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



This important principle does not seem to have been 

 explicitly recognized in the literature of the subject, althouoh 

 Green's treatment of what Kelvin calls the pressured wave 

 involves it *. In 1887 I introduced the complete discussion 

 of the problem in my lectures on Elasticity to my advanced 

 students in the Imperial University of Japan; and made some 

 definite calculations in regard to earthquakes in a paper which 

 was read before the Seismological Society of Japan in 

 February 1888 and published in their ' Transactions.' This 

 is the paper which is reproduced above. 



In the September number of the ' Philosophical Magazine ' 

 for 1888, Lord Kelvin gives the formulse for the case of the 

 incident distortional waves, and discusses a similar problem in 

 the February number for this year (p. 179). 



( I propose now to give the complete solutions for the 

 different cases, the meanings of which are brought out by the 

 definite numerical calculations given above. 



I. Distortional Wav3 at the Interface of two Elastic Solids. 

 The solution is of the form 



yir = Be i6 ( <?x+y+w ' ) + Bie i6( cx+y+at) 



(b = A 1 e i6(- '>' x+y+ ' o;) , 



ty'= -Q' e ib{c'x+y+o,t)^ 



i- in medium mnp ; 



Hum mnp 

 <£' = A' e iHy'x+y+<ot)^ 



The equations of motion in the two media give the relations 



n(c* + l)=pco* = (m + n)( 7 2 +l), 1 ({) 



n' (c' 2 + 1) = p'a, 2 = ( m! + n') (y' 2 + 1) , | 



The quantities c c' 7 7' are evidently the cotangents of 

 the angles of incidence, refraction, and reflexion of the various 

 waves. 



The conditions to be satisfied at the surface, x = 0, are 

 (1) Equality of normal displacements on each side of the 



* To show how completely the principle was neglected, I need but 

 refer to Question 9 on p. 378 of Ibbetson's ' Elasticity' (1887), in which 

 a plane so-called " sound-wave" is assumed to give rise to reflected and 

 refracted waves of like type only, when it impinges on the plane interface 

 of two elastic solids. It is taken for granted that only the normal 

 components of displacement and of stress in the two media are equal at 

 ever} r point of the interface. But no reason is even hinted at why the 

 tangential components should be treated as of no account. In fact, for 

 two solids in slipless contact, all four conditions must be satisfied. 



