88 Mr. C. Chree on Bars and 



assume the ordinarily accepted equation* 



d?w _ P^ <Pw, fw\ 



dz 2 ~ M dt 2 ^° ; 



the axis of the bar being taken for axis of z. This equation is 

 obtained on the hypothesis that the cross section is very small 

 compared to the length, though it is generally supposed to 

 apply to bars of moderate cross section. The consequent 

 solution satisfies the conditions over a free or fixed end of the 

 bar, but not those over the cylindrical surface. The results 

 we shall obtain for a bar of two or three materials, each layer 

 being of a length great compared to the diameter of the cross 

 section, are thus just as satisfactory as those ordinarily ac- 

 cepted. We do not think either that, in extending the results 

 to a continuously varying bar, we introduce any appreciable 

 error, as the principle that the lateral displacements are very 

 small compared to the longitudinal at distances from a fixed 

 end, great compared to the diameter of the bar, is not affected. 

 Assuming in (38) to <x cos kt, we get 



S + 3F-0' (39) 



k 2 p 

 whence, if A and B be constants, and a 2 stand for ~-, we get 



w= co$kt(Acosuz + Bsma.z). . . . (40) 

 If we neglect u, we thus have 



^i + *£ = 0, and therefore T= 0. 

 dz dr 



Thus the only stresses existent in the solid are 



R= (m + n) -T— = (m + w)a(— Asina-s + Bcosoz) cos Jet, . (41) 



ctz 



P = (m — n) -7— = (m — n)a( — A sin clz + B cos otz) cos kt. . (42) 



az 



The surface-conditions over the cylindrical surface, whether 

 circular or not, would require P to be zero, which, save for 

 the altogether exceptional case m = n, it is not in general. 

 This is the defect we referred to above ; but it will not modify 

 the following results to a greater extent than those ordinarily 

 accepted for a bar of one material. 



At a fixed end of the bar we have w=0, at a free R = 



and so -7- = ; thus the natural periods of vibration of a free- 

 as 



* See Lord Rayleigh's ' Sound/ vol. i. § 1-50, putting M for q. 



