Question

Two sound waves having wavelengths of $87\,cm$ and $88.5\,cm$ respectively, when superimposed, produce $10$ beats per second. Find the velocity of sound.

Answer 1

Hint:Wave interference is the phenomenon that occurs when two waves meet while travelling along the same medium. The two waves of nearby frequencies travelling in a medium along the same direction meet at a point called a beat. The velocity of sound in medium can be obtained by using the wavelength and frequency.

Formula used:
\[N = V\left( {\dfrac{1}{{{\lambda _1}}} - \dfrac{1}{{{\lambda _2}}}} \right)\]
Where, $N = $ beat frequency (The count of beats per second is equivalent o the difference in frequencies of two waves is called as beat frequency)
$V = $ Speed of sound in a medium, ${\lambda _1}$ And ${\lambda _2}$ are the wavelengths of sound.

Complete step by step answer:
Given, $N = 10$ Beats per seconds. ${\lambda _1}$ = $87 cm = 0.87 m$.${\lambda _2}$ = $88.5\,cm$ =$0.885\,m$.
The wavelength of the two sound wave and the beats per second is given in the question by using wavelengths and beats per second the velocity of sound can be calculated as follows :-
We known that frequency of the beats is given by,
\[N = \dfrac{V}{{{\lambda _1}}} - \dfrac{V}{{{\lambda _2}}}\]
Taking $V$ common in above equation,
\[N = V\left( {\dfrac{1}{{{\lambda _1}}} - \dfrac{1}{{{\lambda _2}}}} \right)\]
Substituting the given data in the above equation, we get
\[10 = V\left( {\dfrac{1}{{0.87}} - \dfrac{1}{{0.885}}} \right)\]
On simplifying the above equation,
\[10 = V\left( {\dfrac{{0.885 - 0.87}}{{0.87 \times 0.885}}} \right)\]
$\Rightarrow V = \dfrac{{10 \times 0.76995}}{{0.015}}$
$\therefore V = 513.3\,m/s$

Hence, velocity of sound in medium is $513.3\,m/s$.

Note:The beat frequency is different from frequency since beat frequency is the difference in frequency of two waves. It is because of constructive and destructive interference. The S.I unit for wavelength is $m$, we need to first convert $cm$ to $m$ and then the calculation should be done.