SHIP VIBRATIONS: THE VIRTUAL INERTIA OF A ‘SPHEROID IN SHALLOW WATER 
In this case, we have 
$; = Ph (#') Qh (6) cos w’ 
1 
mn) 
lim 
wea == Hin 
Using the expansion (6) in (21), with the point (kK pq) 
being (4; 6; w,) in spheroidal co-ordinates, we obtain 
the coefficient of P! («) P! (¢) cos w as 
Zeal Oo 
n(n + 1)? dp 
el — k?)? P! (k) dk 1 
[@—b2+0—p2+e-arp CP 
M= 
1 
{a — 2) P! (ke) P! (uy) Q! (L,) cos wo de (22) 
=i 
Further, we have 
1 
12 | @—) PL) dh 
2 op | [(k—A)?+p* +47}! 
—1 
Px (#41) Qn (£1) cos wy, = (23) 
Taking the limit of 07/) p* as p->0, this leads to a 
fractional increase of kinetic energy, or of the inertia 
coefficient, given by 
r) T/Ty = : 
—Qn+ IE J2n@ + 1) (GB — 1) Qo) Q (Go) (24) 
with 
1 1 
1—Ah) (1 
z= | | i 
oi 
These integrals can, of course, be evaluated explicitly in 
closed expressions for any given value of m. However, 
the expressions become very lengthy for the higher 
orders and we shall not reproduce them here in general. 
For numerical computation it proved somewhat better 
to express the double integrals in terms of subsidiary 
single integrals. Also one can obtain, either from the 
double integrals or from the explicit expressions, approxi- 
mate forms suitable for q large or qg small. 
10. We give now the results for some special cases. 
n= 1. For motion at right angles to the boundary, 
(18) and (19) give 
C) T/To = 
=f 
= $log {[2 + 4 + @)']/q} 
spa (2a == Ie =2ar)G = GA) 
Prq+ae@ . (7) 
For motion parallel to the boundary, (24) and (25) give 
8 T/T) = —F E[(G — 1) Qi (£0) Qi (Lo) (28) 
— Ky? Pn (A) Pn (Kk) 
= hy = ep? 
dhdk (25) 
—3DK(Q — 1) Qi (4) QS) - (26) 
with 
d=”) —2) 
[kip Hep 
dhdk 
587 
with 
1 
1 
ue @ Pa =2 
= [(k es hy? ae Gg |? 
== 
—$ flog [2 + 4 + @)!]/a} 
f 2 (Fs Gia f 
dhdk 
52 4 
45 45 
q)(4 + 9) 
=59 = 257 (29) 
It is readily verified that as gq — 00, or e 0, we recover 
the known results for a sphere, namely 3 53/8 f? and 
3b7/16f? respectively. We may also find the limiting 
values for a long spheroid with b/a small, by making 
e—>1, ao, while retaining b? = a?(1 —e?). The 
limiting values are the same for the two types of motion, 
as is the case for a circular cylinder; but the value of 
the ratio is 2b7/5f? instead of b7/2f?._ Thus in this respect 
the circular cylinder is not the limit of a long spheroid. 
This value can also be obtained directly by the two- 
dimensional strip method. For this purpose we con- 
sider a circular section of the spheroid of radius y; take 
its contribution to the kinetic energy of the fluid motion 
as proportional to y?(1 + y?/2f?), and integrate along 
the axis of the spheroid. 
n= 2. Considering only motion in a vertical plane 
(18) and (19) give 
8 T/Tp = —15 A/4 (GB — 1) Q5 Go) Q5 (Lo) 
with 
(l—R) (1 —k)hk 
al, Togugsie [é-—yter dhdk (30) 
= For a 2-node vertical vibration 
) oe = —21 A/32 (65 — 1) Q3 (4) Q4 (Co) 
with 
ere h?) (5 h? — 1) (1 — k*) (5k? — 1) 
~ OG (=F = GP 
: (31) 
n= 4. For a 3-node vertical vibration, 
8 T/Ty = —45 A/32 (65 — 1) Qh (Lo) Q4 (£0) 
with 
=< 
(1 —h*) (7 h3 —3 A) (1 —k?) (7 3 —3 k) : 
Pi i (Gaara dhdk 
x (32) 
For a longs Spins! the limiting values of these expres- 
sions are 2, +, 22b°/f* for n= 2, 3, 4 respectively. 
These can also be obtained by the two-dimensional strip 
method, taking into account the distribution of trans- 
verse velocity along the axis of the spheroid. 
