Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/130

Glass rods or tubes may also be made to vibrate longitudinally by means of a moist piece of cloth; but it is advisable to clamp them firmly at the centre, when each half will vibrate according to the same laws as the wooden rods above. The existence of a motion of the particles of glass to and fro in the direction of its length may be well exhibited, by allowing a small ball of stone or metal suspended by a string to rest against one extremity of the rod, when, as soon as the latter is made to sing by friction, the ball will be thrown off with considerable violence.

Part VIII.
Theory of Pipes.

75.The longitudinal vibrations of air enclosed in pipes are of greater practical importance than those of other bodies, because made available to a very great extent for musical purposes. In the flue, horn, trumpet, and other wind instruments, it is the contained air that forms the essential medium for the production of sound, the wood or metal enclosing it having no other effect but to modify the timbre or acoustic colour of the note.

76.In dealing with the theory of pipes, we must treat their air precisely in the same manner as we have dealt with elastic rods vibrating lengthwise, a pipe stopped at both ends being regarded as equivalent to a rod fixed at both ends, a pipe open at both ends to a rod free at both ends, and a pipe stopped at one end and free at the other. When therefore the air within the pipe is anywhere displaced along the length of the pipe, two waves travel thence in opposite directions, and being reflected at the extremities of the pipe, there results a stationary wave with one or more fixed nodal sections, on one side of which the air is at any moment being displaced in one direction, while on the other side it is displaced in the opposite.

Fig. 24.

Hence, when the air on both sides of the node is moving in towards it, there is condensation going on at the node, followed by rarefaction on the reversal of the motion of the air. The full lines in annexed figs. are curved of displacements, the dotted lines curves of velocity and density (vid. § 10 and 14).

As a stopped end prevents any motion of the air, a nodal section is always found there. And as, at the open end, we may conceive the internal air to be maintained at the same density as the external air, we may assume that such end coincides with the middle of a ventral segment.

From these assumptions, which form the basis of Bernouilli's Theory of Pipes, we infer:

Fig. 25.

77.That in a pipe stopped at both ends, as in a rod fixed at both ends, the fundamental note (fig. 25, 1), corresponds to $\lambda =2l$, and therefore to $n={\tfrac {\text{V}}{2l}}$, and therefore to $n={\tfrac {\text{V}}{2l}}$, ${\text{V}}$ denoting the velocity of sound in air, and the overtones to numbers of vibrations $=2n$, $3n$, and so on. Fig. 25, 2, represents the octave.

78.That in a pipe open at both ends the same holds good as in the previous case. For (fig. 26, 1) ${\text{AC}}={\tfrac {1}{4}}\lambda$ $\therefore \lambda =4$ ${\text{AC}}=2l$, and in fig. 26, 2, ${\text{AD}}={\tfrac {1}{4}}\lambda$, and also $={\tfrac {1}{4}}l$ $\therefore \lambda =l$, or ${\tfrac {1}{2}}$ its value for the fundamental; and similarly for the other harmonics.

Fig. 26.

79.That in a pipe open at one end and stopped at the other (or, as it is usually termed, a stopped pipes, case § 77, being purely imaginary), the fundamental note has $n={\tfrac {\text{V}}{4l}}$, and the overtones correspond to $3n$, $5n$. . . .

For, in fig. 27, 1, ${\text{AB}}$ or $l={\tfrac {1}{4}}\lambda$, and in fig. 27, 2, ${\text{CB}}$ or ${\tfrac {1}{3}}l$, whence $\lambda ={\tfrac {4}{3}}l$, which being ${\tfrac {1}{3}}$ of value of $\lambda$ in previous case, shows that the number of vibrations is three times greater. Similarly for the other overtones.

Fig. 27.

80.It follows from above, that a given pipe (whether open or stopped) may be made to emit, in addition to or in combination with its fundamental, a series of overtones, which, in an open pipe, follow the natural numbers, and hence are the octave, twelfth, &c., but, in a stopped pipe, follow the odd numbers, so as to want the octave and other notes represented by the even numbers. The succession of overtones may be practically obtained by properly regulating the force of the blast of air by which the air-column is put into vibration.

81.If the fundamental notes of two pipes of equal lengths, but of which one is open, the other stopped, be compared together, they will be found to differ in pitch by an octave, the stopped being the lower. This fact is in keeping with the theory, for the numbers of vibrations being respectively ${\tfrac {\text{V}}{2l}}$ and ${\tfrac {\text{V}}{4l}}$, are in the ratio of $2$ to $1$.

82.By altering the length of the same pipe, we can vary the pitch of the fundamental at pleasure, since $n$ varies inversely as $l$. This is effected in the flute and some other wind instruments by means of openings along part of the pipe, which, being closed or opened by means of keys and of the fingers, increase or diminish the length of the vibrating air-column. In this manner the successive notes of the scale are usually obtained within the range of an octave. The scale is further extended by bringing into play the higher harmonics.

83.Since in an open pipe $n={\tfrac {\text{V}}{2l}}$, and therefore $l={\tfrac {V}{2n}}$, if for ${\text{V}}$ we put $1090$ ft., and for $n\;264$, which is the number of vibrations per second usually assigned to the note ${\text{C}}$, we get $l=2$ ft. very nearly. This, accordingly, is the length of the so-called ${\text{C}}$ open pipe. The ${\text{C}}$ stopped pipe must, by what has been stated above, be $4$ feet in length.

84.Conversely it is obvious that the velocity ${\text{V}}$ of sound in air, and generally in any gas, may be deduced from the equation ${\text{V}}=2nl$, and that if two pipes of equal length contain respectively air and any other gas, the velocities in the two media being to each other directly as the number of vibrations of the notes they respectively emit, we may, from the well-ascertained value of the velocity in air, determine in this way the velocities in other gases, and thence the values of their coefficients $\gamma$ (vid. § 21).

85.While the inferences drawn by means of Bernouilli's theory agree, to a certain extent, with actual observation, there are discrepancies between the two which point to the existence of some flaw in one or both of the hypotheses on which the theory rests. In truth, the conditions assumed by Bernouilli are such as do not fully occur in