The Effect of Shallow Water on Wave Resistance. 501 
where f(x) is found from the assigned pressure distribution, p = F(a), by 
means of (3). The factor exp. (—4yt) serves to keep the integrals deter- 
minate, so that they give a solution which corresponds to the main part of the 
surface waves trailing aft from the moving disturbance. It is to be noted 
that ultimately ~ is made zero in the final results, and it is only retained in 
the intermediate analysis to a degree sufficient to attain its chief purpose. It 
should be stated also that all the analysis is subject to the usual limitation 
that the slope of the surface is supposed to be always small. 
We take the wave resistance to be the resolved part of the pressure system 
in the direction of motion, or 
Bee [FoR of as, (5) 
taken over the whole surface. 
The disturbance (4) may be analysed into plane waves ranged at all 
possible angles to Ov. Substituting 
rJole{ @+eye+y%}!2] = | gir (+ct)e05 6 Gos (ey sin b) dd, (6) 
0 
we can integrate with respect to ¢, and obtain, after rejecting superfluous 
terms in p, 
s ; ’ V sec? b 
»—1/2ut pixct cos > iS oo ne 
ie i fA Bea Men xe? —g sec? d tanh xh + ipc sec 7) 
Using this in (4), the surface elevation can be expressed in the form 
9 a/2 5 1 hel ex (x cos +y sin ) 
Tp G— (iss sec cond | «f («) tanh ch Koy see tanh Kh + ipe sec 
entk(e cos @+y sin p) 
—————— 8 
o xe?—gq sec? d tanh ch —wc sec (8) 
3. We simplify the calculations by specifying the surface distribution of 
pressure as 
p= F(a) = Al/(P+ 0°), (9) 
where A and / are constants. It follows from (3) that /(«) = Ae-*. Now 
in (8) consider an element makiny an angle ¢ with the axis Oz. Change to 
axes Ox’, Oy’, given by 2’ =xcosd+ysin dg, y’ = ycos¢—asin gd. Then the 
integral with respect to « becomes 
io) tea! 
xe— tanh xh if : i “ - 
0 xe? —g sec? $ tanh Kh + we sec 
en ikz! 
} de. (10) 
ss Ke*—g sec? d tanh xh —ipc sec h 
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