Elastic Equilibrium of Semi-Infinite Solid. W 



More generally, if the boundary conditions are given in the 

 form of an arbitraiy function, e.g. 



7z=f(r,d), ] 



zr = 0, zd r=o] 



at z = 0, the general solution can ])e obtained by making use of 

 the integral theorem 



Fir) = I J„ikr) MJ:fF(a)J„, (Ä;a) ada, (26) 



'o 'o 



provided the function /(r,Ö) caii be expanded into a trigonometrical 

 series of the form 



/(r, ^) = i {Mr)cosmd+fJr)dnmd] , (27) 



where />»(r) and /»«(r) are supposed to satisfy tlie above integral 

 theorem. 



Comparing the expansion (27) with (zz^ which follows from 

 the first formula of (24) by putting z={), we see tliat the functions 

 Zvi{k) and ZmQc) which correspond to the boundary conditions (25) 

 or (27) are the solutions of the integral equations 



Mr) = I Z.,„{}^)J„i'kr)d-k, 







f,u{r)=jz.^{h)J„{hr)dh. 



Looking at the integral theorem (20), the solutions of these integral 

 equations appear easily to be 



/.CO 



Z,X'k) = A,7/,„(aV„,(Av/)«r/«, 



) (28) 



Z.„ik) = Ic/fja)J.,Xka)ada. 



Thus, substituting tliese values of ZmQc) and Zm(k) in the 

 formuke (23) and (24) we get the solution answering to the 



