571 Prof. T. H. Havelock on 
same wave-length as the free wave of the same frequency, 
while the second term may be called the local oscillation. 
If we take f(z)=e-"*, the second term in (9) and (10) 
vanishes, and we regain the expressions for a simple 
progressive wave. 
If we take, more generally, 
VACa Nae ee, Rca wenn OLE) 
over the whole range for z, the second integral in (10) can 
be evaluated explicitly in terms of Cosine and Sine integrals, 
and we obtain 
t= 2o0A SECO) 2oA sin ot ‘6 
Ko +p " wg(ko+P) 
x [Ci(pa) cos pa —Ci(K,x) cos xv + 82( px) sin pax 
—Si(«yx) sin Koa — a (sin pe—sinkyr)]. . (12) 
As we make p smaller we approach the limiting case of 
constant normal velocity over the whole of the plane z=0. 
It is of interest to note that the amplitude of the travelling 
wave remains finite in the limit, but that the amplitude of 
the local osciliation becomes logarithmically intinite at z=0. 
3. A problem of some interest is the decay of the vertical 
oscillations of a floating body due to the propagation of 
waves outwards from it, but a direct attack upon the 
problem is difficult. We may perhaps obtain a rough 
estimate by applying the preceding analysis to a simplified 
form of the problem. Imagine a log of rectangular section 
floating in water with the sides vertical ; let b be the breadth 
and @ the depth immersed. Now suppose the log made to 
execute small vertical oscillations of amplitude a and 
frequency o. Let one of the sides of the log lie in the 
plane 2=0; then the disturbance in the water on that side 
may be regarded as due to a certain oscillating distribution 
of normal fluid velocity over the plane «=0 from z=d to 
z=o. If we make the assumption that this is of the form 
P@)GSGH oo 6 0 5. 0 (C8) 
then, from continuity of flow, we have 
2" /(e)dr=eab. oe oo G4) 
d 
Without attempting to solve the actual problem, let us 
assume 
H@QSACH a 5 2 0 0 0 (lS) 
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