A Pipe Open at Both Ends

solved StandingWaves

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?

Inputs

length

L

30 cm

speed

v

330.0 m/s

source

ν

1.1 kHz

Outputs

harmonic

n

How it's solved

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 web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

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

solution.rb

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

playground.rb

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 it drew on

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)