Two sounding bodies producing progressive waves given by $y_1 = 4 \sin (400 \pi t)$ and $y_2 = 3 \sin (404…

Two sounding bodies producing progressive waves given by $y_1 = 4 \sin (400 \pi t)$ and $y_2 = 3 \sin (404 \pi t)$, where $t$ is in second, are situated near the ears of a person. The person will hear
  1. 2 beats per second with intensity ratio 4/3 between maxima and minima
  2. 2 beats per second with intensity ratio 49 between maxima and minima
  3. 4 beats per second with intensity ratio 49 between maxima and minima
  4. 4 beats per second with intensity ratio 4/3 between maxima and minima

Solution

Beat frequency, $f_b = f_2 - f_1 = \frac{\omega_2 - \omega_1}{2\pi}$ $= \frac{404\pi - 400\pi}{2\pi} = 2\text{ Hz}$ Further, $A_1 = 4$ and $A_2 = 3$. Hence, $\frac{I_{\max}}{I_{\min}} = \left(\frac{A_1 + A_2}{A_1 - A_2}\right)^2 = 49$

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