Question
Two sources of sound placed closed to each other, are emitting progressive waves given by $${y_1} = 4\sin 600\,\pi t$$ and $${y_2} = 5\sin 608\,\pi t$$
An observer located near these two sources of sound will hear
A.
4 beat/s with intensity ratio $$25 : 16$$ between waxing and waning
B.
8 beat/s with intensity ratio $$25 : 16$$ between waxing and waning
C.
8 beat/s with intensity ratio $$81 : 1$$ between waxing and waning
D.
4 beat/s with intensity ratio $$81 : 1$$ between waxing and waning
Answer :
4 beat/s with intensity ratio $$81 : 1$$ between waxing and waning
Solution :
$$\eqalign{
& {\text{Given,}}\,{y_1} = 4\sin 600\,\pi t \cr
& {\text{and}}\,{y_2} = 5\sin 608\,\pi t \cr} $$
Comparing with general equation
$$\eqalign{
& y = a\sin 2\pi ft \cr
& {\text{We}}\,{\text{get,}}\,\,{f_1} = 300\,Hz. \cr
& {\text{and}}\,\,{f_2} = 304\,Hz \cr
& {\text{So,number of beats}} = {f_2} - {f_1} = 4{s^{ - 1}} \cr
& {\text{We know that,}} \cr
& \frac{{{I_{\max }}}}{{{I_{\min }}}} = {\left( {\frac{{{a_1} + {a_2}}}{{{a_1} - {a_2}}}} \right)^2} \cr
& = {\left( {\frac{{4 + 5}}{{4 - 5}}} \right)^2} = 81 \cr} $$