where E is the elasticity (Young's modulus (§ 172) in this case) and the density. Hence, since

In the case of brass, Young's modulus has the value 1.1 × 1012, and the density is 8.7. Hence the velocity of sound in brass is

a number which agrees with that obtained by experiment.

In the case of a rod clamped in the middle, the first overtone is produced by a mode of vibration in which there are three nodes, one of them being, of course, at the middle. The pitch of the note given is nearly three times that of the fundamental note. The next overtone contains five nodes, and the pitch corresponds to nearly five times that of the fundamental, and so on. If a rod is held with one end fixed and the other end free, there must be a node at the fixed end and a loop at the free end. Hence the wave-length of the fundamental, when such a rod is vibrating longitudinally, will be equal to four times the length of the rod, a result which follows immediately from the case of a rod clamped at the middle, for this latter may be regarded as made up of two rods clamped at one end. The positions of the nodes for the fundamental and the first two overtones are shown in Fig. 259, from which it will be immediately seen that the frequencies are as I : : 3:5

N

L

N

L

N

L

NL

N

L

N L

FIG. 259.

305. Torsional Vibrations.- When a rod is clamped at one end, and the side is bowed transversely with a rosined bow, a very high note can be obtained. The vibrations in this case consist of an alternate twisting and untwisting of the rod, and are called torsional vibrations. If the solid is in the form of a rectangular bar, and one face is held horizontal, by strewing sand on this face and bowing the edge of the rod it can be set in torsional as well as transverse vibrations, and the positions of the nodal lines will be shown by the sand.

The character of the nodal lines thus obtained is shown in Fig. 260.

B

A

B'

FIG. 260.

If the vibrations were simply trans

verse, the nodal lines would be at right angles to the edge. Owing,

however, to the production of torsional vibrations, in addition to the transverse vibrations, when the bar is bowed at A and damped at B and E', the nodal lines are inclined, as shown in the figure.

306. Vibrating Columns of Gas. - The column of gas, say air, enclosed in a tube can be caused to vibrate longitudinally in a manner strictly analogous to that of the longitudinal vibrations of rods. Two cases have to be considered, namely, that in which the tube is open at both ends, and that in which the tube is closed at one end.

In the case of vibrating columns of air, at the nodes, which are points where the air particles are at rest, there will be maximum change of pressure, for the particies will alternately be crowded together and separated at these points. The loops, on the other hand, will be places of maximum motion, but of minimum change of density and pressure.

In the case of a closed pipe, there can be no motion of the air particles which are in immediate contact with the closed end, so that the closed end must always be a node. At the open end, where the air column communicates with the external air, the changes of density can only be very small, so that the open end may for the present, at any rate, be regarded as a loop. Hence the fundamental is produced when the

air column vibrates, as at (a), L Fig. 261. The wave-length will be equal to four times the length of the pipe, for it is always equal to four times the L distance between a node and the adjacent loop. The first overtone is produced when there is one node besides that at the closed end, as shown at (b), while the second overtone is produced when there are two additional nodes, as at (c). The air particles on the two sides of a node are always

L

moving in opposite directions, and when a condensation is taking place at one node, a rarefaction is taking place at the adjacent nodes. The wave-length at (b) is equal to twice the distance between consecutive nodes, that is, is equal to 4, where 7 is the length of the pipe. Thus the wave-lengths of the fundamental and of the overtones of a closed pipe are

4/ 47 47
47,
&c.
3' 5' 7'

Since the velocity of sound in the air is the same in all cases, and v=nλ, the frequencies of the fundamental and of the overtones are inversely proportional to the wave-lengths, so that, if the frequency of the funda

mental is taken as unity, the frequencies of the fundamental and of the overtones are 1, 3, 5, 7, &c. In this case, therefore, only the odd

harmonics of the fundamental are present in the overtones.

In the case of a pipe open at both ends, there must be a loop

So that in the case of an open pipe all the harmonics of the fundamental are produced by the overtones.

The positions of the nodes and loops in vibrating columns of air can be investigated by means of an arrangement devised by Koenig, and called a manometric capsule or flame.

A

D

F

CG

A hole is made in the side of the tube AB (Fig. 263), and over this hole is stretched a thin india-rubber membrane C. A small metal, or wooden, capsule D covers the membrane, leaving a small enclosed space G. Ordinary coal gas is supplied through the tube E, and escapes through F, where it is lighted. If the pressure within the pipe alters, the membrane C will be forced in and out, causing the pressure in G to vary also. The results of the variation of the pressure of the gas in G will be to cause the size of the flame to vary, when the pressure in G is increased the size of the flame increases, while when the pressure in G decreases, so also does the size of the flame. Since the variations in the pressure inside the sounding-pipe occur with great rapidity, the changes in size of the flame cannot be observed when the flame is looked at directly, on account of the persistence of vision, for the images of the small and large flames made on the retina overlap. In order to overcome

B

FIG. 263.

this difficulty, the flame, instead of being observed directly, is looked at by reflection in a mirror, shown in Fig. 264, which can be rotated about a

vertical axis, so that the images of the flame are no longer superposed. If the hole in the side of the pipe coincides with a loop, then there will be no variations in pressure, and hence the manometric flame will not vary in size, and when viewed in the rotatory mirror will show as a continuous band of light. If, however, the hole is at a node, the flame, when viewed in the rotatory mirror, will be broken up into a serrated appearance, as shown in Fig. 265.

We have in the above discussion sup posed that a loop was formed exactly at the open end of a pipe. This, however, is not accurate, for it is only at a little distance beyond the end of the pipe that no changes in density occur. If there is a flange at the open end of the pipe,

FIG. 264.

as shown in Fig. 266, the loop occurs at a distance of 0.82 R outside the

end of the pipe, where ' is the radius of the pipe. If there is no flange, the loop is at a distance of 0.57 R from the end. Hence the distance between the open end of a pipe and the nearest node is always less than half the distance between any two consecutive nodes, or less than 1/4, where A is the wave-length of the note given by the pipe. The effect of this correction for the open end is virtually to lengthen the pipe, but this will not alter the relative pitches of the overtones. Since, however, the correction for an open pipe will have to be applied at both ends, its virtual length will be / +2α, where a is the correction for the end, and the wave-length of the note

A

FIG. 266.

emitted will be 2(+2a), while the virtual length of a closed pipe of length will be +a, and the wave-length of the note emitted will be 4(a). Hence the interval between the notes given by an open and a closed pipe of the same length (7) will be

4(1+a) 2(1+2a)'

and this is less than 2. Hence the open pipe, instead of giving the octave of the note given by the closed pipe, as the elementary discussion previously given would lead us to expect, gives a note somewhat lower than the octave.

307. Organ-Pipes.-The most familiar case of the vibration of columns of air occurs in the case of organ-pipes. An organ-pipe consists of two parts: (1) a tube enclosing a column of air which is set in vibration, and which governs the pitch of the note emitted; and (2) an arrangement for setting this column of air into vibration and maintaining the vibrations when started. There are two distinct ways in which the vibrations of the air column can be started and maintained. In one of these air is forced through the channel A, Fig. 267 (a), and the stream of air strikes against the bevelled lip B of the pipe. The stream of air striking this edge sets up vibrations in the air contained within the body of the pipe, in the same way that vibrations can be set up in the air contained in the barrel of a key by blowing across the top.

In the other method the air is set in vibration by means of the transverse vibration of a thin plate of metal, C, Fig. 267 (b), called a reed, which is fixed at one end, and nearly fills the aperture leading from a box, D, to the pipe E. If air is forced into D it will, in escaping, set the plate C in vibration, and the reed in its motion alternately closes and opens the passage from the box D. It is the impulses derived from this intermittent supply of air which sets the column of air in the pipe into vibration.

A

(a)

FIG. 267.

E

(b)

Open organ-pipes are tuned by reducing the size of the open end by bending a sheet of metal so that it covers the opening more or less. The smaller the opening, the lower is the note given. Closed pipes are tuned by forcing in more or less the plug which constitutes the closed end, and thus altering the length of the pipe.

If the pipe is not very narrow, the note given when it is blown gently is very nearly a pure tone. If, however, the pipe is narrow or the wind pressure is great, the pipe will give a note in which the first overtone is very marked. When very strongly blown, the first and second overtones