1903.] Tremors over the Surface of an Elastic Solid. 129 



on the general laws of wave-propagation in an unlimited medium, as 

 developed by Green and Stokes ; but Lord Rayleigh's discovery* of a 

 special type of surface-waves has made it evident that the influence of 

 the free surface in modifying the character of the vibrations is more 

 definite, and more serious, than had been suspected. The present 

 memoir seeks to take a further step in the adaptation of the theory to 

 the actual conditions, by investigating cases of forced waves, and by 

 abandoning (ultimately) the restriction to simple-harmonic vibrations. 

 Although the circumstances of actual earthquakes must differ greatly 

 from the highly idealised state of things which we are obliged to assume 

 as a basis of calculation, it is hoped that the solution of the problems 

 here considered may not be altogether irrelevant. 



It is found that the surface disturbance produced by a single 

 impulse of short duration may be analysed roughly into two parts, 

 which we may distinguish as the " minor tremor " and the " main 

 shock," respectively. The minor tremor sets in at any place, with 

 some abruptness, after an interval equal to the time which a wave of 

 longitudinal displacement (in an unlimited medium) would take to 

 traverse the distance from the source. Except for certain marked 

 features at the inception, and again (to a lesser extent) at an epoch 

 corresponding to that of direct arrival of transversal waves, it may be 

 described, in general terms, as consisting of a long undulation leading 

 up to the main shock, and dying out gradually after this has passed. 

 Its time-scale is more and more protracted, and its amplitude more and 

 more diminished, the greater the distance from the source. The main 

 shock, on the other hand, is propagated as a solitary wave (with one 

 maximum and one minimum, in both the horizontal and vertical 

 displacements) ; its time-scale is constant, and its amplitude diminishes 

 only in accordance with the usual law of annular divergence, so that 

 its total energy, unlike that of the minor tremor, is maintained 

 undiminished. Its velocity is that of free Rayleigh waves, and is 

 accordingly somewhat less than that of waves of transversal displace- 

 ment in an unlimited medium.! 



The paper includes a number of subsidiary results. The various 

 problems are attacked, in the first instance, in their two-dimensional 

 forms. The interpretation of the analytical results is then com- 



* ' Lond. Math. Soc. Proc.,' vol. 17, p. 4 (1885) ; ' Scientific Papers,' vol. 2, 

 p. 441. 



f Compare the concluding passage of Lord Rayleigh's paper : 



" It is not improbable that the surface-waves here investigated play an important 

 part in earthquakes and in the collision of elastic solids. Diverging in two dimen- 

 sions only, they must acquire at a great distance from the source a continually 

 increasing preponderance." 



The calculations show that the preponderance is much greater than would be 

 inferred from a mere comparison of the ordinary laws of two-dimensional and 

 three-dimensional divergence. 



