SHIP VIBRATIONS: THE VIRTUAL INERTIA OF A SPHEROID IN SHALLOW WATER 
the greater part, and a useful approximation is obtained 
by taking only one or two terms. For instance, if we 
take the simplest case n = 1, representing a transverse 
vibration with no node, we have, omitting the time 
factor and any constant multiplier 
$o = A3/Q3 (Co) . P3 (x) Q3 (OQ) sin2 w 
+ A/Q3 (o) . P3(u) Q3(Osin2w +... 
the coefficients being given by (40) with n= 1. 
gives AZ = 5/16, AZ = 1/64,... 
For the kinetic energy for the complete spheroid in 
an infinite liquid, we have the usual surface integral (4), 
in which 0¢/)¢ = + P!(u)cosw@ according as w is 
positive or negative. We obtain 
T= = PE —e)fe. LEQ (Co)/ 
Q3 (Co) + z8z Q5 (o)/Q5 (Go) +. --] (43) 
For the case we have been using for numerical com- 
putation, ae = 105, the terms in the square brackets in 
(43) are 0-9375 + 0-0195 +... Although the rest of 
the series converges rather slowly, much the greater part 
is given by the first term. 
Consider now the same problem in water of depth f. 
To avoid prohibitive complications, we shall take only 
the first term. Although this leaves somewhat uncertain 
the degree of approximation, yet, as we are concerned 
more with relative increase than with absolute value, we 
may expect to get at least the main character of the 
variation with depth of water. Omitting unnecessary 
factors, we begin, in the notation of previous sections, 
with 
(42) 
This 
0 = P3(n) Q3 (¢) sin2w 
and suppose the bed to be given by z = //ae. 
(44) 
We now have an infinite series of image systems 
associated with the points z = + 2s fJae, and we have 
b= —2 E (— 1 PE) BG) sin2, (45) 
To expand a typical system associated with z = qg, we 
have, from (8), 
P3 (5) Q5 (Z,) sin 2 w, 
» (1 — k*) 3 (kA) dk 
 dydz|[@ —kh? +y?4+  —@)’}? 
= 
Y (1 — k*) P3(k) dk 
Ipoq|(@ —kK? +0 —P? +e -—o} 
=1 peulahees (46) 
Wit eG pil —)per) (Gra) COs ay 
q=(1 — -)! (G — 1) sin; we select the required 
= lim 
p—>0 
589 
term using the expansion (6), giving for the coefficient of 
P3 (1) Q3 (0) sin 2 w the expression 
Boo 
288 dpdq 
1 
(1 — k?) P3 (k) P3 (144) Q3 (¢,) sin 2 w, dk 
—1 (47) 
Further we have 
1 
2 (1 — A?) P3(h) dh 
PA) iG) sa Ion = | 3 
Deg Cl) ees aaal 
-1 
(48) 
Putting this into (46), and taking the limit as p — 0, the 
contribution from a typical term may be written as 
= 5 B,, with 
1 1 
¥ (1 —2)2(1 — kK)? dhdk 
Bg [& = AP +P] 
=I 
-1 
(49) 
and the required term in the expansion is 
& E(— 1)°-' B, P3 (u) P3(O) sin2w (50) 
G=i1l 
to which, in order to maintain the normal velocity on 
the spheroid, we add 
= 5 E(— 11 B, PH (Co)/Q (Lo) . PS (H) (CO) sin 2 w 
(51) 
Finally, after using (7), the value of ¢ on the spheroid, 
to this approximation, is 
QB (Go) [1 — 15 E(— 1 BY 
2 (4 — 1) Q3 (Co) Q3 (Eo)] P3 (u) sin 2 w 
Since the value of 0 4/)¢ on the spheroid is unaltered, 
the kinetic energy is increased by the factor in square 
brackets in (52). Hence the relative increase in kinetic 
energy, or in the inertia coefficient is given by 
8 T/T. = — 15 E(— 11 BY/2 G — 1) V (Lo) Q8 (Lo) 
. (53) 
(52) 
with B, given by (49) in which gq = 2s /f/ae. 
References 
(1) F. M. Lewis: Trans. S.N.A.M.E., 37, p. 1, 1929. 
(2) J. Lockwoop Taytor: Phil. Mag., 9, p. 161, 1930. 
(3) C. W. Prouwaska: Bull. Ass. Tech. Marit., 46, p. 171, 1947. 
(4) K. WENDEL: Jahr. Schiffbautech. Ges., 44, p. 207, 1950. 
(5) G. WeEINBLUM: Schiff. u. Hafen, 3, p. 422, 1951; also 
T.M.B. Report 809, Washington, 1952. 
(6) E. W. Hosson: Spher. and Ellips. Harmonics, p. 416. 
(7) T. H. HAveELocK: Quart. Jour. App. Math. and Mech., 
1952. 
