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relatively large changes in the density of the air at the node. At the loop, where the air has its greatest motion, there is practically no change in the density. This mode of longitudinal vibration of the air is similar to the longitudinal vibration of a rod or a stiff spiral spring clamped at the middle point (sect. 334).

The frequency of vibration of an air column can be computed from equation (6). In a tube closed at one end the length of the tube, L, is one fourth of a wave-length, or

λ = 4 L.

Hence, for a closed tube vibrating as in Fig. 206, equation (6) becomes.

n=

V 4 L

In a tube open at both ends and vibrating as in Fig. 207,

λ=2L
V

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FIG. 207

The cases shown in Figs. 206 and 207 are not the only modes of vibration of air columns. It is not difficult to find others, for it is only to be remembered that there must be (1) a node at a closed end and (2) a loop at an open end. In Fig. 208 are shown a few of the different modes of vibration of both open and closed tubes. The relationships between the length of the tube and the wave-length are given, as are also the relative frequencies of the sounds emitted by a tube vibrating in these modes. The student should verify these relations. To do this it is necessary to remember that the distance between any node and the adjacent loop is oné fourth of a wave-length.

In a tube such as is used in a pipe organ or in some other wind instrument, a large number of the possible modes of vibration can be produced simultaneously. In the case of a tube closed at one end, if no is the frequency of the fundamental, the frequencies of the different modes of vibration are no, 3 no, 5no, 7no, 9no, etc.; hence the overtones of a tube closed at one end include only the odd harmonics. In a tube open at both ends the frequencies are no, 2η, 3η, 4no, 5no, 6no, 7no, etc.; hence the overtones of a tube open at both ends include all the harmonics. Since the qual

ity of a sound depends on the overtones, it follows that the qualities given by the two types of tubes are different.

The material of which tubes are made affects the relative intensity of different overtones and hence affects the tone quality. In some wind instru

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produced, the different modes of vibration being obtained by different ways of blowing.

337. Musical intervals. A musician learns to measure musical intervals by his ear, but in many cases a quantitative method of measuring them is necessary. A musical interval is measured by the ratio of the frequencies of the two notes. It is usually expressed as the ratio of the larger frequency to the smaller; thus, the interval between two notes of

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frequencies 500 and 400 is 5/4. The interval is independent of the magnitudes of the frequencies, depending only on their ratio. For example, the interval between notes of frequencies 600 and 400 is the same as that between 1500 and 1000, being 3/2 in both cases. 338. Law of harmonious combinations. The most harmonious combination of notes is that in which the intervals can be expressed by small whole numbers.

The interval 2/1 is expressed by the smallest numbers and is the most harmonious combination. It is called the octave. The interval 3/2 is next in simplicity. It is called the major fifth. The major fourth is the interval 4/3. The major third is the interval 5/4, and the minor third is the interval 6/5.

As the numbers get larger the harmony diminishes. For example, two notes having the interval 9/8 are discordant. This is the interval between do and re in the major scale.

339. The major scale. The intervals used in the natural major scale are as follows:

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If we arbitrarily take 24 as the frequency of the first do, the frequencies of the other notes in the major scale are given in the

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The ratio numbers in the third and following lines show the harmonious combinations. Thus do and sol are harmonious because their frequencies are proportional to 2 and 3. Apparently in the formation of this scale the pitch of each note was so chosen that it would harmonize with the others. For example, if we start with do, the octave do2 would be chosen because of the harmony. The next would be mi and sol, for they are related to do by simple ratios. Then re and ti would be chosen because they harmonize with sol, and so on.

PROBLEMS

+ 1. What is the speed of a deep-water wave the wave-length of which is 30 ft.?

✓ 2. What is the speed of a water-wave of long wave-length if the water is only 1 ft. deep?

3. If the velocity of a sound-wave is 330 m./sec., and the wave-length is 66 cm., what is the frequency ?

4. (a) Compute the wave-length of a sound-wave produced by a high soprano singing a note of pitch 1300. (b) Compute the wave-length of the low bass pitch 60. (The temperature is 20° C.)

/ 5. Thunder was heard 5 sec. after a flash of lightning was seen. How many feet away was the lightning? (Assume the temperature of the air to be 20° C. Do not take into account the speed of light.)

6. A gun is fired, and in 3.2 sec. the echo is heard. How far away is the cliff which reflects the sound? (The temperature is 0° C.) X7. A whistle has a frequency of 300 vibrations per second. What will be the apparent frequency if it is moving away from the observer with a speed of 88 ft./sec. (60 mi./hr.)? (The temperature is 20° C.)

8. Compute the musical interval between the two tones heard, one just before and the other just after a whistling engine has passed at 60 mi./hr. (Assume the temperature of the air to be 0° C.)

9. An open organ-pipe has a length of 40 cm. What will be the frequency of the fundamental at 20° C.? of the first

overtone?

10. A tube, closed at one end, is 50 cm. long. What will be the frequency of vibration, at 20° C., of its fundamental mode? of its first overtone? of its second overtone?

11. Compute the frequency of the fundamental note of a child's tin whistle, consisting of a tube 3 cm. long, closed at one end, when blown with air which has a temperature of 30° C.

12. A tube 42 cm. long, closed at one end, is in resonance with a tuningfork. What is the number of vibrations per second of the fork? (The temperature is 20° C.)

13. In an experiment with the dust figures of a Kundt's tube, at 20° C., it was found that a rod 60 cm. long, clamped at its center, produced airwaves which were 8.2 cm. long. (a) What was the frequency of vibration? (b) What is the velocity of a compressional wave in the rod?

14. At international pitch "violin A" has a frequency of 435 vibrations per second. If C is taken as the do of a major scale, compute the frequency of C, E, and G of the octave containing this A.

PART IV. MAGNETISM AND
ELECTRICITY

CHAPTER XXV

MAGNETISM

Magnets, 340. Magnetic forces act through most substances, 341. Poles, 342. Poles always exist in pairs, 343. Attractive forces have a limited range, 344. The magnetic compass, 345. Two kinds of poles, 346. The magnetic field, 347. Magnetic lines, 348. The earth a great magnet, 349. Declination, 350. Inclination, or dip, 351. Variations in the earth's magnetic field, 352. Importance of a knowledge of the earth's magnetism, 353. Magnetic induction, 354. The tendency of soft iron to move into stronger fields, 355. Retention of induced magnetization, 356. An explanation of the behavior of iron filings in a magnetic field, 357. Other magnetic substances, 358. Effect of changes in temperature, 359. Magnetism and the molecular theory, 360. The common electrical units are based upon magnetic measurements, 361. The quantitative law of attraction between two poles, 362. Strength, or intensity, of a magnetic field, 363. Strength of a magnetic field a vector quantity, 364. Use of magnetic lines to indicate strength of a field, 365. Magnetic moment, 366. Energy in a magnetic field, 367. Summary, 368.

340. Magnets. That magnets in the form of bars of hardened steel exert attractive forces on pieces of iron and steel is well known. Pieces of iron ore that are already magnetized are sometimes found. These are called natural magnets or lodestones. Magnetic attraction has been a familiar phenomenon a very long time, how long we do not know. Certainly Thales (about 600 в.с.), one of the "seven wise men" of early Greece, knew something about it. Some very old tales refer to magnetic action. For example, in one of the Arabian Nights tales it is stated that there was a mountain magnetized so strongly that ships sailing near it were wrecked because the nails were pulled out of the timbers.

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