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 624: 55

Answer

$93\;\rm m$

Work Step by Step

This is an open-closed tube problem. We need to find the length of the stunned where the frequencies of the standing waves inside it are given by $$f_m=\dfrac{mv}{4L}\tag{$m=1,3,5,...$}$$ And since we are dealing with only the odd numbers of $m$ since it is an open-closed tube model, the two next frequencies are at $m$, and $m+2$. So, $$f_m=\dfrac{mv}{4L}$$ $$f_{(m+2)}=\dfrac{(m+2)v}{4L}$$ Hence, $$f_{(m+2)}-f_m=\dfrac{(m+2)v}{4L}-\dfrac{mv}{4L}=\dfrac{2v}{4L}=\dfrac{v}{2L}$$ $$f_{(m+2)}-f_m=\dfrac{v}{2L}$$ Thus, $$L=\dfrac{v}{2[f_{(m+2)}-f_m]}$$ Plugging the two given frequencies and the speed of sound in the tunnel; $$L=\dfrac{335}{2[6.3-4.5]}=\color{red}{\bf 93}\;\rm m$$
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