Two sound waves having displacements $\mathrm{x}_1=2 \sin (1000 \pi \mathrm{t})$ and $\mathrm{x}_2=3 \sin…

Two sound waves having displacements $\mathrm{x}_1=2 \sin (1000 \pi \mathrm{t})$ and $\mathrm{x}_2=3 \sin (1006 \pi \mathrm{t})$, when interfere, produce
  1. 5 beats/s with maximum intensity 25 units
  2. 6 beats/s with maximum intensity 16 units
  3. 3 beats/s with maximum intensity 25 units
  4. 1 beats/s with maximum intensity 5 units

Solution

Given, $\begin{aligned} & \mathrm{X}_1=2 \sin (1000 \pi \mathrm{t}) \\ & \mathrm{X}_2=3 \sin (1006 \pi \mathrm{t}) \end{aligned}$
Comparing with standard equation, $\mathrm{x}=\mathrm{A} \sin \omega \mathrm{t}$ $\begin{aligned} & \omega_1=1000 \pi, \omega_2=1006 \pi ...(i)\\ & \mathrm{~A}_1=2, \mathrm{~A}_2=3 ...(ii)\\ & \therefore \quad \mathrm{f}_1=\frac{\omega_1}{2 \pi}=\frac{1000 \pi}{2 \pi}=500 \mathrm{~Hz} ...[From(i)]\\ & \mathrm{f}_2=\frac{\omega_2}{2 \pi}=\frac{1006 \pi}{2 \pi}=503 \mathrm{~Hz} ...[From(ii)]\end{aligned}$ $\therefore \quad$ Beat frequency $=503-500=3$ beats/second Intensity $\propto \mathrm{A}_{\text {max }}^2$ $\mathrm{A}_{\max }=\mathrm{A}_1+\mathrm{A}_2=2+3=5$ units ...[From(ii)] Intensity $=(5)^2=25$ units

Asked in: MHT CET 2024 (10 May Shift 1)

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