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284                     HANDBOOK OF ACOUSTICS.
(b) Taking the velocity of sound in air to be 1100 ft. per second, find the approximate length of an open pipe giving 256 vibrations per second. In what respects is this calculation incomplete ?
Ant.—(a) See pp. 102, 103, 104 with fig. 54 (A).
1100 550 (b) Approximate length = ^ y 256 = 256 = ^ ft. 2 ins. nearly.
In this calculation it is assumed that the reflection of the sound-wave at the open end of a pipe takes place exactly at the end. This is not the case, and in that respect the calculation is incomplete. 5. (a) Describe and explain the phenomena of beats. (b) If two notes a semitone apart give six beats per second, when sounded together, what are their vibration-numbers ? Am.—(a) See pp. 144, 145, and 156 with fig. 76.
(b) Let x denote vibration number of the lower tone, then x -f- 6 denotes vibration number of the upper tone,
x + 6 _ 16
• ~~ 15
therefore 1 + ~ = 1 + Tc
PC                       i.0
6 - A x ~ 15
and              x = 90
Thus the vibration numbers are 96 and 90.
6.   (a) What are combination tones, and under what conditions are they audible P
(b) Explain the observation that with mounted tuning-forks excited very gently there is but slight dissonance in an impure octave, but that if the forks are excited vigorously the dissonance becomes marked.
Am.—(a) See pp. 128 and 131.
(b) When the forks are vigorously excited, the differential combina­tion tone beats with the lower fork, but when very gently excited, the probability is that the differential is not produced.
7.    Explain Helmholtz's theory of consonance and dissonance, and employ it to prove that a fifth is a more consonant interval than a fourth.
Am.—See Chap. XV and Chap. XVI, p. 191, with fig. 80.
8.    Explain how the need for tempered intonation in the case of keyed instruments arises, and describe the system of equal tempera­ment.
Am.—See pp. 226 to 233 and pp. 238 to 241.