Physics - Sound Wave Question with Solution | TestHub

The given progressive waves are $y_1=4\sin 708\pi t$ and $y_2=3\sin 700\pi t$.

The angular frequencies are ${\omega }_1=708\pi$ and ${\omega }_2=700\pi$.

The frequencies of the two sources are:
$f_1=\frac{{\omega }_1}{2\pi }=\frac{708\pi }{2\pi }=354$ Hz
$f_2=\frac{{\omega }_2}{2\pi }=\frac{700\pi }{2\pi }=350$ Hz

The number of beats heard per second is:
$n=f_1-f_2=354-350=4$

The amplitudes of the two waves are $A_1=4$ and $A_2=3$.

The maximum amplitude during waxing is $A_{max}=A_1+A_2=4+3=7$.
The minimum amplitude during waning is $A_{min}=A_1-A_2=4-3=1$.

The intensity of a wave is directly proportional to the square of its amplitude. The intensity ratio between waxing and waning is:
$\frac{I_{max}}{I_{min}}={\left(\frac{A_{max}}{A_{min}}\right)}^2={\left(\frac{7}{1}\right)}^2=\frac{49}{1}$

The number of beats is $4$ and the intensity ratio is $49:1$.

Answer: $4$ , $49:1$