508 



PROFESSOR CHRYSTAL 



If the square of dp be negligible, so that dk v = k'„ — k v and 3t„ = x„ - r v are both small, 



we have 



dk y = A'„ cos «„t„ + B'„ sin ii v t v } 



3t„ = (B „ cos n v T v - A „ sin ?i„r v )//,\n y ) 



The maximum value of dk v for different values of r v corresponds to tan n v r v = B^/A',, ; 

 and under these circumstances dr v = 0, and dk„ = (A'/ + B'„ 2 )*. 



11. As an example of the application of the above formulae, let us consider the 

 effect on the uninodal seiche of a sudden rise of pressure dp which begins at the 

 negative end of the lake at t = 0. 



Putting Q\(n) =/*, and Q 1 (/«) = ^(/ u2_ 1), we get 



where, if = n x ajv = 2-7ra/vT 1 



A x = -J(2v+l)3pe, 



B 1= +i(2v+l)3j><I>; 



©: 



4> = 



1 - cos 26 _ sin 20 ; 

 sin 20 1 + cos 26 



and 



V when wT > 2a : 



©=-| ^J + -L| C os Wl T-^-_}smn 1 l + _; 

 A f 1 - i(»,T) s , n,T 1 • T |J!,T 11 T 1 



-when vT<2a. 



(37). 



It will be observed that in general and <I> are functions of v alone when i>T>2a. ; but 

 functions of v and T when vT<2«. 



If we restrict ourselves to the case where T = 2x/?i 1 = T l5 then nJTj = 27r, and we get 

 for the case where vT 1 <2a p.e. v<26"9 (mile/hour)] 



<p= - — . 



2 



In the case under consideration, therefore, the functions and <I> are determined 



as follows : — 



„ 1 - cos 26 sin 26 , rs^n^> \ 

 © = - — ^ — when 0<^#<V ; 



= 9 



$ = 



6 2 

 sin 2(9 1 + cos 26 



6 2 6 



- * when 0>tt 



when 0<(9<tt ; 



■ (38). 



The graphs of and <£ are given in figs. 21 and 22. 

 The greatest possible increase of amplitude is given by 



3/Cj = f 3p(© 2 + <E> 2 )*. 



