§ 13 Vibrations of Stopped Pipes: Modes of Division: Overtones
But that the current of air should be thus able to accommodate itself to the requirements of the tube, it must enjoy a certain amount of flexibility. A little reflection will show you that the power of the reflected pulse over the current must depend to some extent on the force of the current. A stronger current, like a more powerfully stretched string, requires a great force to deflect it, and when deflected vibrates more quickly. Accordingly, to obtain the fundamental note of this 24-inch tube, we must blow very gently across its open end; a rich, full, and forcible musical tone is then produced. With a little stronger blast the sound approaches a mere rustle; blowing stronger still, a tone is obtained of much higher pitch than the fundamental one. This is the first overtone of the tube, to produce which the column of air within it has divided itself into two vibrating parts, with a node between them. With a still stronger blast a still higher note is obtained. The tube is now divided into three vibrating parts, separated from each other by two nodes. Once more I blow with sudden strength; a higher note than any before obtained is the consequence.
In Fig. 96 are represented the divisions of the column of air corresponding to the first three notes of a tube stopped at one end. At a and b, which correspond to the fundamental note, the column is undivided; the bottom of the tube is the only node, and the pulse simply moves up and down from top to bottom, as denoted by the arrows. In c and d, which correspond to the first overtone of the tube, we have one nodal surface shown by dots at x, against which the pulses abut, and from which they are reflected as from a fixed surface. This nodal surface is situated at one-third of the length of the tube from its open end. In e and f, which correspond to the second overtone, we have two nodal surfaces, the upper one, x′, of which is at one-fifth of the length of the tube from its open end, the remaining four-fifths being divided into two equal parts by the second nodal surface. The arrows, as before, mark the directions of the pulses.
We have now to inquire into the relation of these successive notes to each other. The space from node to node has been called all through “a ventral segment”; hence the space between the middle of a ventral segment and a node is a semi-ventral segment. You will readily bear in mind the law that the number of vibrations is directly proportional to the number of semi-ventral segments into which the column of air within the tube is divided. Thus, when the fundamental note is sounded, we have but a single semi-ventral segment, as at a and b. The bottom here is a node, and the open end of the tube, where the air is agitated, is the middle of a ventral segment. The mode of division represented in c and d yields three semi-ventral segments; in e and f we have five. The vibrations, therefore, corresponding to this series of notes, augment in the proportion of the series of odd numbers 1:3:5. Could we obtain still higher notes, their relative rates of vibration would continue to be represented by the odd numbers 7, 9, 11, 13, etc.
It is evident that this must be the law of succession. For the time of vibration in c or d is that of a stopped tube of the length x y; but this length is one-third of the length of the whole tube, consequently its vibrations must be three times as rapid. The time of vibration in e or f is that of a stopped tube of the length x′ y′, and inasmuch as this length is one-fifth that of the whole tube, its vibrations must be five times as rapid. We thus obtain the succession 1, 3, 5; if we pushed matters further we should obtain the continuation of the series of odd numbers.
And here it is once more in your power to subject my statements to an experimental test. Here are two tubes, one of which is three times the length of the other. I sound the fundamental note of the longest tube, and then the next note above the fundamental. The vibrations of these two notes are stated to be in the ratio of 1:3. This latter note, therefore, ought to be of precisely the same pitch as the fundamental note of the shorter of the two tubes. When both tubes are sounded their notes are identical.
It is only necessary to place a series of such tubes of different lengths thus together to form that ancient instrument, Pan’s pipes, p p′, Fig. 97 (page 223), with which we are so well acquainted.
The successive divisions, and the relation of the overtones of a rod fixed at one end (described in page 205), are plainly identical with those of a column of air in a tube stopped at one end, which we have been here considering.