Ship Waves. 5 
The quantity H, — Y, is monotonic and decreases to an asymptotic value 
2/x. The symmetrical terms in (14) represent the local disturbance, becoming 
infinite near the origin like 21. The last term in (14) represents the wave 
disturbance in the rear. The expressions are easily calculated from tables 
of the functions, and fig. 1 shows the two parts of the disturbance. 
It will be seen that there is discontinuity at the origin, but that arises from 
extending this particular distribution right up to the free surface. If we 
retain the quantity d used at the beginning of this section, it is easily seen that 
the discontinuity is associated with the last term of (14) ; for any finite value 
of d, this part of the disturbance is zero at the origin. 
4. Consider now a uniform distribution over a finite length of the vertical 
plane y = 0, extending over the range —1 <<a <J. This might be deduced 
from the previous section by integrating with suitable precautions to allow 
for the discontinuities in those expressions; but we shall use the general 
formula (4). Suppose in the first place that the distribution extends from a 
depth d to an infinite depth ; then we have 
M 1 foe) 7 ro Kee Ki tio’ 
=| a d | PO (inepesete CAE IO 2 IN 7 15 
: mile Ja J ik K — Ky Sec” 6 + iu sec 9 5 me) 
For the elevation along the line y = 0, this gives 
6 | 2 0 p—Kd fptx (z—l)cos@ __ ix (x+U) cos d 
ae | sec 040 | corm COT eee ph JG) 
Tu Jo 0 K—k,sec? 6 + iusec 0 
We may put d=0 in (16). Further, the disturbance separates into equal 
and opposite disturbances associated with the front and rear of the system, 
or, as we may call them, into bow and stern systems. Writing q, for x — 1, 
we have to evaluate the real part of 
i ar | 2. () elk Cos 8 
a \ sec 00 | —_—__—___—___———_ (17) 
0 
- K. 
0 K — Ky sec? 8 + tu sec 0 
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