HARMONICS OF A PIPE GOAL Find frequencies of open and closed pipes. PROBLEM A pipe is 2.46 m long. (a) Determine the frequencies of the first three harmonics if the pipe is open at both ends. Take 343 m/s as the speed of sound in air. (b) How many harmonic frequencies of this pipe lie in the

Question

HARMONICS OF A PIPE

GOAL Find frequencies of open and closed pipes.

PROBLEM A pipe is [latex]2.46 \mathrm{~m}[/latex] long. (a) Determine the frequencies of the first three harmonics if the pipe is open at both ends. Take [latex]343 \mathrm{~m} / \mathrm{s}[/latex] as the speed of sound in air. (b) How many harmonic frequencies of this pipe lie in the audible range, from 20 Hz to [latex]20 000 \mathrm{~Hz}[/latex] ? (c) What are the three lowest possible frequencies if the pipe is closed at one end and open at the other?

STRATEGY Substitute into Equation [latex]14.18[/latex]

[latex]f_n = n \frac{v}{2L} = nf_1 \qquad n = 1,2,3, . . .[/latex] [14.18]

for part (a) and Equation 14.19

[latex]f_n = n \frac{v}{4L} = nf_1 \qquad n = 1,3,5, . . .[/latex] [14.19]

for part (c). All harmonics, [latex]n=1,2,3[/latex]... are available for the pipe open at both ends, but only the harmonics with [latex]n=1,3,5, \ldots[/latex] for the pipe closed at one end. For part (b), set the frequency in Equation [latex]14.18[/latex] equal to [latex]2.00 imes 10^{4} \mathrm{~Hz}[/latex].

Accepted Answer

(a) Find the frequencies if the pipe is open at both ends.

Substitute into Equation 14.18, with [latex]n=1[/latex] :

[latex]f_{1}=\frac{v}{2 L}=\frac{343 \mathrm{~m} / \mathrm{s}}{2(2.46 \mathrm{~m})}=69.7 \mathrm{~Hz}[/latex]

Multiply to find the second and third harmonics:

[latex]f_{2}=2 f_{1}=139 \mathrm{~Hz} \qquad f_{3}=3 f_{1}=209 \mathrm{~Hz}[/latex]

(b) How many harmonics lie between [latex]20 \mathrm{~Hz}[/latex] and [latex]20 000 \mathrm{~Hz}[/latex] for this pipe?

Set the frequency in Equation [latex]14.18[/latex] equal to [latex]2.00 imes 10^{4} \mathrm{~Hz}[/latex] and solve for [latex]n[/latex] :

[latex]\begin{aligned}&f_{n}=n \frac{v}{2 L}=n \frac{343 \mathrm{~m} / \mathrm{s}}{2(2.46 \mathrm{~m})}=2.00 imes 10^{4} \mathrm{~Hz} \end{aligned}[/latex]

This works out to [latex]n=286.88[/latex], which must be truncated down [latex]\left(n=287 ight.[/latex] gives a frequency over [latex]\left.2.00 imes 10^{4} \mathrm{~Hz} ight)[/latex] :

[latex]n=286[/latex]

(c) Find the frequencies for the pipe closed at one end.

Apply Equation [latex]14.19[/latex] with [latex]n=1[/latex]

[latex]\begin{aligned}&f_{1}=\frac{v}{4 L}=\frac{343 \mathrm{~m} / \mathrm{s}}{4(2.46 \mathrm{~m})}=34.9 \mathrm{~Hz} \&f_{3}=3 f_{1}=105 \mathrm{~Hz} \quad f_{5}=5 f_{1}=175 \mathrm{~Hz}\end{aligned}[/latex]

REMARKS For a pipe closed at one end, referring to the "second harmonic" can be confusing, because that corresponds to [latex]f_{3}[/latex]. Calling it the first overtone, however, is unambiguous.

Question 14.9: HARMONICS OF A PIPE GOAL Find frequencies of open and closed...

Related Answered Questions

WHY DOES THE PROFESSOR SOUND LIKE DONALD DUCK? A professor performs a demonstration in which he breathes helium and then speaks with a comical voice. One student explains, “The velocity of sound in helium is higher than in air, so the fundamental frequency of the standing waves in the mouth is ...

Verified Answer:

Verified Answer:

RESONANCE IN A TUBE OF VARIABLE LENGTH GOAL Understand resonance in tubes and perform elementary calculations. PROBLEM Figure 14.25a shows a simple apparatus for demonstrating resonance in a tube. A long tube open at both ends is partially sub merged in a beaker of water, and a vibrating tuning ...

Verified Answer:

HOW DO BUGLES WORK? A bugle has no valves, keys, slides, or finger holes. How can it be used to play a song? ...

Verified Answer:

WHY ARE INSTRUMENTS WARMED UP? Why do the strings go flat and the wind instruments go sharp during a performance if an orchestra doesn’t warm up beforehand? ...

Verified Answer:

OSCILLATIONS IN A HARBOR Why do passing ocean waves sometimes cause the water in a harbor to undergo very large oscillations, called a seiche (pronounced saysh)? ...

Verified Answer:

Verified Answer:

Verified Answer:

THE SOUNDS HEARD DURING A STORM How does lightning produce thunder, and what causes the extended rumble? ...

Verified Answer:

Verified Answer: