520 T. H. Havelock. 
and at higher speeds positive. For comparison of the maximum local effect 
with the rest of the surface elevation in two cases discussed later, it may be 
noted that 
rgt? fC) /4M = — 1-346, for cof = 4 
= -+ 0-860, for «of = 0-5. (15) 
The local effect at points other than the origin was not calculated, although 
rough estimates were made for the central line, 6’ = 0, from (12) to verify 
that it falls off in much the same way as in the two-dimensional case ; for this 
purpose the second term in (12) was put into the form 
(dicgM/xcf) i, J sec? 6 dO, (16) 
where 
“9 COS U+ USIN U _ gx 
J =( can oar we ea ttl (BP) yy du 
‘3 z — pe? [P cos {ax/(p/8)} — Q sin {ax/(p/8)}1, 
n 
s 2 
LB ap a NOS 9 ar pee COS 
Cc) °" x é 
QB ea me a a (17) 
with = p+ po2/B, tang =aJ,/(Bp), «= Kor, p=Kofsec?®, B= kof. 
5. Consider now the third term in (12). For computation, we alter the form 
slightly. We take 0’ = x — ¢, so that ¢ is the angle the radius vector makes 
with the negative axis of #, and further we put 
= cot ¢; t = tan 0. (18) 
Then this part of the surface elevation, which we may denote by ¢ — @ is 
given by 
7 ae 4ieg°M =f 2) o—Bl at , 2 12h 
Ghent Gtae br son te (1 + #) e~** sin [a (¢’ — t) (1 + @)/(1 + ¢?)}) dt. 
(19) 
In this form «, or xr, is a positive quantity, r being the distance from the 
origin. The axis of z in front of the disturbance is given by t’ = — , and 
(19) is then zero; for the axis of « in the rear of the origin t’ =-++ ©. The 
usual first approximation to the integral in (19) consists in assuming « large 
enough so that the only appreciable contributions come from the groups of 
terms near the positions of stationary phase of the harmonic constituents ; 
this leads to the familiar pattern of transverse and diverging waves contained 
within radial lines making angles of about 19° 26’ on either side of the negative 
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