Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

Published by Pearson
ISBN 10: 0321740904
ISBN 13: 978-0-32174-090-8

Chapter 21 - Superposition - Exercises and Problems - Page 625: 69

Answer

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Work Step by Step

First, we need to find the wavelength of the three identical sources. $$\lambda=\dfrac{v}{f}=\dfrac{340}{170}=\bf 2\;\rm m$$ $$\color{blue}{\bf [a]}$$ We know that the net amplitude at some point is determined by the interference of the given waves. At the point you stand, the two sources 2 and 3 interact destructively since the path length difference of the two waves is half wavelength. $$\Delta r_{3\rightarrow2}= r_3-r_2 = \sqrt{4^2+3^2}-4=1$$ Now the net amplitude depends only on the first source above where we know that the amplitude of the wave from each speaker is $a$. Therefore, $$\boxed{A=a}$$ $$\color{blue}{\bf [b]}$$ If we canceled the middle source 2, the interference at the given point from 1 and 3 is perfectly constructive since $\Delta r_{1\rightarrow 3}=0$. This makes the net amplitude $2a$ So to have a net amplitude of $3a$, we need to move it to the left one half-wavelength. Then the three waves reach point P in phase. $$\color{blue}{\bf [c]}$$ We know that the sound intensity is given by $$I=CA^2$$ So, when the amplitude is maximum, $A=3a$ $$I_{max}=C(3a)^2=9Ca^2$$ and from a single source, $$I_{single}=Ca^2$$ Hence, $$\boxed{I_{max}=9I_{single}}$$
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