388 Prof. H. Lamb on Waves due 



Putting now fjb = 0, we have the simple expression 



_ <f)(K)e ixx 



CJI 



(12) 



for the wave-train generated by the travelling disturbance. 

 The train follows or precedes the disturbing agent according 

 as U^c. Examples of the two cases are furnished by gravity 

 waves on water, and capillary waves, respectively. 



If there is more than one value of k satisfying (5;, there 

 will, of course, be a term of the above type for each such 

 value. This happens in the case of water-waves due to 

 gravity and capillarity combined, and in the example treated 

 more particularly below. 



If the expansion of a — kc in powers of h' were continued 

 the next term would be ^k' 2 d~U/dk, and it is therefore 

 necessary for the validity of the above approximation that 

 the ratio 



5*£/c*-> 



should be small even when h f x is a moderate multiple of 2tt. 

 The ratio 



dJJ 



dk > V~< 



must therefore be small. Unless U = c exactly, this condition 

 is always fulfilled when x is sufficiently great. It may be 

 added that the results (10) and (11) are accurate in the 

 sense that they give correctly the leading term in the 

 evaluation of (4) by Cauchy's method of residues. 



2. If the value of h which satisfies a = kc also happens to 

 make U = c, the procedure must be modified. We write, 

 then, 



*-H5*' 2 =±<^' 2 - • • • o*> 



say. Taking the case of the upper sign, we have 



'-£?<HL.iW • • • (M) 



This can be evaluated by contour integration. The value of 

 the integral is found to be 



2wt'^- 1 -H>* 2m^ 1 -^ 

 or 



2(-l^-^•)/3 , ° 2(1-0/3 ' 



