18 



PROFESSOR HORACE LAMB ON THE PROPAGATION OF 



If in (73) and (76) we regard only the terms which are sensible at a great distance 

 from the origin, we have, for x positive, 



V- 



V- 



and similarly for x negative we should find 



u n = 



(77); 



(78). 



These formulae represent a system of free Rayleigh waves, except for the 

 discontinuity at the origin, where the extraneous force is applied. The vibrations 

 are elliptic, with horizontal and vertical axes in the ratio of the two numbers 

 H and K, which are denned by (G8) and (71), respectively. To calculate these, we 

 have, since F (K) = 0, 



and therefore 



J^&J 

 = -KF( 



where, by dift'ereutiation of (62), 



2&X (2/c 2 - #) 



, . 



. . (so). 



In the case of incompressibility I find 



H - '05921, K= -10890; 

 whilst on POISSON'S hypothesis 



H = -12500, K = -18349, 



so that the amplitudes are, for the same value of ju, and for the same applied force, 

 about double what they are in the case of incompressibility. 



A similar treatment applies to the formulae (55), which represent the effect of a 

 concentrated horizontal force Pe ipt . Taking account only of the more important terms, 

 I find, for x positive, 



' (81), 



and, for x negative, 

 where 



MO - _ !? HV p(| -">, v = K'e**'-") 



/A /I, 



H , 



' 



K'= - 



= H'e* (l+er) , =:? K'e 

 V- /* 



, 2/1-'^, (2>c 2 - P) 2 

 " 



(82), 



. . . . (83). 



