PHILOSOPHICAL TRANSACTIONS. 



I. On the Propagation of Tremors over the Surface of an Elastic Solid. 



By HORACE LAMB, F.R.S. 



Eeceived June 11, Read June 11, Revised October 28, 1903. 



INTRODUCTION. 



1. THIS paper treats of the propagation of vibrations over the surface of a " semi- 

 infinite " isotropic elastic solid, i.e., a solid bounded only by a plane. For purposes 

 of description this plane may be conceived as horizontal, and the solid as lying below 

 it, although gravity is not specially taken into account.* 



The vibrations are supposed due to an arbitrary application of force at a point. In 

 the problem most fully discussed this force consists of an impulse applied vertically to 

 the surface ; but some other cases, including that of an internal source of disturbance, 

 are also (more briefly) considered. Owing to the complexity of the problem, it has 

 been thought best to concentrate attention on the vibrations as they manifest 

 themselves at the free surface. The modifications which the latter introduces into 

 the character of the waves propagated into the interior of the solid are accordingly 

 not examined minutely. 



The investigation may perhaps claim some interest on theoretical grounds, and 

 also in relation to the phenomena of earthquakes. Writers on seismology have 

 naturally endeavoured from time to time to interpret the phenomena, at all events in 

 their broader features, by the light of elastic theory. Most of these attempts have 

 been based on the laws of wave-propagation in an unlimited medium, as developed 

 by GREEN and STOKES ; but Lord RAYLEIGH'S discovery t 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 theory 

 to actual conditions, by investigating cases of forced waves, and by abandoning 

 (ultimately) the restriction to simple-harmonic vibrations. Although the circum- 

 stances of actual earthquakes must differ greatly from the highly idealized state of 



* Professor BROMWICH has shown (' Proc. Lond. Math. Soc.,' vol. 30, p. 98 (1898)) that in such 

 problems as are here considered the effect of gravity is, from a practical point of view, unimportant, 

 t 'Proc. Lond. Math. Soc.,' vol. 17, p. 4 (1885) ; ' Scientific Papers,' vol. 2, p. 441. 

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