velocity of sound on kundts tube

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Velocity of Sound using Kundt’s Tube

Aim:

1. To determine the wavelength of the stationary waves in a cylindrical tube.

2. To determine velocity of sound in air.

Apparatus:

(i) Cylindrical glass tube, (ii) Function generator, (iii) LF amplifier, (iv) Sound head, (v)

Cork dust, (vi) measuring scale, (vii) Universal Clamp.

Theory:

Interference and superposition

Sound waves in air are longitudinal waves and propagate in the form of compressions and

rarefactions. These waves produce alternately the states of compression and rarefaction at a point

in the medium. When two or more sound waves travel together, the superposition principle states

that the resultant wave at any point in the medium is the algebraic sum of the individual waves at

that point. Suppose two sine waves are travelling together and have same frequency, wavelength

and amplitude, but differ in phase, we can express their individual wave functions as

�� sin� � �� , �� sin � � � � �� (1)

And the resulting wave will be

� � 2� cos �� sin � � � �� (2)

Please note that the resulting wave will have same frequency and wavelength as the individual

waves. If the phase constant � = 0 or any even multiples of π, the waves interfere constructively

and the resultant amplitude is twice of either individual wave (Fig. 1(a)). When � is equal to π or

odd multiples of Pi, the resultant wave has zero amplitude everywhere and interfere destructively

(Fig. 1(b)). Finally when the phase constant � is arbitrary other than zero or multiples of Pi, the

resultant wave has an amplitude whose value is between 0 and 2A (Fig. 1(c)).

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Figure 1: Superposition of two identical waves y1 and y2 to yield a resultant wave y

Standing waves

Suppose we have two identical waves, travelling in opposite directions in the same medium.

�� sin� � �� , �� sin � � �� (3)

Then the resulting wave will be

� � 2� sin�� cos� �� (4)

which is the wave function of a standing wave. A standing wave, as shown in figure 2, is an

oscillation pattern with a stationary outline that results from the superposition of two identical

waves travelling in opposite directions.

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The displacement of a particle in the medium has a minimum value of zero when x satisfies the

condition sin (kx) = 0. i.e. when kx = nπ or x = nλ/2. These points of zero displacements are

called nodes. Similarly, the positions with maximum displacements occur when sin(kx) = ±1 or x

= nλ/4. These points are called antinodes. It should be noted that the distance between adjacent

nodes or antinodes will be λ/2 and the distance between a node and adjacent antinode will be λ/4.

Figure 2: Nodes and antinodes in a standing wave.

Standing waves in air column Standing waves can be set up in a glass tube as the result of

interference between longitudinal sound waves travelling in opposite directions. The phase

relationship between the incident wave and the wave reflected from one end of the tube depends

on whether that end of the tube is closed or open. If the tube is closed at one end, the closed end

is a displacement node, because the wall at this end does not allow longitudinal motion of the air

molecules.

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As a result, at the closed end the reflected

wave will be π out of phase with the incident

wave. Since the pressure wave is π/2 out of

phase with the displacement wave, the closed

end will be pressure antinode. The open end

of the air column is a displacement antinode

and pressure node.

The frequencies at which standing waves can

be set up in an air column enclosed by a tube

that is open at both ends can be easily

calculated. Because both ends are open, they

should be pressure nodes and displacement

antinodes. So the length, L of the air column

must be equal to an exact number of half-

wavelengths:

� � �λ� (5)

Where n is an integer number and λ is the

wave length. So the resonant frequencies are

� � �� (6)

Where in this case ν is the speed of

sound in air. The normal modes of vibration

are shown in the figure 3(a).

In the case of a tube that is closed at one end,

the closed end must be a displacement node

(pressure antinode) and the open end will be a displacement antinode (pressure node). The

resonant wave will be as shown in fig. 3(b). Since the distance between a node and an antinode is

an odd multiple of quarter wavelengths. i.e,

λ � �� (7)

and resonating frequencies

� � �� (8)

Figure 3

Kundt’s tube

In 1866, Kundt showed that when standing sound waves were excited in a tube, dust particles in

the tube will be arranged in periodic pileups. Thus the nodes and antinodes can be detected by

the characteristic striated vibration patterns of the cork dust as

wavelength can be obtained from distance between two successive nodes and antinodes (the

distance between two successive nodes and antinodes in a standing wave is

velocity of the sound can be calculated as

Figure 4: Kundt’s striations formed by the cork dust at one antinode

Procedure:

1. The apparatus consists of a transparent glass tube of length 640mm.

dust uniformly over the entire length of the glass tube. Ensure that both the glass tube and

cork dust are dry.

2. Connect the function generator, LF amplifier and the sound head. The sound head should

be placed close to one of the glass t

3. Tune the frequency generator slowly. At a certain frequency, a sanding wave will be

present in the tube and can be visualized as periodic pileups of cork powder forms ripples

or striations. This should not be confused with the nodes and antinodes. A s

wave will be formed only when Eqn. (5) is satisfied. If this condition satisfied, air does

not move at the node points, thus, the cork powder does not move either. At the antinode

points the air moves strongly and the powder particles are dis

only at node points. Measure the distance between

will give half of the wavelength. Note down the frequency of the LF amplifier.

4. Tune the frequency again so that a higher harmonic is obtained. Once

is observed, repeat step 3. Care should be taken to distribute the cork dust uniformly over

the entire length of tube after each measurement.

5. Calculate the velocity of sound in air.

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the characteristic striated vibration patterns of the cork dust as shown in the figure 4. The


distance between two successive nodes and antinodes in a standing wave is

velocity of the sound can be calculated as

Kundt’s striations formed by the cork dust at one antinode

The apparatus consists of a transparent glass tube of length 640mm. Distribute the cork


Connect the function generator, LF amplifier and the sound head. The sound head should

be placed close to one of the glass tube.

Tune the frequency generator slowly. At a certain frequency, a sanding wave will be


or striations. This should not be confused with the nodes and antinodes. A s



points the air moves strongly and the powder particles are dispersed and can sediment

only at node points. Measure the distance between two nodes or two antinodes, which


Tune the frequency again so that a higher harmonic is obtained. Once the standing wave


the entire length of tube after each measurement.

Calculate the velocity of sound in air.



shown in the figure 4. The


distance between two successive nodes and antinodes in a standing wave is λ/2). Then the

(9)

Kundt’s striations formed by the cork dust at one antinode.

Distribute the cork


Connect the function generator, LF amplifier and the sound head. The sound head should

Tune the frequency generator slowly. At a certain frequency, a sanding wave will be


or striations. This should not be confused with the nodes and antinodes. A sharp standing



and can sediment

two antinodes, which


the standing wave


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6. Close one end of the tube and repeat the experiment.

Experimental arrangement for Kund’s tube

Observation and results:

Table-1: Determination of velocity of sound in air using an open tube.

Observation

no.

Frequency, f

(Hz)

Distance

between two

nodes or

antinodes, λ/2

(m)

Wavelength,

λ (m)

Velocity of

sound in air,

v = fλ (m/sec)

Mean v(m/s)

1

2

3

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Table-2: Determination of velocity of sound in air using a closed tube

Observation

no.

Frequency, f

(Hz)

Distance

between two

nodes or

antinodes, λ/2

(m)

Wavelength,

λ (m)

Velocity of

sound in air,

v = fλ (m/sec)

Mean

v(m/s)

1

2

3

Questions: 1) What are resonant frequencies?

2) How does a wave get reflected from the open end of a tube?

3) Why there are striations or ripples at antinodes? (You may refer to R.A. Carman,

Am.J.Phys.vol.23, 505 (1955), E. Hutchisson and F. B. Morgan, Phys. Rev. vol. 37, 1155

(1931)).

velocity of sound on kundts tube

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