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Chapter 14: Problem 76

A 1.5 -m-long pipe has one end open. Among its possible standingwave frequencies is \(225 \mathrm{Hz} ;\) the next higher frequency is \(375 \mathrm{Hz}\) Find (a) the fundamental frequency and (b) the sound speed.

Short Answer

Expert verified
Thus, (a) the fundamental frequency of the tube is 225Hz and (b) by calculation using the given formula, we can find the speed of sound in the pipe.

Step by step solution

01

Analyze the provided frequencies

According to the problem, we know that 225Hz and 375Hz are standing wave frequencies of a tube which is closed at one end and open at the other end. Following the properties of such a tube, the frequencies of its normal modes (harmonics) follow the pattern \(f, 3f, 5f, ...\) where \(f\) is the fundamental frequency (first harmonic). Comparing with the given frequencies, we can see that 225Hz and 375Hz could be the first and second harmonics respectively, indicating that the fundamental frequency is 225Hz.
02

Calculation of fundamental frequency

Based on analysis in Step 1, the fundamental frequency (first harmonic) or the lowest possible frequency of the tube is 225 Hz.
03

Calculation of sound speed

The formula to calculate the speed of sound using the frequency of the first overtone (here it's 375 Hz, or the third harmonic, since this is an open-closed tube) and the length of the tube is \[V = 4f_{overtone}L\] where \(V\) is the speed of sound, \(f_{overtone}\) is the frequency of first overtone and \(L\) is the length of the tube. Here, we can substitute \(f_{overtone} = 375 Hz\) and \(L = 1.5 m\) into the equation to find \(V\).

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