SOUND 



585 



or harmonium any note and its fifth are struck 

 together beats are heard. It is indeed by getting 

 rid of these beate that we finally effect a perfect 

 toning of say two contiguous violin-strings. These 

 beats on the tempered instruments are due to the 

 fact that the interval is not a true fifth, so that the 

 higher harmonics which are present cause beating 

 combinations. This is shown by comparing the 

 first three harmonics of notes which form (1) a 

 perfect fifth and (2) a tempered fifth. 



PERFECT FIFTH. 



c. o. B<t. 



622 783 

 1044 

 1566 1566 



&C. &C. 



TEMPERED FIFTH. 



C. 



522 

 1044 

 1566 



O. 

 782-1 



1564-2 



&c. 



1-8 



Similar results may easily be obtained for all tem- 

 pered intervals, if the higher harmonics exist in 

 sufficient strength. We may, however, make use 

 of the combinational tones in producing beats. 

 For instance, by the rule given above, we get for 

 the first combinational tones 260-1 and 261-9, which 

 beat also 1 -8 times a second. The second combina- 

 tional tone is intensified by the actual presence of 

 the second harmonic of C ; and we can make the 

 beating still stronger by sounding the octave along 

 with C and G. 



All vibrations in air of sufficient intensity and 

 suitable frequency produce sound-sensations. The 

 highest frequency which gives an audible sound is 

 about 70,000. This is much higher in pitch than 

 the highest notes used in music. For example, 

 the frequency of the highest note on the pianoforte 

 falls short of" 4000. Very high pitched sounds near 

 the limit of audibility are very disagreeable to the 

 ear ; and sounds which are heard py one person 

 may be, because of their high pitch, quite in- 

 audible to another. There are many noises of 

 whose pitch it is impossible to say anything 

 definite. They are no doubt confused mixtures of 

 tones of fluctuating pitch. Konig has pointed out 

 that the peculiar quality or timbre of trumpet 

 notes and other notes of piercing quality is in 

 great measure due to a fluctuating character in the 

 vibration. The various overtones, both harmonic 

 and anharmouic, do not combine with the prime in 

 a steady periodic manner. Anharmonic overtones 

 must of necessity produce fluctuations in the 

 periodic character of the note ; and the theory of 

 quality cannot be complete without taking their 

 existence into account. In this direction Ilelni- 

 holtz's theory requires extension. 



Of great value and interest are Helmholtz's 

 investigations on the forms of vibration of strings, 

 especially of violin strings. If we can experiment- 

 ally determine the form of the wave into which the 

 string is thrown when bowed, we can by Fourier's 

 mathematical process calculate the harmonics 

 that enter in. Helmlipltz solved the problem by 

 viewing through a microscope the motion of a, 

 small white speck at different points of the string, 

 the microscope itself being attached to the vibrat- 

 ing end of a tuning-fork, and being so made to 

 execute a simple harmonic motion at right angles 

 to the direction of vibration of the white speck. 

 The principle of the method is identical with that 

 introduced by Lissajoux in obtaining what are 

 known as Lissajoux's Figures. Two tuning-forks, 

 with bright reflecting surfaces fixed to their vibrat- 

 ing ends, are arranged so that while one vibrates 

 to and fro vertically, the other vibrates to and fro 

 horizontally. A l>eam of light is reflected from the 

 one upon the other, and after a second reflection 

 is focussed sharply upon a screen or viewed in a 

 telescope. When either fork is vibrating alone 

 the image on the screen will vibrate also along a 

 straight line. But when both are vibrating to- 

 gether the light spot on the screen will execute 



the motion which is the resultant of the two 

 mutually perpendicular vibrations. Definite figures 

 are obtained only when the two forks are tuned 

 accurately in unison, or by some simple interval 

 apart. For example, if the two forks give the same 

 note the figure on the screen will be an ellipse or 

 one of its extreme forms, the circle or straight line. 

 If the forks are very slightly out of tune imper- 

 ceptibly so to the ear the ellipse will gradually 

 change shape, passing from the circle to the straight 

 line. The figures obtained when the ratio of the 

 frequencies is as 1 : 2 (the octave), 2 : 3 (the fifth), 

 3 : 4 ( the fourth ), and so on, are very beautiful ; 

 and if the interval is just short of perfect tuning 

 the gradual passing of the figure through a series 

 of related but slightly differing forms is very in- 

 structive. The experiment is valuable as giving 

 an optical demonstration of the laws of combina- 

 tion of vibratory motions in lines inclined to one 

 another. Compare the explanation of elliptic and 

 circular Polarisation (q.v. ) of light. 



Although sounds come to our ear through the 

 air or other fluid as waves of compression and 

 rarefaction, they may have their origin in vibra- 

 tions of quite a different type. In solid substances 

 there are distortional waves as well ascompressional 

 waves. In a pure distortional wave the substance 

 changes form and does not change bulk. Its 

 existence depends upon the Rigidity (q.v.) of the 

 substance. In gases and liquids only compres- 

 sional waves can exist, and these depend upon the 

 compressibility ( see COMPRESSION ). \\ hen a 

 bar, stretched string, or plate is set into vibra- 

 tions a certain kind of elasticity is brought into 

 play, and a certain strain produced, which in 

 general involves both change of form and change 

 of bulk. Corresponding to this there is an appro- 

 priate stress whose ratio to the strain is the co- 

 efficient or modulus of elasticity upon which the 

 velocity of the wave depends. Calling this modulus 

 E, we find for the velocity of a vibratory wave the 

 expression V = VE/D, where D is the density of 

 the substance set into vibration. 



This formula was first applied by Newton to the 

 case of air. Assuming Boyle's Law (see GAS), 

 he obtained the expression VP/D for the velocity 

 of sound, where P is the pressure. But when the 

 proper numbers are put in the velocity is found to 

 fall short of its true value by about 180 feet per 

 second, or by one-sixth of the whole. Newton's 

 assumption is in fact false. Boyle's Law holds 

 only when, throughout the changes of pressure 

 and density, the temperature remains constant. 

 But in the rapid condensations and rarefactions 

 which accompany audible sounds the temperature 

 varies, increasing during condensation and decreas- 

 ing during rarefaction. Now rise of temperature 

 means increase of pressure and fall of temperature 

 decrease ; so that the result of these temperature 

 changes is to increase the forces at work i.e. to 

 increase the elasticity. Laplace first made tins 

 necessary correction to Newton's calculation. We 

 must in fact multiply the pressure by 1'41, which 

 is the square of 1-2 nearly. The complete theory 

 gives for the velocity of sound in dry air at a tem- 

 perature of t F. the value 



V= 1122 + 1 -09 (t- 60) feet per sec., 



which is as near as may be to the mean of experi- 

 mentally determined values. 



The quantity E is the same for all the true gases. 

 Hence the velocity of sound in gases varies in- 

 versely as the square root of the density. In other 

 words, a given wave-length will vibrate propor- 

 tionately faster. Thus, if an organ-pipe be blown 

 with hydrogen gas in it instead of air, the note 

 will leap up nearly two octaves in pitch, since tho 

 density of air is 14-4 times that of hydrogen. 



