70 Prof. C. G. Knott on Reflexion and Refraction of 



below the surface. Lord Rayleigh's formulae also show that 

 the amplitude of the displacement is directly as the wave- 

 length ; so that for vibrations of short period the surface 

 motions are proportionally small. 



If we consider the features of earthquake motions, we find 

 that the vertical motion when it is appreciable is always very 

 much smaller than the horizontal motion. Hence we cannot 

 have here merely the surface disturbance discussed by Lord 

 ftayleigh. If his investigation touches upon any earthquake 

 phenomenon, this phenomenon is never met with by itself alone. 

 Horizontal displacements exist, at any rate along with it, of a 

 magnitude greater far than Lord Rayleigh's result requires. 

 The simple conclusion is that ordinary earthquakes cannot 

 be regarded as due to the propagation of surface-waves. 

 Milne has, at various times, speculated upon the existence 

 of such surface-waves outstripping the vibrations transmitted 

 through the mass. There never has seemed to me sufficient 

 reason for calling in the aid of these surface-waves, as distinct 

 from the mass-waves. Lord Rayleigh's investigation shows 

 besides that the velocity of a surface disturbance is somewhat 

 less than the velocity of the distortional plane wave travelling 

 through the mass. There is no evidence of a quickened 

 velocity. These two facts, namely, the comparative minute- 

 ness of the vertical motion in all earthquakes, and the somewhat 

 slower speed of Lord Kayleigh's surface-wave, seem to show 

 that we can expect very little towards the elucidation of earth- 

 quake phenomena, by taking into account the so-called 

 surface-wave. 



I now pass to the consideration of the reflexion and refrac- 

 tion of plane waves at the surface of separation of two elastic 

 media. In doing so I shall direct more especial attention 

 to the case in which the one medium is rock and the other 

 water. The case in which both media are solid substances 

 is a good deal more troublesome to deal with ; and so far I 

 have not had time to work out any detailed calculations con- 

 cerning it*. A few general considerations will show the 

 nature of the problem. 



The reflexion and refraction of plane waves at the bounding 

 surface of two media have been very closely studied by many 

 mathematicians. Especially have their efforts been directed 

 towards the explanation of the ordinary phenomena of light 

 upon a purely dyuamic basis. Oauchy, Green, Maccullagh, 

 Lorenz, Rayleigh, Thomson [Kelvin], may be mentioned in 

 this connexion, it is sufficient here to point out that, when 

 the problem is worked out for the case of two incompressible 

 * [Now worked out below, Part II.]. 



