## Pipes

Pipes can determine pitch, and can act as resonators, as in many musical instruments.

Like strings, pipes produce harmonic overtones, and are widely used to make music. They can serve as acoustic waveguides, as speaking tubes, and limited sections of pipe make resonators. The cross-sectional shape of the pipe is not important, but is commonly circular or square. A pipe, by definition, has one dimension much larger than the ones transverse to it; the unique direction is called the axis of the pipe. Ideally, the walls of the pipe are rigid, smooth and do not conduct heat. The pipe is usually cylindrical, but in some cases may be conical. The pipe is also considered to be straight, although gentle curvature will have no effect on the analysis.

There are two well-worn paths to finding out about pipes. The simplest is analogous to the treatment of strings. The displacement of the air is assumed to be in the axial direction, as in a plane wave. In fact, what we have is merely a delimited portion of a plane wavefront. Acoustic plane waves are studied in another paper, and we shall use the results from that source. Follow the link for details. The second, and more general, road uses the vector potential to treat waves in three dimensions. This method, too, is explained in another paper, Wave Dynamics, to which the reader is referred.

The equation of motion in the plane-wave approach is easily found. If y(x,t) is the displacement of the cross-section that is at x when at rest, the condensation s = dy/dx, and the net force on a slice of width dx and unit cross-sectional area will be dF = γp(d2y/dx2)dx, since γp is the compressibility of the medium, assumed to be an ideal gas. This must equal the inertial force ρ(d2y/dt2)dx. As in the case of the string, the result is the simple wave equation, with phase velocity c = (γp/ρ)1/2. This is no surprise whatsoever.

Consider a length L of pipe closed by rigid walls at each end. This is precisely analogous to the case of the string fixed at both ends. Such a pipe is not very practical, since there is no way to excite vibrations, and no way for the energy to radiate. When an aperture is made for blowing the pipe, this end acts more like an open end than a closed one. The boundary condition, as in the case of the string, is that y(0,t) = y(L,t) = 0, and we find precisely the same solutions. The frequencies of oscillation are f = nc/2L, n = 1,2,3,..., and the corresponding wavelengths are λ = 2L/n.

A more useful pipe has one end open, and is called a stopped pipe. Sound can radiate more copiously from this open end, which usually has the air jet and lip used for exciting the vibrations. What boundary condition should we apply at the open end? This really demands a more careful analysis, but maybe we can make a useful approximation for the sake of argument. This is that the dilatation vanishes at the open end, or dy/dx = 0. This is nearly true, since the pipe cannot increase the pressure in the large volume of air outside it. Our solution is still A sin (ω/c)x, but the boundary condition at the open end is now cos (ωL/c) = 0. This means that (2πL/&lambda) = π/2, 3π/2, 5π/2, etc., or λ = 4L/(2n - 1), n = 1,2,3,.... The fundamental has twice the wavelength of the fundamental of the stopped pipe, and the frequencies are as 1,3,5,.... The even harmonics are missing. All of these properties are borne out in practice. The only difference is that the effective length L of the pipe is a little longer than the actual length. An increase of about 0.6 of the radius of the pipe seems to give good results. The amount of the end correction depends on the surroundings of the end. With an infinte flange, the correction increases to about 0.83.

An open pipe has both ends open. Now we apply the boundary condition dy/dx = 0 at both ends. The solution is C cos (ωx/c), with sin (ωL/c) = 0. This is the same relation as for the closed pipe, and so the same frequencies result. The fundamental has wavelength 2L, and the frequencies are in the ratio 1,2,3,4,.... The end correction must be applied at both ends, of course. The difference in harmonic content gives the sounds from open and stopped pipes different qualities.

These matters are so easily explained on the basis of the ordinary propagation of sound waves that the frequencies of stopped and open pipes appear in school physics.

The same analysis applies to a liquid in a tube, but the elasticity of the tube has some effect, since liquids are not nearly as compressible as gases, and the pressures become higher. Lamb quotes an analysis that yields c2 = c02/(1 + 2κa/hY), where κ is the compressibility of the liquid (2.22 x 1010 dyne/cm2), a the radius and h the thickness of the tube, and Y is Young's modulus for the material of the tube (glass, Y = 6.03 x 1011 dyne/cm2). For water in a glass tube whose wall thickness is 1/10 the radius, the wave velocity is reduced to 75.9% of the usual value. When flowing water is suddenly turned off, a pressure pulse results that can bounce back and forth in the pipe, creating water hammer.

To take proper account of the end effects, we must use the more general analysis based on the velocity potential φ. The particle velocity is the negative gradient of φ, and the condensation s is the time derivative of φ, divided by c2. Suppose we have an open pipe extending indefinitely to the right. Inside the pipe, we have plane waves approaching the mouth, and the reflected wave returning. Outside the pipe, outgoing spherical waves carry energy away. Take the origin inside the pipe, where the waves are still plane. The velocity potential for x>0 can be written φ = Ceikx + De-ikx, assuming a harmonic time dependence. At x = 0, φ = C + D, and the velocity is ik(C - D). Outside the pipe, φ = 0 (since s must vanish). There is an analogy with electric circuits such that the difference in velocity potential equals the flux times a resistance in a medium of unit resistivity. This is shown in the Figure. In the present case, this means that C + D = (α/A)ikA(C - D), where α is the length of tube of area A that would provide whatever actual resistance exists at the end of the pipe. This relation can be written D/C = -(1 - ikα)/(1 + ikα). If we set kα = tan kβ, this becomes D/C = -e-2ikβ. Now φ can be written φ = Ce-ikβ(eik(x + β) - e-ik(x + β)). At x = -β, φ = 0, which is the boundary condition assumed in the simple plane-wave theory. When kα is small, then β is about equal to α. On the other hand, if kα is large, as it would be if the mouth were greatly restricted, kβ approaches π/2, and so C = D, nearly. The boundary condition is now like that at a closed end.

We can now apply this theory to a pipe of finite length. If the pipe is closed at x = L, φ = C cos k(L - x), so that v = -dφ/dx = 0 at the closed end. The flux at the mouth is kAC sin kL, so C cos kL = (α/A)C sin kL, whence cot kL = kα. This equation gives the frequencies of vibration. If kα is small, we have approximately kL = (n + 1/2)π - kα, or k(L + α) = (n + 1/2)π, exactly as in the simple theory, if L is increased by α. Should kα be larger, the overtones are no longer exactly harmonic. If α is even larger, the pipe begins to resemble a resonator, and k2 = 1/Lα.

The frequencies of a pipe will rise and fall with the temperature, because of the dependence of the speed of sound on the temperature. An open pipe is tuned by changing the aperture at the open end (which changes the effective length), and a stopped pipe is tuned by moving the stop. Because air is such a light medium, there is little energy stored in the vibration. If the note is to be sustained, the pipe must be constantly driven. On the other hand, this means that a note will die away quickly, so the music can be rapid.

The mouth of a stopped pipe acts like a simple source. If we write the velocity potential as φ = C cos k(L - x), the maximum kinetic energy is T = (ρAL/4)k2C2, which is the energy in the mode, and the rate of radiation is W = (ρc/8π)k4A2C2, which is the rate of decay -dT/dt of the energy. Then, dC/dt + C/τ = 0, with τ = 4πL/(k2Ac). The ratio τ/T = 4L2/πA for the lowest mode.

A very important part of an organ pipe is the means used to excite the vibrations. This is usually done by an arrangement that acts like a valve, periodically opening and closing to admit air to the pipe to sustain the vibrations at the desired strength. The valve is automatically controlled by the oscillation itself, like the escapement in a clock, which drives the pendulum or balance wheel at its resonant frequency. One means is used in the flute pipe, where a jet of air impinges on a sharp edge, as shown in the Figure. This is similar to the action of a flute, where the jet of air comes from the lips. The fundamental vibration exerts the strongest influence on the air jet, but since the admission of air is not sinusoidal, the overtones are also excited. It is interesting that in a wide pipe, where the harmonic relations of the overtones are disturbed, the fundamental alone is excited to any degree. A narrow pipe may give a note rich in harmonics. If the air pressure is increased, the first overtone may control the jet instead of the fundamental, and this may extend to higher harmonics as the pipe is more and more strongly blown. The sharpness of the edge has some influence on the harmonics that are excited, a sharp edge favouring higher harmonics, as does a metal pipe. The effect is similar to the excitation of vibrations in a resonator or jug by blowing across its mouth.

Soon after the discovery of hydrogen, it was found that a hydrogen flame burning in an open tube could excite a strong sound. This was called the singing flame, and was a popular demonstration in the 19th century. An oscillation would be reinforced if more heat were added near the time of greatest condensation, and less at greatest rarefaction. Of course, the air would stream through the tube as through a lamp chimney while this took place. In fact, a lamp chimney makes a good resonator. Since excess pressure would cause the hydrogen to be pushed back, diminishing the flame, and the opposite would happen at a rarefaction, it might seem that the effect would be impossible. However, this does not consider resonance in the hydrogen supply tube. On the high-frequency side of resonance, the phase would be correct for reinforcement, while on the other it would damp any oscillations. It is easy to see why the demonstration was somewhat tricky to get working properly. There is an interaction between the oscillation in the tube, and forced oscillations in the supply pipe, that make it work. Other forms of resonator may also be used, and other gases. Wheatstone showed that the flame was indeed intermittent, by using a rotating mirror. When the flame is observed with the eye alone, this cannot be seen, of course. A related phenomenon (Rijke's) occurs when a wire gauze closes the bottom of a pipe, or is pushed some ways up. The gauze is heated by a flame, and when the flame is removed, a strong sound is produced. The sound can be maintained if the gauze is heated electrically. Here, the effect depends on the combination of the steady air motion upwards and the vibratory motion in communicating heat in the proper phase, bringing cool air to the gauze at the proper time, which occurs only in the lower half of the tube.

The sensitive flame is a different phenomenon, but since it may be confused with the singing flame, this is an appropriate place to discuss it. A jet of gas issuing from a small hole can be adjusted to make a narrow, tall flame when lighted, perhaps a foot long. When a high-frequency sound is applied to the jet, the flame flares, or becomes broad and short and unstable. This effect can be used to detect ultrasonic waves. The flame has nothing to do with the phenomenon, only making it visible. The same thing is observed in smoke jets. What happens is that the sound causes the jet to become unstable and break up, and this occurs near the nozzle. Similar effects are noticed in liquid jets, but here surface tension plays a dominant role. Surface tension is absent in gases, so a different mechanism must be sought. The break-up seems to be sinuous, so it is probably related to the effects of viscosity, rotation being imparted by the walls of the orifice. A water jet breaks up into drops.

A steam whistle is excited like a flute pipe by an edge tone. The whistle may be like a bell, with the lip of the whistle oscillating with four nodal meridians in its lowest mode, but its length and volume may also have an effect, like an organ pipe or an acoustic resonator. The elasticity of the metal must have a large effect on the tone produced, although the air or steam included may well have an influence. Beyond this, I know very little about whistles at the moment, and there is nothing substantial in the references.

Pipes excited by reeds are found in organs and harmonicas. The reed serves as an air valve, and its elasticity and inertia control the frequency of the sound produced. The current of air interrupted by the reed is periodic and rich in harmonics. The pipe or tube associated with the reed is a resonator that reinforces one of the harmonic components in the current of air produced by the reed. If the temperature rises, the sound falls in pitch because of the reduced elasticity of the reed. In a flute pipe, the increased speed of sound causes a rise in pitch. A reed pipe is tuned by adjusting the reed. A cylindrical pipe will reinforce the odd harmonics (the reed acts like a closed end), while a conical pipe will reinforce both odd and even harmonics. A note with only odd harmonics has a 'nasal' quality, it is said. Higher harmonics are discouraged because they are out of resonance with the harmonics. Organ reeds are closed by the current of air, and are called in-beating.

There are also out-beating reeds, whose frequency is controlled mainly by the air in the resonant chamber. The opening is greatest when the reed swings outward with the current of air. This includes the larynx, and musical instruments such as the trumpet and trombone. The clarinet and oboe are similar, though the reeds are in-beating. The whole idea is to introduce puffs of air in the correct phase to maintain the vibration, by interaction between the resonance of the pipe and the resonance of the reed.

Rayleigh says the frequencies that can be produced by whistling with the mouth range from about 500 Hz to 4200 Hz. The mouth and the opening of the lips make a resonator, which is excited by an instability in the jet of exiting air. Variations in volume adjust the frequency. Vibrations of the lips do not enter, but perhaps the teeth have some effect in the normal method of whistling. Tubes can be excited by simply blowing through an aperture with sharp edges. The effect is similar to that in a steam whistle.

Two pipes of about the same frequency can give surprising effects when sounded simultaneously. Taken together as a single system, there are two degrees of freedom. One of the two fundamental normal modes is symmetrical, the two pipes vibrating in phase, which resembles independent motion, and the other normal mode is antisymmetric, with the two pipes vibrating 180° out of phase. In the latter case, which almost invariably occurs, the air pushed out by one pipe is immediately drawn into the other, and very little sound is produced. Moreover, the end correction is changed in the antisymmetric mode, and the frequency raised, which discourages mixing with the symmetric mode. This occurs even with a rather small interaction between the two pipes.

### References

H. Lamb, The Dynamical Theory of Sound, 2nd ed. (London: Edward Arnold, 1925), pp. 173-177, 270-274, 280-287.

Lord Rayleigh, The Theory of Sound, 2nd ed. (London: Macmillan, 1926)