CONTENTS.

CHAPTER I.

SIMPLE HARMONIC VIBRATION.

1. Vibration of tuning-fork. 2. Pitch independent of amplitude. 3.

Elastic resistance proportional to displacement. 4. Simple har-

monic motion defined. 5. Uniform movement in auxiliary circle.

6. Amplitude and period. 7. Calculation of period; isochronism;

Hooke's law. 8. Uniform circular movement compounded of two

S.H. motions. 9, 10. S.H. motion expressed by an equation.

Simple harmonic function. 12. Simple harmonic curve.

Modes of tracing it. 14. Vibrations of a pendulum are simple

harmonic. 15. Cycloidal pendulum

II.

13.

PAGE

I

CHAPTER II.

COMPOSITION OF MOTIONS OF TRANSLATION.

16. Definition of resultant of two motions. 17. Construction for

resultant of two given motions. 18. Motion of middle point of

joining line is half the resultant. 19. Resultant motion ex-

pressed by co-ordinates. 20. Motion of centre of mean position

of n points is of resultant of their motions. 21. Motion of a

n

point which divides joining line in a fixed ratio. 22. The panta-

graph can be used to give the resultant of two given motions on

a scale of one-half. 23. Resultant of n motions can be obtained

on a scale of one nth by a combination of n − 1 pantagraphs.

a

12

CHAPTER III.

COMPOSITION OF VIBRATIONS OF THE SAME PERIOD.

24. Two uniform circular motions in same direction. 25. Two S.H.

motions in same straight line. 26. Two equal circular motions in

opposite directions. 27. Two unequal circular motions in oppo-

site directions; elliptic harmonic motion. 28. Any two S.H.

motions of same period. 29. Projection of S.H. motion upon any

straight line. 30. Projection of elliptic harmonic motion upon

any straight line and upon any plane. 31. Acceleration in elliptic

harmonic motion. 32. Resultant of any number of S.H. motions.

33. Amplitude of resultant of two S.H. motions in same line. 34.

Effect of slight inequality of the two periods. 35. Mean value of

square of amplitude. 36. Beats. 37. Spring and neap tides. 38.

Composition of two S.H. motions at right angles to each other.

39. Effect of slight inequality of their periods. 40. Illustration

from Blackburn's pendulum. 41. General principle on which the

experiment depends. 42-45. Analytical confirmations of fore-

going results on composition of S.H. motions. 42. Motions in

same line. 43. Motions at right angles with 90° difference of

phase. 44. Any two S.H. motions at right angles of same period.

45. Effect of slight inequality of period. 45*. Composition of two

S.H. motions obliquely inclined

CHAPTER IV.

WAVES.

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18

46. Simple harmonic undulation. 47. Meaning of wave-length.

48. Velocity of propagation in terms of wave-length and period.

49. Equation to wave-curve at any time for transverse vibration.

50. Velocities of particles at different points of a wave. 51. Appli-

cation to longitudinal vibration

CHAPTER V.

COMPOSITION OF TWO S.H. UNDULATIONS OF EQUAL WAVE-LENGTH.

52. Composition of undulations defined. 53. Two undulations in

same direction. 54. Effect of slight difference in wave-lengths.

55, 56. Verification from the equations. 57. Application to sound.

58. Two equal undulations in opposite directions with transverse

38

vibrations. 59. With longitudinal vibrations. 60. Nodes. 61. Antinodes. 62. Verification from the equations. 63. Transverse vibrations. 64. Stationary undulation. 65, 66. Composition of undulations in different planes. 67. Effect of slight inequality of wave-length

PAGE

46

CHAPTER VI.

COMPOSITION OF TWO S.H. MOTIONS OF DIFFERENT PERIODS.

68. Motions at right angles with periods as I to 2. 69, 70. With periods as m to n. 71. Specimens of the curves. 72. Geometrical construction for them. 73. Periods approximately as m to n. 74. Experimental methods of tracing the curves; Blackburn's pendulum. 75, 76. Kaleidophone. 77. Tisley's pendulum apparatus. 78. Lissajous' experiment. 79. Optical tuning. 8o. Appearance as of curve on rotating cylinder. 81. Composition of two parallel S.H. motions represented by the ordinates of a curve. 82. Periods of components approximately as two small integers .

CHAPTER VII.

MECHANICAL ILLUSTRATIONS.

83. Subject of chapter. 84. Crank and slot. 85. Crank and connecting-rod. 86. Link motion for reversing. 87. New method of obtaining S.H. motion by jointed parallelogram. 88. Objection to another construction. 89. S.H. motion obtainable by one circle rolling inside another. 90. Another proof, showing that elliptic harmonic motion is also obtained. 91, 92. Two theoretical methods of obtaining S.H. motion. 93. Donkin's harmonograph for composition of two parallel S.H. motions. 94. Adaptation to the composition of two rectangular S.H. motions. 95, 96. Thomson's tide predicting machine.

CHAPTER VIII.

PROPAGATION OF SONOROUS UNDULATIONS.

ΙΟΙ.

97. Propagation in air. 98. Co-efficient of elasticity. 99. Effect of
heat of compression. 100. Formula for velocity in air.
Effect of barometric and thermometric changes. 102. Open spaces,
and speaking tubes. 103. Liquids and solids. 104. Strings

91

CHAPTER IX.

REFLECTION OF SONOROUS UNDULATIONS.

105. Reflection from a closed end of a pipe. 106. From an open end.

107. Echo in Kentish Town well. 108. Resonance of open and

stopped pipes. 109. Vibrations of wind instruments. IIO. Tones

of pipes. III. Correction for width. 112, 113. Tones of a string.

114. Sounding boards

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97

CHAPTER X.

DYNAMICAL INVESTIGATION.

115. Stationary vibration of string investigated dynamically. 116.

Stationary vibration of column of air. 117. General dynamical

equation for propagation along column of air. 118. Application

to S.H. waves

106

CHAPTER XI.

ENERGY OF VIBRATIONS.

119. Energy of vibration of a common pendulum. 120. Calculation

for simple pendulum. 121. The mean working force in small

vibrations is half the maximum force. 122. Proof by integral

calculus. 123. By graphical construction. 124. Energy of station-

ary vibration of string. 125. Energy of stationary vibration of

column of air. 126. Energy of S.H. waves. 127. Variation of

energy in beats. 128. Law of inverse squares.

affected by pitch

129. Loudness as

113

CHAPTER XII.

SIMPLE AND COMPOUND TONES.

130. Ohm's principle and Fourier's theorem. 131. Analysis by reso-
132. Helmholtz's resonators. 133. Combination of pipes

nance.

in organs

CHAPTER XIII.

MUSICAL INTERVALS.

134. Pitch and musical scale. 135. Temperament. 136. Musical
values of harmonics. 137. Absolute pitch. 138. Nature of dis-
cord

PLATES I.-IV

123

127

To be placed at the end of the book.

VIBRATORY

AND

MOTION

SOUND.

CHAPTER I.

SIMPLE HARMONIC VIBRATION.

1. WHEN the prongs of a tuning-fork are squeezed between the fingers and suddenly released, they spring back not only to their original position, but to a nearly equal distance on the other side, and swing backwards and forwards a great number of times before they finally come to rest. This is an example of vibration.

The time occupied in swinging from one side to the other and back again is called the periodic time, or the period of vibration, or simply the period; and the distance that any particle of the fork travels, first to one side and then to the other side of its position of equilibrium, is called the amplitude of vibration for this particle.

2. A tuning-fork, when well started, usually makes several thousand vibrations before coming to rest. Their amplitudes gradually decrease, and hence the sound

B