540 Prof. H. Lamb on Wave-Patterns 



If, on the other hand, /*i<0, the first term on the right- 

 hand side o£ (2) is absent, since the region considered has 

 now no singular point. We have then 



f m eaxdk=0 > (6) 



with the same approximation as in (5). 



Next, suppose x to be negative. We now take as our 

 contour that of the infinite quadrant for which k is positive 

 and m negative. Thus, if fii<0, 



1 ~7Tjie lkr dk= * . m i. e lKX -^ x -l\ - , v . / e mx dm, 



Jo JW j {fc + ifiv Jo J{-im) 



or, in the limit, 



/(k) e dk ~ f( K ) «....-•• w 



when x is considerable. If //i>0, we have 



I 



'n 



f(k) 



kx dk = 0, (9) 



in the same approximate sense. 



It is to be remembered, however, that the integrals which 

 we have neglected may be important for small values of x, 

 and may even become infinite for x = 0. The infinity may 

 be avoided by the insertion of a factor e~ zb in the value of 

 F(z); this leads to a factor e~ vmb under the integral sign in 

 the last term of (2), and so secures convergence by fluctu- 

 ation. This artifice has moreover a physical justification, as 

 enabling us to represent the effect of a force which is diffused 

 about a point, instead of being absolutely concentrated. 



The integrals, as thus modified, will still be sufficiently 

 important to be taken into account when x is small. In the 

 application to one-dimensional wave -propagation this is of 

 no great consequence, as the matter only relates to the state 

 of things in the immediate neighbourhood of the source of 

 disturbance. In the problems now to be considered, however, 

 the point requires attention. 



2. Proceeding now to our question of wave-propagation 

 in two dimensions, an impulse of the type J (&ot), where w 

 denotes distance from a point Q, will give rise to an annular 

 wave-system of the type 



e=j {kwF)4>(k)#* (io) 



about Q as centre. The form of <p(k), and of a as a function 



