A Pipe Open at Both Ends

Q. A pipe, 30.0 cm long, is open at both ends. Taking the speed of sound in air as 330 m/s, which harmonic mode of the pipe resonates a 1.1 kHz source? expects 2.0

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The web of concepts

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The equations it used

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Check an answer

Q. A pipe, 30.0 cm $L = 30.0$ cm long, is open at both ends. Taking the speed of sound in air as 330 m/s $v = 330$ m/s, which harmonic mode of the pipe resonates a 1.1 kHz $ν = 1.1$ kHz source?

The playground

Inputs

length

L

30 cm

speed

v

330.0 m/s

source

ν

1.1 kHz

Outputs

harmonic

n

The solution

Solved with StandingWaves

Given

$L = 30.0$ cm
$v = 330$ m/s
$ν = 1.1$ kHz

Find $n$ .

Working

$L = 30.0$ cm $= 0.3$ m
$ν = 1.1$ kHz $= 1100$ Hz
harmonics of a pipe open at both ends(pipe) $ν = \frac{n \cdot v}{2 \cdot L}$
$1100 = \frac{n \cdot 330}{2 \cdot 0.3}$
$n = 2$
$n \approx 2.0$ to 2 significant figures

Conclusion

An open pipe sounds every whole multiple of its fundamental, the speed over twice its length; the source is twice that fundamental.

The Ruby

1require_relative "../physics"23question :a_pipe_open_at_both_ends do4  kind :numerical5  describes "A pipe, %{length} long, is open at both ends. Taking the speed of sound in air as %{speed}, " \6            "%{harmonic} a %{source} source?"7  given length: "30.0 cm"8  given speed: "330 m/s"9  given source: "1.1 kHz"10  asks harmonic: "which harmonic mode of the pipe resonates"11end

1require_relative "../questions/a_pipe_open_at_both_ends"2require_relative "../science/waves/standing_waves"34solve :a_pipe_open_at_both_ends, using: StandingWaves do5  objects :pipe67  take length: pipe.length_of_the_pipe8  take speed: v9  take source: pipe.frequency1011  harmonic = pipe.harmonic12  answer harmonic:, figures: 21314  conclusion "An open pipe sounds every whole multiple of its fundamental, the speed over twice its " \15             "length; the source is twice that fundamental."16end

1require_relative "../solutions/a_pipe_open_at_both_ends"23play_with :a_pipe_open_at_both_ends do4  title "A Pipe Open at Both Ends"5end

The laws

quantity	written	units	within	from
speed of sound	$v$	m/s	—	StandingWaves
length of the pipe	$L$	m	—	StandingWaves
harmonic	$n$	—	—	StandingWaves
frequency	$ν$	Hz	0 … ∞	StandingWaves
frequency of ground	$ν_{g}$	Hz	0 … ∞
harmonic of ground	$n_{g}$	—	—
length of the pipe of ground	$L_{g}$	m	—
frequency of pipe	$ν$	Hz	0 … ∞
harmonic of pipe	$n$	—	—
length of the pipe of pipe	$L$	m	—

ν_{p} = \frac{n_{p} \cdot v}{2 \cdot L_{p}}

harmonics of a pipe open at both ends

ν_{g} = \frac{n_{g} \cdot v}{2 \cdot L_{g}}

harmonics of a pipe open at both ends(ground)

ν = \frac{n \cdot v}{2 \cdot L}

harmonics of a pipe open at both ends(pipe)