1909. } The Wave-making Resistance of Ships. 293 
consideration the depth of water in the tank is not known. The deepest 
experimental tank appears to be the U.S. Government tank at Washington, 
which has a maximum depth of about 14:7 feet. Now in that tank, with 
a 20-foot model, there would be a “critical” condition near the value 
¢ = 2-9; before and up to that point the residuary resistance curve would 
rise sharply and abnormally. This effect is discussed more fully in the next 
section, and curves are given in fig. 11, with which fig. 8 may be compared. 
It appears, then, as far as one is able to Judge, that it is possible the 
resistance curve in fig. 8 is complicated by the effect of finite depth of the 
tank. 
§ 6. The Effect of Shallow Water. 
We saw in the first section that the wave-making resistance R can be 
written in the form 
BR = hwa? (v—u)/2, 
where wis the group-velocity corresponding to wave-velocity v. For deep 
water w = 30, and the formule are comparatively simple. But for water of 
finite depth / the relation between uw and v depends upon the wave-length 
(27/x). We have 
v= WA (2 tanh Kh) 5 
K 
ne ¢ (cv) = $v (1+ 2h/sinh 2xh), 
K 
Consequently we find 
ils o[4___ 2Kh ) 
Be ae (1 sinh 2xh/ (CM) 
As v increases from zero to ,/(gh), R diminishes from 4wa* to 0, provided 
the amplitude remains constant. But as Prof. Lamb remarks,* the 
amplitude due to a disturbance of given character will also vary with the 
velocity. It is the variation of this factor that we have to examine in 
the manner used in the previous sections for deep water. 
If a symmetrical line-pressure system F(a), suitable for Fourier analysis, 
is moving uniformly with velocity v over the surface of water, the surface 
disturbance 7 is given by 
Twn =} [@ [eve («) sin « {z+(v—V) #} de 
—t [a [eve («) sin « {z+(v+V) t} dk, (22) 
where $(«) = | F (@) cos kw do. 
*H. Lamb, ‘Hydrodynamics,’ (1932 edn. p. 415). 
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