i: 



[ 539 ] 



LXV. On Wave-Patterns due to a Travelling Disturbance- 

 By Prof. H. Lamb, F.R.S.* 



1. 'l^HE procedure of a former paper f may be adapted 

 JL to the case of propagation in two dimensions, as- 

 when a pressure-point advances over the surface of water. 

 Although nothing very novel is to be looked for in the way 

 of results, the generalized treatment of a somewhat intricate 

 problem may be acceptable. 



The method, it may be recalled, depends ultimately on 

 integrals of the type 



^ e izx d* 



A*) ' 



taken round suitable contours. The functions F(c),/(^) are 

 usually algebraic, and f(z) has one or more simple roots of 

 the form /c + i^, where k is positive and fi u which depends- 

 on an assumed frictional coefficient, is in the end taken to 

 be infinitesimal. For simplicity of statement it is assumed 

 for the most part that there is only one root of f(z) of the 

 above type. 



First suppose that x is positive. We put z =k + im, and 

 integrate round the boundary of the infinite quadrant in 

 which k, m are positive. If fii>0, this region contains the 

 singular point K + ifi^ which must be excluded. Thus 



Jo /(*) /'(*-MA*i) Jo X*"0 



The latter integral diminishes rapidly with increasing x. 

 In particular, if 



J -p^=A + Bm + CW+..., ... (3) 

 j\im) y ' 



its asymptotic value is 



^ + p + -^- + w 



Hence, when /-tj-K), we have, for sufficiently large values 

 of oc, 



Jo JW /(*) ' 



* Communicated by the Author. 



f Phil. Mag. (6) vol. xxxi. p. 386 (1916). 



