Question

Two waves of same amplitude superimpose to produce two beats per second. What is the ratio of maximum loudness to that of one of the waves?

• A. $2$
• B. $4$
• C. $8$
• D. $\infty$

Answer 1

The correct option is B $4$

let the waves be

$S_1=Asin2\pi f_{1}t$

$S_2=Asin2\pi f_{2}t$

$f_1-f_2=2$

then the superimposition of waves

$S=S_1+S_2=Asin2\pi f_{1}t+Asin2\pi f_{2}t=2Asin\pi t(f_1+f_2)cos\pi t(f_1-f_2)$

maximum Amplitude of the resultant is 2A

we know that intensity is proportional to $A^2$

so the ratio of resultant loudness and one of the wave loudness is

$\frac{I_1}{I_2}$ proportional to $\frac{(2A{)}^2}{A^2}=4$