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 using suitable diagram, the formation of standing waves in a closed pipe. How this may be used to determine the frequency of the source of sound?

Accepted Answer

When two progressive waves of the same amplitude and frequency travel in a bounded medium with the same speed but in opposite direction, then their superposition results in a stationary or standing wave.

When we place a source of sound near the open end, the wave travels towards the closed end and after the reflection forms a standing wave with the superposition with the incident wave. Because an open end acts like a free end, while a closed end acts like a fixed end, the standing waves for a pipe that is open at only one end have anti-nodes at the open end and nodes at the closed end. The following normal modes are possible

First normal mode:In this mode, a node is formed at the closed and an anti node is formed at the open end as shown in the figure. If the wave length of stationary wave is $λ_{1}$ , then the length of closed pipe is given by

$L = \frac{λ_{1}}{4}$ and frequency $ν = \frac{v}{λ_{1}} = \frac{v}{4 L}$ where $v$ = velocity .

Second normal mode of vibration : in this mode two nodes and two anti nodes are formed. If the wave length of the wave is $λ_{2}$ , then

$L = \frac{3 λ_{2}}{4} or λ_{2} = \frac{4}{3} L$ and frequency= $ν_{2} = \frac{v}{λ_{2}} = \frac{3 v}{4 L} = 3 ν_{1}$ .

i.e frequency of first over tone is three times that of the fundamental.

Third mode of normal vibration frequency. In this mode three nodes and three anti nodes are formed. If $λ_{3}$ is the wavelength of the standing wave then.

$L = \frac{5 λ_{3}}{4} or λ_{3} = \frac{4 L}{5} and ν_{3} = \frac{v}{λ_{3}} = \frac{5 v}{4 L} or 5 ν_{1}$ .

Determination of speed of sound by resonance tube:

A resonance tube (a hollow cylindrical tube) partially filled with water and forced into vibration by a tuning fork. This tube acts as a closed pipe organ. To get the resonance position, the vibrating tuning fork is held near the mouth of the tube and the position of water level is adjusted to get maximum loud sound .

At the first resonance condition, the length of the air column above the water is equal to one fourth of the wavelength of the sound, as shown in the diagram.Water is again lowered in the tube to get the second position of the resonance. At this condition, the length of the air column is $\frac{3 λ}{4}$ . The difference of these reading will give exact value of $\frac{λ}{2}$ .The speed of the sound is calculated by the relation:

$c = ν \times λ$ , where $ν$ = frequency of the running fork used. By putting the known value of the speed of sound, the frequency of the source can be determined.

Explain using suitable diagram, the formation of standing waves in a closed pipe. How this may be used to determine the frequency of the source of sound?

Important Questions on Waves

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