Question

When the wavelength of sound changes from \[1m{\text{ to 1}}{\text{.01}}m\] , the number of beats heard are 4. The velocity of sound is-
A. \[404m/s\]
B. \[4.04m/s\]
C. \[414m/s\]
D. \[400m/s\]

Answer 1

Hint:There is one relation between the number of beats heard, wavelength and the velocity of the sound. By using that particular formula, we can easily find the value of velocity as change in wavelength and number of beats are given.

Formula Used:
\[b = {\upsilon _1} - {\upsilon _2} = v\left( {\dfrac{1}{{{\lambda _1}}} - \dfrac{1}{{{\lambda _2}}}} \right)\]------- (i)
Here b is number of beats and also we know that \[\upsilon = \dfrac{v}{\lambda }\]
Here V is the velocity,\[\lambda \] is the wavelength and \[\upsilon \] is the frequency.

Complete step by step answer:
Given: Number of beats heard (b) =4 and \[{\lambda _1},{\lambda _2}\] is given- \[1m{\text{ to 1}}{\text{.01}}m\] respectively. Now substitute all the given values in the equation (i), we get-
\[
4 = v\left( {\dfrac{1}{1} - \dfrac{1}{{1.01}}} \right) \\
\therefore v = 404m/s
\]
So, the velocity of the sound is \[v = 404m/s\].

Hence the option A is correct.

Additional information:
A wave not only travels in space but it also propagates through time, so if the two waves can produce interference by overlapping in space they should also produce an interference pattern when they overlap in time, this phenomenon is called beats.The beat frequency is when two sound waves with different frequencies come across each other, then their amplitude gets added and subtracted alternatively for a given time period. This leads to the growth of the sound to louder and softer.

Note:As we know that, \[c = \nu \lambda \], here c is the speed of light, is the frequency and is the wavelength. For finding the frequency we can use this relation.An important occurrence of the interference of waves is in the phenomenon of beats. In the simplest case, beats result when two sinusoidal sound waves of equal amplitude and very nearly equal frequencies mix. The frequency of the resulting sound (F) would be the average of the two original frequencies: \[f = \dfrac{{{f_1} + {f_2}}}{2}\]