542 Prof. H. Lamb on Wave-Patterns 



The effect of an impulse Bt delivered at Q is therefore 

 given by 



the factor e~^ being introduced as in the former paper to 

 represent the effect of slight dissipative forces. Integrating 

 from £ = to t = co, we have 



^~~ 47r 2 J J.,, /* — i(a-kccos^) 



3. The approximate integration with respect to & is 

 effected by means of the formulae of § 1. Let k be a value 

 of k satisfying the equation 



cr = kc cos i/r, (17) 



where t|t must of course lie between +7r. That is, 27r//c 

 is the wave-length corresponding to the wave- velocity 

 c cos ty. It may of course happen in particular cases that 

 there is no such value of k, or there may be more than one, 

 ..as in the case of waves in superposed liquids treated in the 

 former paper, or in that of waves under the joint influence 

 of gravity and capillarity. In the latter event each such 

 value will give rise to a separate term in the value of f. 



Taking for definiteness the case of a single root of (17), 

 the denominator in (16) will vanish, if jjl be small, for 

 k = K-\-ifju 1 , approximately, where 



yL6 l = /x/(cCOS^ — U), (18) 



U being the value which the group-velocity (da/dk) assumes 

 for k = K. Hence, referring to (5) and (9), we have, if 

 U<c cos yjr, 



^~27rJ CCOSi/r-U ' ' * ' ^ 



where the range of integration is to include only such values 

 •of yfr, within the limits +^7r, as make 



a cos yjr + y sin yjr>0 (20) 



If, on the other hand, U>c cos-v^, we have 

 JL_ (Y" (xcos ^ sin, W/c<£(«:)^ 

 *~2ttj U-ccos-f ' • • • (wl ) 



the range now including such values as make 



#cos^r + ?/sin^r<0 (22) 



