Vibrations of Air Cavities.—Columns of Air: Organ pipes. The simplest case of a vibrating mass of air in a solid enclosure is that of a parallel cylindrical pipe the ends of which may be closed or open. The vibrations of such a column are analogous to the longitudinal vibrations of a solid rod. Provided the diameter of the pipe is not too small, so that viscous drag at the boundary is unimportant, and not too great compared with the length of the pipe and the wave-length of the sound, we can assume at any instant, the motion of particles in any particular cross section to be the same. That is, we are dealing with plane waves of sound in the pipe. As in the case of transverse vibra tions of strings and longitudinal vibrations of rods, stationary waves are produced in the air column due to the combined effects of the direct and end-reflected waves. Thus the equation of wave motion in the pipe is a2uat2--E/p.320x2, indicating a wave travelling with a velocity p) where E and p are the appro priate values of adiabatic elasticity and density of the gas con tained in the pipe. Assuming a simple harmonic wave E= a cosnt the solution becomes E = [A cos(nx/c)+Bsin(nx/c)] cosnt. With the appropriate end conditions E = o at a node (a solid end) and 8E/ax =o at an antinode (an open end) the various modes of vibration are readily determined.
Remembering that a pulse of compression is reflected as a rarefaction at an open end and as a compression at a closed end, it will be evident that a wave must travel twice the length of a pipe open at both ends, and four times the length of a pipe closed at one end, before the wave repeats itself, i.e., in one period. If X is the wave length and N the frequency in the stationary wave then N=c/X. For a pipe open at both ends each open end must be an antinode and the length of the pipe a multiple of X/2, that is, / = sX/2 where s= I, 2, 3, etc. We have in this case there fore a complete harmonic series of partials whose frequencies are given by N = the corresponding wave-lengths being 2/ 1 p proportional to I, 1/2, 1/3, 1/4, etc.
For a pipe closed at one end the closed end must be a node and the length of the pipe an odd multiple of X/4, that is 1= sX/4 where s= I, 3, 5, etc. The partials therefore form an odd har monic series the frequencies being N. = where s is an 4/ P odd integer. The corresponding wave-lengths are proportional to I, 1/3, 1/5, etc. A closed pipe resonates to the same fundamental frequency as an open pipe of twice the length.
The position of the nodes n and antinodes a for a few of the partials of "open" and "closed" pipes are indicated in fig. 5.
Correction at Open End.—The approximate theory indicated above (due to Bernouilli) assumes an antinode at the open end.
in the length of the air column with a corresponding lowering of pitch. Rayleigh has shown that this effective increase of length approximates to o.6 times the radius r of the tube if the latter is unflanged. This correction applies at each open end, and is there fore i •2r for an open tube, and o.6r for a closed one. It should be observed also, as in the case of rods, that the simple theory only applies when the diameter of the tube is small relative to the length of a loop X/2.
Resonating Liquids in Metal Tubes.—Columns of liquid en closed in metal tubes may also be set into resonant vibration.
On account of the yielding of the walls of the tube, however, the velocity Al(E/ p) of the wave is slightly modified since the coeffi cient of elasticity involved is dependent on the extent of such yielding. The subject has been examined theoretically by Lamb (Sound, p. 174) and by Green (Phil. Mag., 45, May 1923) who has also determined experimentally the change of velocity pro duced in various liquids by this yielding.
Organ Pipes.—One of the most important applications of the vibration of air columns is found in the organ pipe. This usually
takes the form of a cylindrical metal tube or a wooden pipe of square section. One end of the tube is specially constructed so that a suitable blast of air will set up resonant vibrations in the column of air. In the open "flue" organ pipe the blast of air im pinges on a thin lip which forms the upper edge of a narrow slit opening into the tube. When the blast is correctly adjusted the pipe "speaks" and the air column resonates. The fundamental tone is sounded when the blast is moderate, and by increasing the power of the blast the harmonics can successively be produced.
In another form, known as the "reed" pipe, the blast of air impinges on a reed which controls the amount of air entering the pipe. The reed is set in vibration and puffs of air are admitted to the pipe which resonates under the correct conditions. The reed cannot be regarded as a freely vibrating spring for it is affected by the air blast and the resonance in the pipe. "Organ" pipes have been made covering a range of frequencies 8 to 16,000 cycles per second, corresponding to lengths of 64 feet to t inch respectively. Fifes, flutes, oboes, etc. are other examples of musical instruments employing resonant air columns.
Small Air Cavities: Helmholtz Resonators.—The vibrations of air cavities almost completely enclosed were first studied by Helmholtz and have considerable practical importance. A vol ume of enclosed air having only a small " neck" connecting with the external air radiates very little energy when set into resonant vibration. The damping is therefore very small and the tuning is very exact. The device is therefore very sensitive to a narrow range of frequencies. The motion of the air in the cavity is almost negligible compared with that in the narrow neck. Con sequently we may regard the air in the neck as a piston having mass m = pIS (p density, I length, and S area of neck), whilst the air in the cavity functions as a spring of "strength" where E is the elasticity ( =7p) of the gas and v the volume of the cavity. The frequency N of such a mass and spring is Vilm/27 A true antinode is a point of zero pressure-variation and maxi mum displacement-amplitude. At the open end of a tube the stationary plane waves inside are changing to spherical progressive waves outside. In other words, the tube radiates sound energy in all directions from the end of the tube. On account of this radia tion, the intensity of the reflected waves from the open end is somewhat less than that of the incident waves, which explains the rapid damping of the vibrations when the forcing ceases. The body of air at the open end has the effect of adding inertia to the air in the tube. Consequently there is a virtual increase S/1 is called the " conductivity " of the neck. For a circular aperture in a thin wall, Rayleigh shows that this quantity is equal to the diameter (2a) whence 271N=c-V(2a/v). The fre quency is independent of the shape of the cavity provided its linear dimensions are not comparable with a wave length of the sound to which it resonates. A " spherical " Helmholtz resonator is shown in fig. 6. A series of resonators of graduated frequencies (i.e., of varying volume or area of mouth) may be used in the fre quency analysis of complex sounds. Resonance may be detected by the ear or by means of a sensitive manometric capsule applied to the small pipe at the base of the resonator, or by means of a Tucker hot-wire microphone (see p. 31) placed in the mouth. Resonators of large volume, and correspondingly low frequency, fitted with hot-wire microphones were employed in the detection and location of guns during the war. (See SOUND RANGING.) Large resonant microphones are also used in the detection of aircraft.