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XL. On the Application of Force within a Limited Space, 

 required to produce Spherical Solitary Waves, or Trains of 

 Periodic Waves, of both Species, Equivoluminal and Ir ro- 

 tational, in an Elastic Solid. By Lord Kelvin, G.C. V.O., 

 P.R.S.E. 



[Concluded from the August Number.] 



§ 33. "! ET us now work out some examples such as one 

 _Li suggested in an addition of March 6, 1899, to 

 Lecture XLV. of my Baltimore Lectures now in press*, with 

 the simplification of assuming a rigid massless spherical lining 

 for the cavity, which for brevity I shall call the sheath 

 (see § 43 below). But first let us work out in general the 

 problem of finding what force in simple proportion to velocity 

 must be applied to a mass m mounted on massless springs as 

 described at the commencement of my paper, " Application 

 of Sellmeier's Dynamical Theory to the Dark Lines D : , D 2 

 produced by Sodium-Vapour " (Phil. Mag. March 1899), to 

 keep the sheath vibrating in simple harmonic motion A sin tot, 

 and therefore to do the work of sending out the two sets of 

 waves with which we have been concerned. Let 7 be the 

 required force per unit of velocity of m; so that ry'e is the 

 working force that must be applied to m, at any time when 

 e is its displacement from its mean position. Now the 

 springs, which must act on the sheath with the force P, of 

 (72') above, must react with an equal force on m because 

 they are massless ; so that the equation of motion of m is 



"£— p +»s m- 



And, by the law of elastic action of the springs, we have 



F = c(e— h sin tot), (83), 



* " Imagine a homogeneous mass of rock — granite or basalt, for 

 " example — as large as the earth, or as many times larger as you please, 

 '• but with no mutual grayitation between its parts to disturb it. Let 

 " there be, anywhere in it very far from a boundary, a spherical hollow 

 " of 5 cms. radius, and let a violin-string be stretched between two hooks 

 " fixed at opposite ends of a diameter of this hollow, and timed to vibra- 

 " tions at the rate of 1007 per second. Let this string be set in vibration 

 ,: (for the present, no matter how) according to its gravest fundamental 

 '• mode, through a range of one millimetre. Let the elasticity of the 

 " striae and of the granite be absolutely perfect, and let there be no air 

 "in the hollow to resist the vibrations. They will not last for ever. 

 " Why not ? Because two trains of waves, respectively condensational- 

 " rarefactional and purely distortional, will be caused to travel outwards, 

 '• carrying away with them the energy given tirst to the vibrating string." 



