WAVE PATTERNS AND WAVE RESISTANCE. 9 
we have supposed this to be very large, or theoretically infinite. Let us suppose now that 
the model is of uniform horizontal section down to a depth d below the surface and is then 
cut off by a horizontal plane. For our present descriptive purpose, we may make some 
simplifying assumptions in deducing the formule for the wave system, but these need not 
be investigated here; it is sufficient to state the general result. The effect of making the 
model of draught d, instead of infinite draught, is simply to introduce into each of the 
ternis for the component patterns, in say (13) or (15), an additional amplitude factor, 
namely 
ie STC a een REPT ee hon eal oy uate nt AG) 
Since xd =gd/c?, the value of this factor depends upon the speed. Fig. 7 shows 
curves of this quantity (16), for different values of «d, for the half range of values of 
6 from 0° to 90°. 
From inspection of this diagram we see at once that, for a given speed, if the draught 
is diminished the transverse waves of the pattern become less important. We may put 
1:00 
“75 
“50 
15 30° 45° 60° 75° 90° 
Fic. 7—GRAPHS OF 1] — e—*@°® FoR DIFFERENT VALUES OF «cd. 
alongside this a remark drawn from observation; for instance, in Taylor’s Speed and Power 
of Ships there is the statement: “Narrow deep ships have wave patterns whose transverse 
features are more strongly accentuated than those of broad shallow ships.” 
This might, of course, be anticipated without any mathematical expressions. or the 
effect of a plane wave on the surface is only appreciable down to a depth of, say, half its 
wave-length. But of the component plane waves which combine to make the pattern, 
that which is travelling in the same direction as the ship has the greatest wave-length, and 
in inclined directions the wave-length is proportional to cos? 6 and diminishes to zero at 90°. 
Thus as we diminish the draught, for a given speed, the first components to be affected are 
those of longest wave-length and those are the components which provide the transverse 
waves of the pattern. However, the mathematical expressions enable us to obtain at least 
a rough quantitative estimate of the effect. 
WaveE RESISTANCE. 
10. We turn now to the calculation of wave resistance, and for this purpose it is 
essential to have a knowledge of the wave patterns we have been considering. Throughout 
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