b
(b)

Formula of intensity is given by

\(I =\frac{1}{2} \rho \nu \omega^2 A ^2 \text { or, } I = 2 \pi^2 f ^2 \rho \nu A ^2\)

Hence, \(I \propto f ^2 A ^2\)

Here \(f\) is the frequency, \(A\) is the amplitude of wave, \(v\) is the speed of wave and \(\rho\) is the Density of medium.

Now, Given,

\(y_1=2 \sin (20 \pi t-0.8 \pi x) \Rightarrow f_1=10, A_1=2[\because y=A \sin (2 \pi f \pm k x)]\)

\(y_2=4 \sin (40 \pi t-1.6 \pi x) \Rightarrow f_2=20, A_2=4\)

Now, \(\frac{ I _{\max }}{ I _{\min }}=\frac{\left( f _1 A _1+ f _2 A _2\right)^2}{\left( f _2 A _2- f _1 A _1\right)^2}[I\) didn't give calculations how to find this formula, you should memorize ]

put the values,

Imax/Imin \(=(10 \times 2+20 \times 4)^2 /(20 \times 4-10 \times 2)^2\)

\(=(20+80)^2 /(80-20)^2\)

\(=(100 / 60)^2\)

\(=(5 / 3)^2=25: 9\)

Hence \(\frac{I_{\max }}{I_{\min }}=\frac{25}{9}\)