reinforces is only twice the length of the tube, or λ=21. Hence an open tube which is to resound to a given note or given tuning-fork must be twice as long as the corresponding closed tube. Imagine the closed tube in Fig. 21 replaced by an open tube twice as long, and let the prong of the fork be moving downwards. In the time that it takes to go from a' to a" the pulse of condensation will have travelled the length of the tube. It is reflected from the open end with change of type, i.e. as a pulse of rarefaction, and in the time that the prong takes to go from a" to a' this pulse will have travelled to the top of the tube. Here it is reflected as a pulse of condensation which coincides with that produced by the fork as it begins its next vibration. Thus the conditions for resonance are fulfilled. The open ends of the tubes are antinodes and there is a node at the middle of the tube.

Hints for Experiments.-As a closed resonator of adjustable length you may use a glass tube closed at one end by a cork (Fig. 21), which can be pushed in or out until the maximum resonance is obtained. For an adjustable open resonator make a paper tube which will just slip over the glass tube, and slide this in or out as may be required. Verify by experiment the statement that the length of the open resonator is double that of the closed one. For a c fork (n=256) you will find that the length of the closed resonator is about 33 cm. while that of the open resonator is about 66 cm. Both results indicate that the wave-length in air of the note c is 132 cm. (4 × 33 = 2 × 66=132).

Assuming that the vibration-number of the fork is correct, you can proceed to calculate from your experiment the velocity of sound in air. For we have seen (Art. 16) that the velocity in any medium is given by the equation vnλ. Here n=256, λ=132 cm., and .. v=256 × 132= 33,792 cm. per second.

ข

λ

The same equation shows that n = As the velocity in any given medium (say air) is constant and independent of the pitch, it follows that the vibration-number of a given note is inversely proportional to its wavelength in that medium. Hence also the vibration-number (or pitch) of the note to which a tube resounds is inversely proportional to the length of the tube.

Now this note is precisely that which the air-column emits on its own account when it is thrown into a state of stationary vibration. Such vibrations are easily produced by blowing across the edge of a tube,—say a glass tube about 1 cm. in diameter. Take a tube 33 cm. long, close the lower end with your thumb and blow across the upper edge: it gives the note c. A tube twice as long open at both ends gives the same note. The latter tube (66 cm.) closed at one end gives the note C, an octave below.

63. Organ-Pipes.-Fig. 22 illustrates the construction of

a common form of organ-pipe with a 'flue' or 'flute' mouthpiece, which is very much like that of an ordinary whistle. The air passes from the wind-chest through the conical tube at the bottom of the pipe, escapes through a narrow horizontal slit and strikes against the sharp bevelled edge opposite, which is called the lip. The air escapes in an intermittent manner, with a rushing noise, due to a mixture of vibrations of different frequencies. The air-column selects out of these the particular one which it reinforces, and when this happens the pipe speaks or emits a note.

The pipe shown in the figure is an open pipe. It is open to the air at the bottom (below the lip) as well as at the top, and both of these places are antinodes. There is a node in the middle of the tube (see Fig. 23).

A pipe which is closed at one end (the top) is called a closed or stopped pipe. There is always a node at the closed end of the pipe, for the air there cannot move. The open (or mouth) end is an antinode, for the density of the air there remains constant and equal to that of the air outside. The note emitted by a stopped pipe is always an octave below that emitted by an open pipe of the same length.

Fig. 22.

64. Overtones of Pipes-Quality.-The modes of vibration above described are those of pipes producing their lowest or fundamental note. But by blowing into a pipe more strongly it may be made to 'jump' or produce one or more overtones. The series of overtones produced by overblowing a stopped pipe is not the same as for an open pipe. In discussing the possible overtones in either case we must remember that, as in the case of a string vibrating in segments, the nodes and antinodes follow

each other at regular intervals, and that the distance from a node to the nearest antinode is quarter of a wave-length.

Open Pipes (Fig. 23). The ends of an open pipe are always antinodes. When the fundamental note (I) is sounded there is a node

(N1) at the centre, and the length of the pipe is half the wave-length of the note. When the first overtone (II) is produced, a fresh antinode appears at A, with nodes at N1 and N2. The air-column divides into two equal Ni

vibrating segments, and the wave-length is half that of the fundamental. When the second overtone (III) is produced the column divides into three equal vibrating segments, and the wave-length is one-third that of the funda

mental. Thus the possible overtones (together with the fundamental) include the whole harmonic series of Art. 49 (1, 2, 3, 4, 5, 6 . . .).

Stopped Pipes-(Fig. 24). In the case of a stopped pipe there must always be a node at the stopped end, and an antinode at the open end. Thus when the fundamental (I) is produced, the length of the pipe is a quarter-wave-length. The even harmonics in the series of Art. 49 (i.e. the successive octaves, 2, 4, 6 . . .) are absent from the possible overtones. The first possible division into segments is that shown in Fig. 24, II, with nodes at N1 and N, and antinodes at A1 and A,; the pipe then includes three quarter-wave-lengths. The next division is shown in Fig. 24, III, when the pipe includes five quarter-wave-lengths. The corresponding vibration-numbers are 1, 3, 5, 7. ・ ・ ・

Some of the overtones are generally present, together with the fundamental note of any pipe, and upon their number and relative strength depends the quality of the tone produced. We may clearly expect the quality of tone of a stopped pipe to be different from that of an open pipe. Again, the quality depends upon the form of the pipe: e.g. a narrow pipe more readily yields harmonics than a wide pipe, especially a wide stopped pipe.

65. Vibrations of Rods.-Rods of glass, wood, brass, etc., may be made to vibrate longitudinally and produce musical notes by rubbing them in the direction of their length. A glass rod may be thrown into a state of vibration by drawing a wetted cloth quickly along it; a rod of wood or brass by means of a cloth dusted over with powdered resin. The vibrations produced are stationary vibrations similar to those of organpipes and following the same laws. If the rod is clamped at the centre it vibrates like the air in an open pipe with a node at the centre and antinodes at the ends. If it is clamped at one end, there is a node at that end and an antinode at the free end, as in a stopped pipe.

EXAMPLES ON CHAPTERS VII-VIII.

1. Upon what physical properties do (1) the loudness, (2) the pitch of a musical note, depend? Two organ-pipes of the same length are one of them open and the other closed; how are their notes related as regards pitch?

2. Describe the state of disturbance of the air in a pipe closed at one end, when it resounds to a tuning-fork which is held over it. State the relation between the length of the pipe, the pitch of the note, and the velocity of sound in air.

3. State how the air moves in different parts of a tube 1 ft. long, open at both ends, when sounding its fundamental note. What note does it give?

4. A glass rod 5 ft. long is clamped at its centre, and rubbed longitudinally with a wet cloth. State how it vibrates when thus treated, and calculate the velocity of sound in the glass, if told that the above rod makes 1295 complete vibrations every second.

5. Find the length of a closed organ-pipe which when blown at 15° gives the note c (256 vibrations per sec.)

ANSWERS TO EXAMPLES

CHAPTERS I-IV (p. 254)

1. See pp. 24, 25, 233. 2. See Art. 10. 3. See Arts. 23, 37. 6. See pp. 246, 249.

CHAPTERS V-VI (p. 268)

5. They yield the same note. 6. 80 lbs. 7. 25 lbs.; an additional II lbs. (i.e. total weight = 36 lbs.)

CHAPTERS VII-VIII (p. 279)

1. See Arts: 7, 10, 40, 63. 2. See Arts. 62, 63. 3. The note depends upon the velocity of sound in air, and this again upon the temperature. If we take it to be 1116 ft. per second (p. 246) the vibration-number of the note would be 558. 4. The wave-length of the note in glass is 10 ft. (twice the length of the rod). The velocity in glass is v=nλ=1295 × 10= 12,950 ft. per second. 5. 1.09 ft. or 33.2 cm. See Ex. 3 and p. 246.