Question
Two sources of sound placed close 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 beats per second with intensity ratio $$25 : 16$$ between waxing and waning.
B.
8 beats per second with intensity ratio $$25 : 16$$ between waxing and waning
C.
8 beats per second with intensity ratio $$81: 1$$ between waxing and waning
D.
4 beats per second with intensity ratio $$81: 1$$ between waxing and waning
Answer :
4 beats per second with intensity ratio $$81: 1$$ between waxing and waning
Solution :
$$\eqalign{
& 2\pi {f_1} = 600\,\pi \cr
& {f_1} = 300\,......\left( {\text{i}} \right) \cr
& 2\pi {f_2} = 608\,\pi \cr
& {f_2} = 304\,......\left( {{\text{ii}}} \right) \cr
& \left| {{f_1} - {f_2}} \right| = 4\,beats \cr
& \frac{{{I_{\max }}}}{{{I_{\min }}}} = \frac{{{{\left( {{A_1} + {A_2}} \right)}^2}}}{{{{\left( {{A_1} - {A_2}} \right)}^2}}} = \frac{{{{\left( {5 + 4} \right)}^2}}}{{{{\left( {5 - 4} \right)}^2}}} = \frac{{81}}{1},\,{\text{where}}\,{A_1},{A_2}\,{\text{are amplitudes of given two sound wave}}{\text{.}} \cr} $$