8.5, what must be the relative lengths of a brass and an iron wire of equal size in order that they may have the same pitch under the same tension? 5. Find the ratio of rate of vibration of the following wires of the same material: No. 18 wire 120 cm. long, under tension of 64 kgm.; 130 " 66 66 6. "Middle C" is the tone of 256 vibrations per second, and is produced by a wire 90 cm. long under a tension of 81 kgm. What will be the length of a similar wire to give the pitch E = 320 under tension of 49 kgm. ? 7. What change in diameter of a wire will exactly counterbalance the effect of doubling its length? 8. If a wire 110 cm. long gives "Middle C," what must be its length to give the next octave higher and lower? 9. A sonometer wire is stretched by a force of 25 kgm. and sounds the tone G (384 vibrations per sec.). What must be the tension to cause the wire to sound C of 512 vibrations per second? 10. It is desired to substitute a No. 10 wire for a No. 12 wire in a certain instrument. What change in length must be made in order that, all the other conditions remaining unchanged, the No. 10 wire may produce the same tone? 11. If the length of the No. 10 wire is kept the same as that of No. 12, what must be its tension compared to that of the No. 12 to keep the same tone? 12. Show that the laws (c) and (d) for the vibration of strings may be combined into one, as follows: (e) The number of vibrations varies inversely as the square root of the weight of the wire per unit length. SUGGESTION. Find an expression for the weight per unit length in terms of the diameter, d, and density, D, and show what position in the formula it will occupy. 13. Find the vibration-numbers of the first three harmonics of a note whose fundamental vibration-number is 256. 14. The second harmonic of a certain note is found to correspond to 512 vibrations per second. Find rate of vibration of the fundamental. 15. The third harmonics of two notes have the ratio 18. What is the ratio of their fundamentals? 16. A wire is stretched by a force of 4 kgm., and then has a vibration number for its fundamental note of 320. Calculate the rate of vibration of its first harmonic and also the rate of vibration of the fundamental and first harmonic when the tension is increased to 16 kgm. 17. What is the ratio between the fourth overtone of a note whose fundamental is 256 and the third overtone of one whose fundamental is 320? 18. What overtone of a note whose fundamental is 160 has the same rate of vibration as the fourth overtone of a note whose fundamental is 384? 24. Laws of Vibration of Air-columns.—(a) The vibrationnumbers of air-columns vary inversely as their lengths. (b) The pitch of a closed pipe is an octave below that of the open pipe of the same length. (c) In a closed tube only those overtones are present whose vibration-numbers correspond to the odd multiples of that of the fundamental. In an open tube all the overtones corresponding to the fundamental may be present. XLV. VIBRATION OF AIR-COLUMNS. 1. Draw diagrams to illustrate (a) vibration (fundamental) in a closed tube and an open one of same length (b), fundamental in tubes twice as long as in (a). (c) Fundamental in closed tube and its first two overtones. (d) Same in open tube. 2. If a pipe of length 5 ft. gives a note whose vibrationnumber is 54.5, what will be the vibration-number of a similar tube of length 7 ft.? 3. If a tube 5 ft. long gives a note whose vibration-number is 54.5, find the length of tube to give a note whose vibration number is 162.5. 4. A closed tube gives a note whose vibration-number is 200. What will be the vibration-number of an open tube of similar dimensions? 5. In a closed tube the fundamental has a vibrationnumber of 512. Find vibration-numbers of the first three harmonics. 6. The fundamental note of an open pipe has a vibrationnumber 256. Find vibration-numbers of first three harmonics. 7. Taking the velocity of sound in air as 33300 cm. per second, find (a) the wave-length of a note of 435 vibrations; (b) length of open pipe to give this note; (c) length of closed pipe. 8. How long should a closed tube be to respond to the note of a tuning-fork making 435 vibrations per second? (No allowance being made for the diameter of the tubes.) 9. Repeat Prob. 8 for an open tube. XLVI. BEATS. 1. How many beats per second are produced by two sounds of 200 and 204 vibrations per second respectively? NOTE." Beats are explained by the principle of interference. As the wave-lengths of the two notes are slightly different, while the velocity of propagation is the same, the two systems of waves will, in some portions of their course, agree in phase, and thus strengthen each other; while in other parts they will be in opposite phase, and will thus destroy each other." (Deschanel.) For example, let one of the notes have 50 vibrations per second, and the other 52. The first will perform one vibration in of a second; the other in of a second. The last will thus perform vibrations while the first is performing one. That is, the second will gain of a vibration for every vibration of the first. In 50 vibrations of the first, therefore, it will gain 50 × or 2 whole vibrations per second. That is, the second will " overtake" the first twice every second, and two beats therefore result. It will be noticed that this number of beats will always equal the difference between the two vibration numbers. 2. Two tones are sounded and produce 5 beats per second. One has a frequency of vibration of 435. What is the frequency of the other? 3. Show what harmonics of notes of 312 and 465 vibrations per second will with each other produce 6 beats per second. 4. Explain why two tones of frequencies 200 and 202 produce beats and an unpleasant impression, while two tones of frequencies 256 and 384 are in consonance. XLVII. SCALES-INTERVALS. DEFINITIONS. -Scale; major diatonic scale; minor diatonic scale; interval; chromatic scale; minor and major second; minor and major third; fourth; fifth; major chord; minor chord. as a standard. "INTERNATIONAL PITCH" assigns to A, 435 vibrations 1. If A, has 435 vibrations per second, compute the value of the other seven tones above. 2. If Cs is 256 vibrations per second, find the rate of vibration of two other tones, such that C:X: Y = 4:5:6. NOTE. The chord whose tones have the ratios described in Prob. 2 is called a major chord. 3. Find vibration-numbers of a major chord whose third tone is 512. 4. Show how the major diatonic scale may be built up upon C of 256 vibrations. 5. What is the vibration-number of a tone two octaves below A, of 435 vibrations? 6. What is the vibration-number of a tone one octave above G of 384 vibrations ? 7. Find the vibration-number of a tone which is the upper tone of a "major third" of which G (384) is the lower. 8. If C, (256) is the lower note of a "fifth," what is the higher note? 9. If C, (512) is the higher note of a "fifth," what is the lower note? 10. Find the notes which form "major thirds" with E (320), A (426), and B (480) as their higher tones. 11. What overtone of E (320) forms an interval of a "major third" with C, (512)? |