§ 14 Vibrations of Open Pipes: Modes of Division: Overtones
From tubes closed at one end, and which, for the sake of brevity, may be called stopped tubes, we now pass to tubes open at both ends, which we shall call open tubes. Comparing, in the first instance, a stopped tube with an open one of the same length, we find the note of the latter an octave higher than that of the former. This result is general. To make an open tube yield the same note as a closed one, it must be twice the length of the latter. And, since the length of a closed tube sounding its fundamental note is one-fourth of the length of its sonorous wave, the length of an open tube is one-half that of the sonorous wave that it produces.
It is not easy to obtain a sustained musical note by blowing across the end of an open glass tube; but a mere puff of breath across the end reveals the pitch to the disciplined ear. In each case it is that of a closed tube half the length of the open one.
There are various ways of agitating the air at the ends of pipes and tubes, so as to throw the air-columns within them into vibration. In organ-pipes this is done by blowing a thin sheet of air against a sharp edge. You will have no difficulty in understanding the construction of an open organ-pipe, from this model, Fig. 98, one side of which has been removed so that you may see its inner parts. Through the tube t the air passes from the wind-chest into the chamber, C, which is closed at the top, save a narrow slit, e d, through which the compressed air of the chamber issues. This thin air-current breaks against the sharp edge, a b, and there produces a fluttering noise, and the proper pulse of this flutter is converted by the resonance of the pipe above into a musical sound. The open space between the edge, a b, and the slit below it is called the embouchure. Fig. 99 represents a stopped pipe of the same length as that shown in Fig. 98, and hence producing a note an octave lower.
Instead of a fluttering sheet of air, a tuning-fork whose rate of vibration synchronizes with that of the organ-pipe may be placed at the embouchure, as at a b, Fig. 100. The pipe will resound. Here, for example, are four open pipes of different lengths, and four tuning-forks of different rates of vibration. Striking the most slowly vibrating fork, and bringing it near the embouchure of the longest pipe, the pipe speaks powerfully. When blown into, the same pipe yields a tone identical with that of the tuning-fork. Going through all the pipes in succession, we find in each case that the note obtained Fig. 100. by blowing into the pipe is exactly that produced when the proper tuning-fork is placed at the embouchure. Conceive now the four forks placed together near the same embouchure; we should have pulses of four different periods there excited; but out of the four the pipe would select only one. And if four hundred vibrating forks could be placed there instead of four, the pipe would still make the proper selection. This it also does when for the pulses of tuning-forks we substitute the assemblage of pulses created by the current of air when it strikes against the sharp upper edge of the embouchure.
The heavy vibrating mass of the tuning-fork is practically uninfluenced by the motion of the air within the pipe. But this is not the case when air itself is the vibrating body. Here, as before explained, the pipe creates, as it were, its own tuning-fork, by compelling the fluttering stream at its embouchure to vibrate in periods answering to its own.
The condition of the air within an open organ-pipe, when its fundamental note is sounded, is that of a rod free at both ends, held at its centre, and caused to vibrate longitudinally. The two ends are places of vibration, the centre is a node. Is there any way of feeling the vibrating air-column so as to determine its nodes and its places of vibration? The late excellent William Hopkins has taught us the following mode of solving this problem: Over a little hoop is stretched a thin membrane, forming a little tambourine. The front of this organ-pipe, P P′, Fig. 101, is of glass, through which you can see the position of any body within it. By means of a string, the little tambourine, m, can be raised or lowered at pleasure through the entire length of the pipe. When held above the upper end of the pipe, you hear the loud buzzing of the membrane. When lowered into the pipe, it continues to buzz for a time; the sound becoming gradually feebler, and finally ceasing totally. It is now in the middle of the pipe, where it cannot vibrate, because the air around it is at rest. On lowering it still further, the buzzing sound instantly recommences, and continues down to the bottom of the pipe. Thus, as the membrane is raised and lowered in quick succession, during every descent and ascent, we have two periods of sound separated from each other by one of silence. If sand were strewed upon the membrane, it would dance above and below, but it would be quiescent at the centre. We thus prove experimentally that, when an organ-pipe sounds its fundamental note, it divides itself into two semi-ventral segments separated by a node.
What is the condition of the air at this node? Again, that of the middle of a rod, free at both ends, and yielding the fundamental note of its longitudinal vibration. The pulses reflected from both ends of the rod, or of the column of air, meet in the middle, and produce compression; they then retreat and produce rarefaction. Thus, while there is no vibration in the centre of an organ-pipe, the air there undergoes the greatest changes of density. At the two ends of the pipe, on the other hand, the air-particles merely swing up and down without sensible compression or rarefaction.
If the sounding pipe were pierced at the centre, and the orifice stopped by a membrane, the air, when condensed, would press the membrane outward, and, when rarefied, the external air would press the membrane inward. The membrane would therefore vibrate in unison with the column of air. The organ-pipe, Fig. 102, is so arranged that a small jet of gas, b, can be lighted opposite the centre of the pipe, and there acted upon by the vibrations of a membrane. Two other gas-jets, a and c, are placed nearly midway between the centre and the two ends of the pipe. The three burners, a, b, c, are fed in the following manner: through the tube, t, the gas enters the hollow chamber, e d, from which issue three little bent tubes, shown in the figure, each communicating with a capsule closed underneath by the membrane. This is in direct contact with the air of the organ-pipe. From the three capsules issue the three little burners, with their flames, a, b, c.
Blowing into the pipe so as to sound its fundamental note, the three flames are agitated, but the central one is most so. Lowering the flames so as to render them very small, and blowing again, the central flame, b, is extinguished, while the others remain lighted. The experiment may be performed half a dozen times in succession; the sounding of the fundamental note always quenches the middle flame.
By blowing more sharply into the pipe, it is caused to yield its first overtone. The middle node no longer exists. The centre of the pipe is now a place of maximum vibration, while two nodes are formed midway between the centre and the two ends. But if this be the case, and if the flame opposite the node be always blown out, then, when the first overtone of this pipe is sounded, the two flames a and c ought to be extinguished, while the central flame remains lighted. This is the case. When the first harmonic is sounded the two nodal flames are infallibly extinguished, while the flame b in the middle of the ventral segment is not sensibly disturbed.
There is no theoretic limit to the subdivision of an organ-pipe, either stopped or open. In stopped pipes we begin with 1 semi-ventral segment, and pass on to 3, 5, 7, etc., semi-ventral segments, the number of vibrations of the successive notes augmenting in the same ratio. In open pipes we begin with 2 semi-ventral segments, and pass on to 4, 6, 8, 10, etc., the number of vibrations of the successive notes augmenting in the same ratio; that is to say, in the ratio 1:2:3:4:5, etc. When, therefore, we pass from the fundamental tone to the first overtone in an open pipe, we obtain the octave of the fundamental. Fig. 103. When we make the same passage in a stopped pipe, we obtain a note a fifth above the octave. No intermediate modes of vibration are in either case possible. If the fundamental tone of a stopped pipe be produced by 100 vibrations a second, the first overtone will be produced by 300 vibrations, the second by 500, and so on. Such a pipe, for example, cannot execute 200 or 400 vibrations in a second. In like manner the open pipe, whose fundamental note is produced by 100 vibrations a second, cannot vibrate 150 times in a second, but passes, at a jump, to 200, 300, 400, and so on.
In open pipes, as in stopped ones, the number of vibrations executed in the unit of time is inversely proportional to the length of the pipe. This follows from the fact, already dwelt upon so often, that the time of a vibration is determined by the distance which the sonorous pulse has to travel to complete a vibration.
In Fig. 103, a and b (at the bottom) represent the division of an open pipe corresponding to its fundamental tone; c and d represent the division corresponding to its first, e and f the division corresponding to its second overtone, the dots marking the nodes. The distance m n is one-half, o p is one-fourth, and s t is one-sixth of the whole length of the pipe. But the pitch of a is that of a stopped pipe equal in length to m n; the pitch of c is that of a stopped pipe of the length o p; while the pitch of e is that of a stopped pipe of the length s t. Hence, as these lengths are in the ratio of 1/2:1/4:1/6, or as 1:1/2:1/3, the rates of vibration must be as the reciprocals of these, or as 3:2:1. From the mere inspection, therefore, of the respective modes of vibration, we can draw the inference that the succession of tones of an open pipe must correspond to the series of natural numbers.
The pipe a, Fig. 103, has been purposely drawn twice the length of a, Fig. 93 (p. 215). It is perfectly manifest that to complete a vibration the pulse has to pass over the same distance in both pipes, and hence that the pitch of the two pipes must be the same. The open pipe, a n, consists virtually of two stopped ones, with the central nodal surface at m as their common base. This shows the relation of a stopped pipe to an open one to be that which experiment establishes.