Rubens' flame tube

The Rubens flame tube (after Heinrich Rubens ) is an instrument for making standing sound waves visible .

Rubens' flame tube.

Schematic representation of the flame tube. The orange area represents the flame height in normal operation compared to the wavelength of the sound wave. ${\displaystyle \lambda }$

Animation of the time course of the pressure in the pipe (red line) and position of the pressure nodes (red points) of the standing wave compared to the flame height in the schematic representation.

construction

The Rubens flame tube consists of a tube with a series of small holes of the same diameter on its upper side. One end of the tube is closed with a thin membrane , the other with a sliding piston . Through an inlet opening, combustible gas , e.g. B. propane gas , passed into the inside of the pipe, which flows out through the holes on the top and is ignited there.

functionality

Without the influence of sound, an even row of small flames forms over the holes. If the membrane is made to vibrate by a sound source , the resonance frequency of the air column contained in the pipe can be adjusted by moving the piston in such a way that a standing sound wave is formed inside the pipe. Alternatively, the pipe length can remain constant and the tone frequency can be varied instead.

The shape of the standing sound wave corresponds to the height of the gas flame:

• To the pressure node of the standing wave, ie at the points where the pressure of the combustible gas in the tube is constant, flows most gas out. That is where the flames burn at their highest.
• The pressure bellies , so at the points at which the periodic change of the pressure is the combustible gas in the pipe greatest, flows the least gas from. There the flames are smaller.

The wavelength of the sound can be determined from the distance between the pressure bulges or the pressure knots . The speed of sound in the pipe can thus be determined for a known frequency.

Explanation

Creation of resonance

→ Main article : Standing wave

The sound source generates a sound wave in the pipe, which is reflected at the piston at the other end of the pipe and runs back in the pipe in the opposite direction. The wave traveling back is superimposed on the original sound wave from the membrane according to the principle of interference . The higher the tone, i.e. the faster the vibration of the membrane, the shorter the wavelength of the sound wave; the speed of sound is constant.

If the pitch and thus the wavelength of the sound in the pipe has a suitable relationship to the length of the pipe (or the position of the piston), resonance occurs . A standing wave with nodes and antinodes is then formed in the pipe. Vibration nodes are points with destructive interference at which the opposing waves cancel each other out. Antinodes of oscillation are points with constructive interference , at which the amplitudes of the opposing waves add up to an oscillation with a greater amplitude.

• The end of the pipe on which the membrane is located is a sound-open end. There is a pressure knot here because the sound velocity at the membrane assumes its maximum values.
• At the closed fixed end there is a pressure belly because the speed of sound is zero because the end is rigid and does not resonate.

If the pipe were not closed (which is not possible with the pipe lying horizontally because of the escaping gas), there would also be a pressure knot at the open end, which, for example, would be outside the pipe, as is the case with organ pipes .

The boundary conditions mean that for a given wavelength, resonance occurs only with certain pipe lengths : For a pipe with one open and one fixed end, the length of the pipe must be a multiple of half the wavelength , minus a quarter wavelength : ${\displaystyle \lambda }$ ${\displaystyle l}$ ${\displaystyle \lambda /2}$ ${\displaystyle \lambda /4}$

${\displaystyle {\begin{aligned}l&=n\cdot {\frac {\lambda _{n}}{2}}-{\frac {\lambda _{n}}{4}},\quad \quad n\in \mathbb {N} \\&=(2n-1)\cdot {\frac {\lambda _{n}}{4}}\end{aligned}}}$

For the resonance frequency , inserting with the speed of sound gives : ${\displaystyle f_{n}}$ ${\displaystyle \lambda _{n}={\tfrac {c}{f_{n}}}}$ ${\displaystyle c}$

${\displaystyle \Rightarrow f_{n}=(2n-1)\cdot {\frac {c}{4l}}}$

Anharmonicity at low frequencies

At low sound frequencies, the resonance frequencies of the Rubens flame tube are shifted to higher frequencies than predicted by the above model with two boundary conditions. This effect can be explained by the small holes in the flame tube, which act as Helmholtz resonators. ${\displaystyle f_{n}}$

Flame height during normal operation

Figure 1: Measured values ​​of the flame height (y-axis) determined by experiment on a Rubens flame tube without sound waves for different volume flows of natural gas (x-axis). The dashed line is a linear regression line.

Figure 2: Measured values ​​of the square root of the pressure difference (y-axis) inside and outside a Rubens flame tube without sound waves for different volume flows of natural gas (x-axis). The dashed line is a linear regression line.

The explanations in this section are based on research by Ficken and Stephenson. As described in the next section, under certain conditions other effects can play a greater role than the conditions in "normal operation".

At the position along the pipe, the standing sound wave currently generates a pressure of ${\displaystyle x}$ ${\displaystyle t}$

${\displaystyle p_{\text{Schall}}=p_{\text{a}}\sin \left({\frac {2\pi }{\lambda _{n}}}x\right)\sin \omega t}$

with an amplitude and angular frequency . Since the mean value of the pressure over time is the same at all points, this does not explain the different heights of the flames. ${\displaystyle p_{\text{a}}}$ ${\displaystyle \omega =2\pi f_{n}}$ ${\displaystyle p_{\text{Schall}}}$

As can be seen from Figure 1, the flame height is proportional to the mass flow

${\displaystyle {\dot {m}}=\varrho Av}$

of the outflowing gas, which is the product of density , opening cross-section and flow velocity . According to Bernoulli's law , the flow velocity of the gas flowing out through the holes is not proportional to the pressure difference between the pressure inside and outside the pipe, but rather proportional to the square root of this pressure difference. This applies to the pressure inside and outside the pipe ${\displaystyle \varrho }$ ${\displaystyle A}$ ${\displaystyle v}$ ${\displaystyle v}$ ${\displaystyle p_{\text{in}}}$ ${\displaystyle p_{\text{out}}}$

${\displaystyle p_{\text{in}}=p_{\text{out}}+{\frac {1}{2}}\varrho v^{2}}$.

This is shown in Figure 2 for a Rubensche flame tube without a sound wave. The pressure difference

${\displaystyle p_{\text{in}}-p_{\text{out}}={\frac {1}{2}}\varrho v^{2}=p_{\text{g}}+p_{\text{Schall}}}$

consists of a constant overpressure and a part that is temporally modulated by the standing sound wave . Insertion of , dissolve after gives for the amount ${\displaystyle p_{\text{g}}}$ ${\displaystyle p_{\text{Schall}}}$ ${\displaystyle p_{\text{Schall}}}$ ${\displaystyle v}$

${\displaystyle v(x,t)={\sqrt {{\tfrac {2}{\varrho }}\left(p_{\text{g}}+p_{\text{a}}\sin \left({\frac {2\pi }{\lambda _{n}}}x\right)\sin \omega t\right)}}}$

and insert into the definition of the mass flow results

${\displaystyle {\dot {m}}(x,t)=A{\sqrt {2\varrho \left(p_{\text{g}}+p_{\text{a}}\sin \left({\frac {2\pi }{\lambda _{n}}}x\right)\sin \omega t\right)}}}$.

If the mass flow is integrated over a period of oscillation, then the time average is the mass

${\displaystyle {\bar {m}}(x)=\int _{0}^{\frac {2\pi }{\omega }}{\dot {m}}\;\mathrm {d} t}$

of the escaping gas, the greater the amplitude . This time average value is lower at the pressure bulges than at the pressure nodes, so lower flame heights can be observed at the pressure bulges. ${\displaystyle p_{\text{a}}}$

Reversal effect at low pressures

If the gas supply is switched off or reduced so much that the overpressure in the pipe falls below a certain value, it can be observed that the flame height is reversed. It was observed that the alternating pressure at the pressure bellies caused air and burned gases to be sucked in from the edges of the flame and distributed in the pipe. As a result, the net mass flow of combustible gases at the pressure bulges is greater than at the pressure nodes.

Measurements on a typical test setup resulted in a static overpressure of with an amplitude of at the pressure bulges during normal operation . The reverse effect could be observed at a static overpressure of . ${\displaystyle p_{\text{g}}=1{,}8\;\mathrm {Pa} }$ ${\displaystyle p_{\text{a}}=22{,}5\;\mathrm {Pa} }$ ${\displaystyle p_{\text{g}}=0{,}2\;\mathrm {Pa} }$

history

August Kundt , doctoral supervisor of Heinrich Rubens, showed in 1866 with the help of bear moss spores and cork dust and the Kundt pipe named after him that sound waves can form standing waves in a pipe. Heinrich Rubens then, together with his colleague Krigar-Menzel, designed the Rubens flame tube named after him, which they presented in a publication in 1905. This consisted of a four meter long metal tube with 100 holes two millimeters in diameter.

Although the Rubens flame tube is an “effective” demonstration attempt, the Kundt tube is more often used to make standing sound waves visible in schools, so as not to have to handle flammable gas.

Remarks

1. ↑ One can imagine the point of maximum flow velocity, i.e. at the outer end of the holes, where the flames arise. ${\displaystyle v}$
2. ↑ Signed for where is the sign function . The same applies to the following and must be taken into account. ${\displaystyle p_{\text{g}}<p_{\text{a}}:v(x,t)={\sqrt {2/\varrho }}\operatorname {sgn}(p_{\text{g}}+p_{\text{Schall}}){\sqrt {p_{\text{g}}+p_{\text{Schall}}}}}$ ${\displaystyle \operatorname {sgn} }$ ${\displaystyle {\dot {m}}}$ ${\displaystyle {\bar {m}}}$

Individual evidence

1. ↑ Michael D. Gardnerb and Kent L. Gee: An investigation of Rubens flame tube resonances . In: The Journal of the Acoustical Society of America . tape 125 , 2009, pp. 1285-1292 , doi : 10.1121 / 1.3075608 .
2. ↑ ^{a b c d e} George W. Ficken and Francis C. Stephenson: Rubens flame-tube demonstration . In: The Physics Teacher . tape 17 , 1979, pp. 306-310 , doi : 10.1119 / 1.2340232 .
3. ↑ Duan Jihui and Charles TP Wang: Demonstration of longitudinal standing waves in a pipe revisited . In: American Journal of Physics . No. 53 , 1985, pp. 1110 , doi : 10.1119 / 1.14050 .
4. ↑ George F. Spagna (Junior): Rubens flame tube demonstration: A closer look at the flames . In: American Journal of Physics . tape 51 , 1983, p. 848 , doi : 10.1119 / 1.13133 .
5. ↑ ^{a b} Kent L. Gee: The Rubens tube . In: Proc. Mtgs. Acoust. tape 8 , 2009, p. 025003 , doi : 10.1121 / 1.3636076 .
6. ↑ August Kundt: About a new kind of acoustic dust figures and about the application of the same to determine the speed of sound in solid bodies and gases . In: Annals of Physics and Chemistry . tape 203 , no. 4 , 1866, pp. 497-523 , doi : 10.1002 / andp.18662030402 .
7. ^ Heinrich Rubens and Otto Krigar ‐ Menzel: Flame tube for acoustic observations . In: Annals of Physics . tape 322 , no. 6 , 1905, pp. 149-164 , doi : 10.1002 / andp.19053220608 .