NA TURR 



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THURSDAY, JANUARY 24, 1878 



HAY LEIGH'S ''THEORY OF SOUND" 



The Theory of Sound. By J. W. Strutt, Baron Rayleigh, 

 F.R.S. Vol. I, (London : Macmillan and Co., 1877.) 



THE author, who already, by a series of interesting 

 treatises belonging to different branches of mathe- 

 matical physics, has acquired a respected name in the 

 domain of science, undertakes to give a complete and 

 coherent theory of the phenomena of sound in the work 

 above mentioned, the first volume of which has recently 

 been published ; and he does this with the application of 

 all the resources furnished by mathematics, since without 

 the latter a really complete insight into the causal con- 

 nection of the phenomena of acoustics is altogether im- 

 possible. We must confess that, even in spite of the 

 most intense exertion of the powers of mathematical 

 analysis, in the present state of its development several 

 problems remain unsolved, for which, indeed, the condi- 

 tional equations are known, but for which it has not yet 

 been found possible to carry out the calculation. 



The author will merit in the highest degree the thanks 

 of all who 'study physics and mathematics if he con- 

 tinues his work in the manner in which he has begun 

 it in the first volume. The separate treatises in which 

 the acoustic problems that have been solved hitherto are 

 discussed, are for the most part dispersed in the publica- 

 tions of academies or of scientific societies, which can be 

 found only in larger libraries, and which frequently are not 

 at all easily traced. But even if one has found a treatise of 

 this kind and reads it, it happens often enough that 

 the author refers in his quotations to other works quite as 

 difficult of access, the knowledge of which is necessary for 

 understanding his treatise. Thus the zeal of the student is 

 paralysed by a number of purely external difficulties, and 

 the ordinary result at which an intelligent student arrives 

 after a few attempts in this direction, is that for problems 

 in which he takes great interest he prefers starting anew 

 to find the solution, rather than trying to hunt for it in 

 libraries. Even if we must admit that the insight into the 

 essence of a problem for which one has found the solution 

 oneself is much deeper and clearer than when one has 

 obtained the solution from some ether author, yet an 

 enormous amount of time is thus lost, and the survey of 

 the whole extent of solvable problems remains incomplete. 

 A survey of this kind, however, is necessary for all who 

 wish to work at the progress of science themselves. 

 For in order to obtain decisive results by new scien- 

 tific investigations it is necessary above all things to be 

 quite clear with regard to the question for which forms of 

 experiment or of observation the theoretical deduction 

 from principles can be carried through as purely as 

 the experiment itself. I know by experience that a 

 number of young physicists lose their time and their 

 zeal by trying to solve problems which, taken by them- 

 selves, are very interesting, but for which at present the 

 deductions from the theoretical principles for the given 

 case can only be drawn in coar.<-e approximation, and 

 where the experiments cannot be freed from important 

 sources of error. 

 While praising Lord Rayleigh's book as a means of 

 Vol, XVII.— No. 43© 



overcoming the difficulties described, I do not at all wish 

 to designate it as a mere compilation. On the contrary, 

 it is a perfectly coherent deduction of the special facts 

 from the most general principles, according to a uniform 

 method and in a consequent manner. The mechanical 

 principles of the doctrine of minute oscillations are con- 

 tained in the present volume and are developed in greater 

 generality than in any other book known to me. For this 

 purpose the author in the first chapter explains the 

 general physical principles of sound, of its propagation, of 

 pitch and its dependence on the rapidity of vibration, ot 

 the musical scale, of the quality of sound and its depend- 

 ence on the harmonic over-tones ; and in the second 

 one the doctrine of the composition of harmonic motions 

 of either equal, or nearly equal, or consonant numbers 

 of vibrations, and further illustrates them by the descrip- 

 tion of the physical phenomena and methods in which 

 the principles developed are applied, and to which 

 belong the doctrines of musical beats and of the physical 

 methods to render the forms of vibrations visible. 



Then follows the development of the most general 

 peculiarities of oscillating motions, first, in the third chap- 

 ter, for mechanical systems to the motion of which only one 

 degree of freedom is allowed, and then, in the fourth chap- 

 ter, for systems with a finite number of degrees of freedom. 

 There is a great multitude of peculiarities common to 

 the most heterogeneous sounding bodies, which up to the 

 present have mostly been found in certain instances only, 

 but which can also be deduced from the most universal 

 form of the motion-equations of systems of one or more 

 degrees of freedom of motion. The author in the form of 

 the equations and in the manner of denotation, closely 

 follows the " Natural Philosophy " of Thomson and Tait ; 

 in fact the whole manner of treatment of the mathematical 

 problems corresponds so closely to that adopted in the 

 work just mentioned, that Lord Rayleigh's book may be 

 looked upon as the acoustic part of the excellent hand- 

 book of the two celebrated physicists named. 



With all systems of this kind if there are no exterior 

 forces acting upon them, we find, on the whole, a number 

 of proper tones equal to the number of degrees of free- 

 dom, and the pitch of which does not depend on the 

 amplitude of the vibrations as long as this one remains 

 small enough. Exceptionally, however, several of these 

 proper tones may be of equal pitch. If there is no 

 friction or dissipation of energy the amplitude of every 

 kind of oscillation remains constant. To each separate 

 proper tone a certain form of motion of the whole sys- 

 tem belongs ; so that the directions and magnitudes of 

 the displacement of the separate points of the system are 

 different in each case. Each arbitrary motion of the sys- 

 tem produced in any arbitrary manner, may be regarded 

 as a superposition of these forms of vibrations belonging 

 to the various proper tones of the system. In order to 

 find the amplitude and phase of these different vibrationg 

 for a given original displacement and of given velocities 

 of its different parts, quite similar methods are adopted 

 as those which are employed to develop a given 

 periodical function into one of Fourier's series ; only 

 the whole method here becomes far more intelligible 

 and has a thoroughly certain foundation, because we 

 have to do with a finite number of unknown factors 

 instead of with the infinite number of continuously 



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