: 6 min 4 s
Description: In this video, we will learn about the standing wave formation in a string fixed at both ends and a few calculations on it for a better understanding.

Question

Show that only odd harmonics are present in the vibrations of air column in a pipe closed at one end.

Accepted Answer

Consider a narrow cylindrical pipe of length $I$ closed at one end. In a cylindrical pipe closed at one end, sound waves are reflected at the closed end with a phase reversal and at the open end without a phase reversal. Under the right conditions Interference between incident and reflected waves creates stationary waves in the air column.
At this point stationary waves in the air column must satisfy two boundary conditions ie; a node at the closed end and an anti node at the open end.

The corrected length of air column $L =$ length of air column in pipe $l +$ end correction at the open end.
$∴ L = l + e$
The first mode of vibrations of air column closed at one end is as shown in Figure.

Let $v$ be the speed of sound in air. This is the simplest mode of vibration of air column closed at one end and an anti node at the open end. The distance between a node and a subsequent anti node is equal to $\frac{λ}{2}$ where $λ$ is the wavelength of fundamental mode of vibrations in air column.

Length of the air column,

$L = \frac{λ}{4} ∴ λ = 4 L$

If $n$ is the fundamental frequency, we have
$v = n λ ∴ n = \frac{v}{λ} = \frac{v}{4 L} = \frac{v}{4 (l + e)}$

The fundamental frequency is also known as the first harmonic.

In the second mode of vibration of the air column, two nodes and two anti nodes are formed. If $n_{1}$ is the frequency and $λ_{1}$ is the wavelength of wave in this mode of vibrations in air column, we have,

Length of the air column,

$L = \frac{3 λ_{1}}{4} λ_{1} = \frac{4 L}{3}$

The velocity of the second wave,

$v = n λ_{1} n = \frac{v}{λ_{1}} = \frac{3 v}{4 L} n = \frac{3 v}{4 (l + e)}$

This frequency is called the third harmonics or first overtone. Similarly, during the third mode of vibration of the air column, three nodes and three anti nodes are formed.
The next higher mode of vibrations of air column closed at one end is as shown in Figure.

Here the same air column is made to vibrate in such a way that it contains a node at the closed end, an anti node at the open endwith two more nodes and anti nodes in between. If $n_{2}$ is the frequency and $λ_{2}$ is the wavelength of the wave in this mode of vibrations in air column, we have

The length of the air column,

$L = \frac{5 λ_{2}}{4} λ_{2} = \frac{4 L}{5} n_{2} = \frac{v}{λ_{2}} n_{2} = \frac{v}{(\frac{4 L}{5})} n_{2} = \frac{5 v}{4 L} n_{2} = \frac{5 v}{4 (l + e)}$

This frequency is called the fifth harmonic or second overtone.
Thus, we see that the frequencies of the modes of vibrations are in the ratio $n : n_{1} : n_{2} = 1 : 3 : 5$ . This shows that only odd harmonics are present in the modes of vibrations of the air column closed at one end.

Show that only odd harmonics are present in the vibrations of air column in a pipe closed at one end.

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