Question

Two identical speakers emit sound waves of frequency 660 Hz uniformly in all directions. The audio output of each speaker is 1 m W and the speed of sound in air 330 m/s. A point P is a distance 2m from one speaker and 3m from the other. Find
a) Intensities $ {I_1} $ and $ {I_2} $ from each speaker from P separately.
b) If they are in coherence, the resultant I at P.
c) The resultant I at P if the two waves differ by a phase of $ \pi $ at the time of emission.

Answer 1

Hint
Use the formula $ {I_1} = \dfrac{P}{{4\pi {R^2}}} $ to take out individual intensity due to speakers. Then use the formulas $ I = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} \cos \Delta \;\phi \;\; $ and $ \,\;\phi \;\; = \dfrac{{2\pi }}{{\;\lambda }}\Delta x $ to solve the problem. For part b) take the phase difference as 0.

Complete step by step answer
Given, Frequency = 660 Hz
Speed of sound in air = 330 m/s
Power of each speaker = 1 m W
a) Intensities $ {I_1} $ and $ {I_2} $ from each speaker from P separately.
$\Rightarrow {I_1} = \dfrac{P}{{4\pi {R^2}}} $
$\Rightarrow {I_1} = \dfrac{{1 \times {{10}^{ - 3}}}}{{4\pi \times {{(2)}^2}}} $
$\Rightarrow {I_1} = 19.9 \times {10^{ - 6}}{\text{W/}}{{\text{m}}^{\text{2}}} $
(putting the values from the question)
For second speaker:
$\Rightarrow {I_2} = \dfrac{P}{{4\pi {R^2}}} $
$\Rightarrow {I_2} = \dfrac{{1 \times {{10}^{ - 3}}}}{{4\pi \times {{(3)}^2}}} $
$\Rightarrow {I_2} = 8.85 \times {10^{ - 6}}{\text{W/}}{{\text{m}}^{\text{2}}} $
b) If they are in coherence, the resultant I at P.
$\Rightarrow I = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} \cos \Delta \;\phi \;\; $
$\Rightarrow {\text{and}}\,\;\phi \;\; = \dfrac{{2\pi }}{{\;\lambda }}\Delta x $
We know
$\Rightarrow v = f\;\lambda $
$\Rightarrow \;\lambda = \dfrac{v}{f} = \dfrac{{330}}{{660}} = \dfrac{1}{2} $
$\Rightarrow \Delta x = 3 - 2 = 1 $ m
Hence,
$\Rightarrow \Delta \;\phi \;\; = \dfrac{{2\pi }}{{\dfrac{1}{2}}} = 4\pi $
Hence the net intensity becomes,
$\Rightarrow I = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} \cos \Delta \;\phi \;\; $
$\Rightarrow I = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} \cos (4\pi ) $
$\Rightarrow I = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} $
Putting the values we have,
$\Rightarrow I = 19.9 + 8.85 + 2\sqrt {19.9 \times } 8.85 $
$\Rightarrow I = 55.3 \times {10^{ - 6}}W/{m^2} $
c) The resultant I at P if the two waves differ by a phase of $ \pi $ at the time of emission.
Here, $ \;\phi \;\; = \pi $
Hence resultant intensity becomes,
$\Rightarrow I = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} \cos \Delta \;\phi \;\; $
$\Rightarrow I = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} \cos (\pi ) $
$\Rightarrow I = {I_1} + {I_2} - 2\sqrt {{I_1}{I_2}} $
Putting values we get,
$\Rightarrow I = 19.9 + 8.85 - 2\sqrt {19.9 \times } 8.85 $
$\Rightarrow I = 28.7 \times {10^{ - 6}}W/{m^2} $.

Note
That the formula $ {I_1} = \dfrac{P}{{4\pi {R^2}}} $ is only applicable for point sources. If the source is a line source or a sheet, then we can’t use this formula. We will have to use different formulas for taking out intensity.