If two sound waves, `y_1 = 0.3 sin 596 pi [t - x//330]` and `y_2 =0.5 sin 604 pi[t - x//330]` are superposed, what will be the (a) frequency of resultant wave (b) frequency at which the amplitude of resultant waves varies (c ) frequency at which beats aer produced. Find also the ratio of maximum and minimum intensities of beats.

Explanation

Comparing the given wave equation with `y=A sin omega[t-(x//v)][as k//w=1//v]` we find that here `A_(1)=0.3 and omega_(1)=2pi f_(1)=596 pi`,
i.e, `f_(1)=298 Hz and A_(2)= 0.5 omega_(2)=2pi f_(2)=604 pi, i.e., f_(2)=302 Hz`
So (a) The frequency of the resultant wave
`f_(av)=(f_(1)+f_(2))/(2)=((298+302))/(2)=300Hz`
(b) The frequency at which amplitude of resultant wave varies `f_(A)=(f_(1)-f_(2))/(2)=((298-302))/(2)=2Hz`
(c) The frequency at which beats are produced `f_(b)=2f_(A)=f_(1)-f_(2)=4Hz`
The ratio of maximum to minimum intensities of beat
`(I_("max"))/(I_("min")) =((A_(1)+A_(2))^(2))/((A_(1)-A_(2))^(2))=((0.3+0.5))/((0.3-0.5))=(64)/(4)=16`