Name: Stationary Wave in Closed and Open Organ Pipe
Uploaded: 2019-07-19
Duration: 11 min 39 s
Description: Organ pipes are musical instruments that produce a musical sound by blowing air into the pipe. Let's study the formation of stationary waves in organ pipes.

Question

What are beats? Prove that the number of beats per second produced by two sources of sound is equal to the difference in the frequencies of the two sources

(n_{1} ~ n_{2})

.

Accepted Answer

When two sound waves of nearly equal frequencies are produced simultaneously, then the intensity of the resultant sound produced by their superposition increases and decreases alternately with time. This rise and fall in the intensity of sound is called the phenomenon of ’beats’.

Let the frequency of the two waves be $ν_{1}$ and $ν_{2}$ . The equation for the waves are written as:

$s_{1} = a \cos 2 π ν_{1} t and s_{2} = a \cos 2 π ν_{2} t$

On superposition, their resultant can be written as:

$s = s_{1} + s_{2} = a \cos 2 π ν_{1} t + a \cos 2 π ν_{2} t$

$\Rightarrow s = 2 a \cos π (ν_{1} - ν_{2}) \cos π (ν_{1} + ν_{2})$

As the difference in their frequency $(ν_{1} - ν_{2})$ is very small, the resultant wave can be taken as a wave with a frequency $= (ν_{1} + ν_{2})$ and amplitude $2 a \cos (ν_{1} - ν_{2})$ .

For the maximum amplitude, we have:

$\cos 2 π (\frac{ν_{1} - ν_{2}}{2}) t = \pm 1$

$\Rightarrow (ν_{1} - ν_{2}) t = k$ where $k = 0, 1, 2, 3, . . .$
$\Rightarrow t = \frac{k}{ν_{1} - ν_{2}}$

Substituting $k = 0, 1, 2, 3, . . .,$ we get

$t = 0, \frac{1}{ν_{1} - ν_{2}}, \frac{2}{ν_{1} - ν_{2}}, \frac{3}{ν_{1} - ν_{2}}, . . .$

Hence, at these instants the amplitude will be maximum and so the intensity of sound will also be maximum. We can see that the time-interval between two consecutive maximum intensities is $\frac{1}{ν_{1} - ν_{2}}$ second. Hence, in $1$ second the intensity will be maximum $ν_{1} - ν_{2}$ times and $ν_{1} - ν_{2}$ beats will be heard in 1 second. Number of beats per second (beat frequency)

$= n_{1} - n_{2} =$ difference of the frequencies of sound-sources.

What are beats? Prove that the number of beats per second produced by two sources of sound is equal to the difference in the frequencies of the two sources $(n_{1} ~ n_{2})$ .

Important Questions on Superposition of Waves-1 : Interference and Beats

Learn and Prepare for any exam you want !

What are beats? Prove that the number of beats per second produced by two sources of sound is equal to the difference in the frequencies of the two sources n1~n2.

Important Questions on Superposition of Waves-1 : Interference and Beats

Learn and Prepare for any exam you want !

What are beats? Prove that the number of beats per second produced by two sources of sound is equal to the difference in the frequencies of the two sources $(n_{1} ~ n_{2})$ .