2 T. H. Havelock. 
are given of the surface elevation along the line of motion. A similar analysis 
is given for the distribution corresponding to the model described above, and 
the connection between the distribution and the model is indicated. 
Finally, the results are generalised to give the central surface elevation for a 
model, of infinite draught, of any sectional form. The general expressions 
are of simple character and some deductions can be made from their form. 
In addition, they are suitable for the numerical or graphical calculation of the 
profile for any required model of this type. A brief analysis of a parabolic 
model is made to illustrate the general results. 
2. Consider a doublet of moment M at « depth f below the surface of water 
and moving horizontally with constant velocity wu. For the present applica- 
tions we need only the expressions when the axis of the doublet is horizontal 
and in the direction of motion ; further, we take moving axes with Ow in the 
direction of motion, O in the free surface, Oz vertically upwards, so that the 
position of the doublet is the point (0,0, —f). The velocity potential of 
the fluid motion is given by* 
@ 
cos 00 | em KH) FiRm pe de 
Ml (i cos 040 [ Be irae (0) 
AES 0 K — Ky sec? 0 + iusec 0 
where 7 = x2cos 0+ ysin 0 and ky=g/u*?. The real part of (1) is to be 
taken. The first term expresses the velocity potential of the given doublet in 
a form valid for z + f > 0, that is for points above the doublet. In the second 
term pis a small positive constant which is ultimately made zero. The surface 
elevation ¢ is given by 
mio ea (2) 
This gives 
T te) 2 p—Kftine 
¢=Lim >| ao | SiMe Cm ak (3) 
u>0TUS-, Jo K—kgsec? 0+ iu sec 0 
In this form € is finite and continuous, and the expression may be generalised 
by summation or integration for a distribution of doublets. We shall consider 
here the distribution to be in the vertical plane y = 0. If M(h, f) is the moment 
per unit area at the point (h, 0, —f) we have 
1 [o} °oM ] dl d 7 d0 oa) Ke ews Hino’ d (4 
== (sf) ; ile [, kK — Ky Sec” 6 + au. sec 0 - ! 
* © Proc. Roy. Soc.,’ A, vol. 121, p. 518 (1928). 
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