Question

Two superimposing waves are represented by equation y1=2sin2pi(10t-0.4x) and y2=4sin2pi(20t-0.8x). The ratio of Imax to Imin is

(1) 36:4 (2) 25:9 (3) 1:4 (4) 4:1

Answer is (2). How do we come to it?

Answer 1

Formula of intensity is given by
$\bold{I = \frac{1}{2}\rho\nu\omega^2A^2}\:\: or,\bold{I=2\pi^2f^2\rho\nu A^2}$
Hence, $\bold{I\propto f^2A^2}$
Here f is the frequency , A is the amplitude of wave , ν is the speed of wave and ρ is the Density of medium.

Now, Given,
y₁ = 2sin(20πt - 0.8πx) ⇒f₁ = 10 , A₁ = 2 [ ∵ y = Asin(2πf ± kx) ]
y₂ = 4sin(40πt - 1.6πx)⇒f₂ = 20 , A₂ = 4

Now, $\bold{\frac{I_{max}}{I_{min}}=\frac{(f_1A_1+f_2A_2)^2}{(f_2A_2-f_1A_1)^2}}$ [ I didn't give calculations how to find this formula , you should memorize ]
put the values ,
Imax/Imin = (10 × 2 + 20 × 4)²/(20 × 4 - 10 × 2)²
= (20 + 80)²/(80 - 20)²
= (100/60)²
= (5/3)² = 25 :9

Hence, $\bold{\frac{I_{max}}{I_{min}}=\frac{25}{9}}$

Answer 2

Answer:

this is the best i can do☄☄☄☄