TUtIE. Ci' T1F
VE RESISTkNTO THE }L
OF SHIPS
(Part II  EffFcts c' ?oticr
aheae on Wave Re
.stacce to Rdllingt
y Ton
.i.hica
original
aper wa
puhiishe
in the journal of th Eocity
c
t'aia1
.httct
f .iap. Vol.
7.
This is th
t.raysiat.i.'r
:f the criinal paper, reviied
anì en1ared by the author foi'
cirf.fl if th çoirS isussd
by
orNn1r
ft with a
sist.: parer "flolkr
riments
with e1iPrope1ir.g
odei
htp.
Thi
blihi in the journal of th
Kana;
Arcflitects cf Japar.. Vol.
72).
CJ!TENTS
Intcictior.
...
'aveMakir.g 1esi'tace cf ar. ¿llipsoid
ol' PjcliC.
3.
Disctssions of the above results.
.
Effcts
f Lirited width of Water.
.Disc
sior.
he aov
Results.
olli.ng
xperthentS witl. selfPropelling Model
h2ps.
7.
_{Suinuary.}
.ppenuix.
IrtrdUctiO.
[t i
well knowr, fact that wave resistance
to sh
;.Luing
increases '.i.ih ìncreang her forwaro
speed.
This p
nomencn has
bn examined both thec.retìally an
x:erimentally by ïny
investiLat:rs
among t.hn, W. White
(1), A.W. Johns (2), E. Brtin
3),
.F. Fayne (4)
lì. Legendre (5), R. 5ar.tis and N. Russe (b,.
.
S. Baker (7), N. Kato (8) and so on.
riowever, thorough
explaration .of the rrchanis cf augmentaexplaration of resistance to rolling
still recainS to he done.
The author disuseS Ìre the resLlts öf
rofling exj.erimers
'sing sefpropelling model ships with and without bilge kc'is
as well as the theoretical
crnpitatio
cf wave resitr.ce t.
rolling of an ellipsoid of re.îc1ution in
notion ahead.
2.
Waret'aKing t&istance of an ELlipsd
tieV?lution.
e tae the x, y and zaxes as in F'i. 1. Ari ellipsoid
of revolutin wt.h a major axis 2a and ai equatorial radtus h
rcves thrcu
water carI1eì to the fr
surface with a
contant:peed U in the drtr of x posìtve.
If
he ellipsoid
rolls about. a longitudinal axis
abovethe
ajor axis, the
transvere velocity V may be expressed by Eq. (2), provided
that
tw rolling angle is defined by
8
t9 sinrt,
= r
9o coscrt,
& is the amp1tde of rclling, t tii
and r0 the height
th roing axi
above the major axis.
The ve.ccity potential ' for the fluid motion is gi7en by
ti
real part cf (rfer to Appendix T)
ae
_{2ezh2\hJ}

Dx os 9dB dk
tJ(1h2)dhJLkk(f)(b)9
.fYSiì.GJjifl8d9dk
(3)
00aZe _JZ
_{)dh}
_{k?(k,e}
+Y5ifl8] CO5bdBdk
(a1
h1 )dh
_{j}
kG(k)8 )expk(fz )+ik t(xh)co&
wr1er
ac
+Ysin&] sir&d8dk
i
r&0a'
e
i
l.ie'
.,1+e
i2e
'
'°gT
icg1
2e12
(22ufrt+v22
_{n}
aU1
F(k,)=
!+(g/U')sec6+i(P/U)sec
k(g/i, )iec+i(/L/U)sec6
'
o . .g:4(cos6)ucos4ifr)
gkpUkcos)(r Jk)
_{cos}
(1)
f2.
\ I v1/4.
and G are functions to be dterniind by the condition of the
free surÇce given belo.
IT.;=O,
ULL5o.O.
}
5)
(4or
There fore
In Ihese eqationS
is the elevation o.f the free
surface and
p Rayleigh's internal resistance.
It is evident that the first
and third teriis or the righthand side of (3) havé no effect or.
wave resistance to rolling nor
do the second and f ot3rth terris
on speed ahead.
Taking therefore only the second and fourth
terrs we write again
utbtng z=O in the real part of
*fco
,.(
F1 cos(kcosO)sin(rsin&)4F25in(C0)5in(uh18)
kddk.
Similarly
JJicos(
xcos
in(kysin8)+G2sinUxcos)sin(kysi48) kd6dk,
co
where h_{)dhÇ°os(khcos6)}
=
i_ae e h1 )dhkeSinO,.
(_XR+SY)cosrt(SX+RY)Sir3f t X°
_{_(_xR+SY)sinrt(SX+RY)cosrt;} R= gk_+2Urkcos8U1k1coS & S=,u.(rUkcosB), X (rUkcos8)2 ,Energy dissipated per unit time by wavemaking3s given by the formula Pfl
l.í
Elirnj
J J(9g)
dxdydt', (1.0) .where T is a rolling perio&
However, since
jfrx,Y)
47jd9jCF,G1
4FG2)kdk.
(11)
ws now have
F1 G1 4F2G=
xcos(khcosû)cos'(krncos6)dhdxn.
00 xF=ìin4flNlgf
d8.f
dk( dhÇ dm(a2 e1_hl)(a2!n2)
3 i ).ee
j3e
(13) LSifl26
_{h} _{)} (iou a) R1iS2 ijae f ae 211f(a2
e2 h2 )(a e2
2)k e
siI?8 'X
jaej
ae
B *5
Transforming by a contour integratthn (refer to
_{Appendix II),}
we have
Cr3 LN'.
Çk,))).
4
1 2(1+2)cos8+/1449CPBß
v'l4cos&
_{42jCo!l}
)xexp( 2f .l+2+4)cOs8)+il
r
23, .ose
,'. il+2'CosD/1+vcOs6 )exp(
2f i+2Pcos.Ji+4'cos
4)COS
12Xcosô
sìn2
X
cos&5
with
91Z for
_{2Ecos11}
_{for}_{i..}
(17)L in (14) to (16) approxivteìv represente _{the length of}
th ellipsoid if it is long andslender 1ike _{a ship.}
_{¿Vis a}
rtio of th
speed of ahead motion to the propagaticn _{velocity} of a wave ripple and_{X}
_{a Fróude 's number.}Although is the function of 'and)., it _{may be considered} as that o',\ and k0L, for
¡' _{/g.}
And kL/2means the ratio of L
to thelength of tw
_{dimensional}
waves of the period T. _{cor.puted by the numerical integration}i _{shown in Fig. 2 in which 2f/L is assumed to be 1/50.}
3. Discussions of the above Results.
It appears in Fig. 2 that wave.aaking reiistanee incrèäses
witn speed, very
gradually at firstbut rapidly at Froude's
numbers over 0.1. _{There is an inf1ction point at V= 1/4, at}which 9, passeu from 7C to ic . _{As to k0L, the smaller it} _{is,}
the greater the resistance.
When an ellipsoid has no speed, wavemaking resistance is due to
progressive waves.
_{The energy per unit tire}
transported
 by these waves is of an order c, A1 L, provided the mean hèight
of the waves is 2A _{and e,} _{a proportional constant.}
When an
ellipsoid is in motion ahead, it moves through undisturbed _{ter} constantly iving
it energy.
_{Some of the energy is left behind}
the e11ipojd, and some reriains _{in a conser'rtive system of} staniiñg waves around the ellipsoid. _{Let us suppose the}o

2ee2)' ;
À=, t=2ae.
'(16)
' a Bessel function. The
upper limit of the
jiwan htLght of the standing waves to be 2A nd the energy left behind per unit ti is c2Â U.
Then tn
rate
of increase in wave resistance is c1A U/c1 A L. The faster is U or the shorter L the larger is the rate.Furthermore, A5 varies with U. If the ellipsoid bas no speed, the height of
standing waves is
comparativelygreat on
both sides of the ellipsoid and it is small in the fore and aft body. But when the ellipsoid r'ioves ahead, the ditributionof wave heights relative to it
changes:
i.e.,
dense at the bow .;hile thin at the stern, forelementary waves,
created6y all
parts of the ellipsoid which compose a standing wave, are crowded at the bow.I
Therefore, the increint of lost energr changes gradually up to a certain Speed because the height of the standing
waves
left behind the ellipsoid is coiarative1y low, although its length increases with U.When speed reaches the
magnitude corresponding to V = 1/4, the whole system of standing waves is left behind the ellisoid. Athen becomes constant.
The
incrernt of the resistance is linear'y proportional to U.The physical meaning of y= 1/4 may be
explained
as follows: For small U, although the ellipsoid is ina wave systeii
generatedby itself; with incréasing U, the wave shifts
back relative to
t.
At )=U/g=l/4, wave speed becomes
equal to U.
gain, when
the stem of the ellipsoid is at A in
Fig. 4, the principal part
of the waves generate.d by the stemis expressed as
wre r is the distance from A to the
princia1 part. of wave
ripples. Mter the period 2/, when the stem comes to S, ir.
order that the principal part may hev reached B, r must be
U./r.
And we get
=2*:
¿r tWç
07g
_{= 1/4.}
This value of 's"
is independent of the dimension and form
of
the moving obstacle. For speeds U.over
g/4a, the obstacle
outruns ripples, as shownifl
Fig. 5.We see these properties of wave resistance at U in the experimental results on "Conte di Savoia". Fig. 6 shows this
n comparison with an
ellipsoid, where resiStance to the rolling
o"Conte di Savoia II
includes slight resistance duetó very
srmll bUge keels.
The same tendency nay.be found in the
experiments on Amodel ("Royal Sovereign without bulges nor
bilge keels) after
Payne (4), although we cannot make conparative
study because the particulars of the ship are not
known.
5
h.
Effect of Limited Width of Wter.
A motion ahead always increases wave
_{eSitèLnce to the}
rolling as discussed above.
ut there. are soi
reports showing
contrary results made by Legendre (5), Kato (8), etc.
The author has guessed tt such cofltrary resultsy
possibly $ a.ttributed to sidewall effects 1 tbe expériintal
tanks. Accordingly he examines the sìdeÑ11'effects ònan ellipsoid of revolution.
Approximately, we consider an aigebx'iÍ supe osition of
an infinite number of velocity potentials f'or each of iirga
ellipsoids which áre assumed to be in an infinite number and
separated one from the other by a distance q representing the
tank width.
The disturbance on the surface of thè ellj.ps
oid coming from the images, will be small if q is modertelylarge conared
with the transverse sectional length of the. 'ellipsoid and
wemay explain general. tendencies (réfer to Appeñdix III).
Velocity potential for each image is
y replaced by y+i
(rn=±l, *2,. ...) on theighthand side of Eq. (3) and
multipiled by (l)'..
Rewriting similarly in .(7).and (8) and suflning
with respect to m (including rn=O), we have
rn J
)
F1cos(coS8)sin(ksi(4s):).pF2sjri(.}ocOs&)
rn0
_{\}
(J,)"
xsin(ksins(Y+mq))jkd&di(I
r°
{).
)frt
cos(kxcos8)sin(1cysin8)+F2s](1Keos)sin(kysjfle)j
Ofl
x1+2
(_lfLcos(nqksin8) kdôdkl
,
(n
1,2,3 . .
_{.}_{). (19)}
tffl'Io
À similar forti way be had for..
Eqs. (12) and l3) change therefore to :those
withintegrands
multiplied by [1+2(_l)tcos(knqsjie)j2
_{and made tcnd to no°}
ni
As this multiplier inc1ues.only circular
functions, if weconsider it together with cos(khcbs&)(csknIós8) in (13), it is
transformed Into a form
consisting only of circularîunctìcns
like the righthand
member of Eq (x) in Appendix II _{Accordingly} with respect to the integration of.E.,1also, the first and second
groups in (xvi) in Appendix II must be multiplied by the same multiplierwithin which k is replaced by k1
_{or k2 .} _{Consequently} we have . corresponding tó of Eq. (25):{2n:ltt
(9)jls2$1(1)tcos(k,nqsiz1e)3
e(_l)1tco(k1nqsin6)*JJ]
(20)
fl]
where f, and f are the first and second terms
01the irite.grar.d in
Eq. (15) respectively.
Ising a wellknown surcation forrala of a trigonometrical
series,
l2
(l)'cos(k1nqsjn5)
_{cos(n+)k,qsine}
rl
I2n+1
_{J}
L(2nf1)c0s2 k,qsine
This is very s11 for every value of 9
, excepting
ir and
â11which make cok,qsin&/2)zero 1ik kjqsin4= (2r+1)7L
and
k1qsin 8,',.
2rt (r..O,l,2,.,.) wider liiitations
_{¿9, >6,,.>o and}
b
> o
When k2 replaces k, taking
or
we have
f
( I. i C)ri,
ì
,(&,r ).
)
2niI 8sr 2fr.,)Z2(2) (,
2+l
1+2
(1)cos(k2nqsiri
ne].
))h10.
2tPutting
_{'qsin& =o<.}
2 j n1+2
_{(1)"cos(k, nqsin8)}2d6}
ri=l
t+1)(1)"cos(n)
t2n+1T
L'2r
tl+2.'co
k1+4Vco
_{.5mG.}
L2cos'8
For such r, that for whjh
&,,. or
_{coincides with either}
integration i.mit
or ò, the values under
_{in the above}
equation iiust be ha1ve.
n
(1+4
.'
n1
_{2C}
2n + ,1 2I
[(kqsin)1
r, (
'r
) +¡(k2qsin)18
f2(
2,j
b,,.(21)
Resúlts of nwr.erical coutatior1
_{for k0L=4 are shown above}
the abscissa J=2qk in Figs. 8
_{for J}
11, in
hich
represents
O,
and
_{for ti}
_{o,}
O Y
_
.
When the speed U isciiparativèly
low, wavemald.ng
resistance Is very great if the tank width q is about odd :ies
of hail' the length 'ji
a two dimensional.waé corresponding tc
the rolling pericd, ihile wavemaking resistance. decreases
rapidly as q cones off these widths decrease to less than
or there.
The widths correspond to wave lengths for free transverse
oscillations oí' the
tank water. For snail U, wave creats areaccumulated.
The lower is U th narrower
are the widths.
It
is considered fromthis
that reflectedwaves
sent by the ellipsoid'sentrance in the direction of its sides, superpose. themélves
on the waves chót in the sa
direction by the run, and. the
effective widths. of water increases to. some extent.
When the tank dLdth is any multiple of the wave length of
the characteristic osrillation, resistance takes the least values.
esistance vanishes if
thelength of the ellipsoid. is infinite,
except where the tank width equals odd times of half the wave
length. if the ellipsoid ! of
finite length, however, portionS
of the eleruitary waves it creates propagate obliquely toward
side wal1
reulting in resistance to some degre.
At the
satie time the energy giver. b
an ellipsOid
t!motion is absorbed
to create standing ives
in the water that comes in contact
with the eJpsoid as the latter advances through it.For a constant speed U, with an increase of th width,
t he curves of / repeate s mular Thrrr. The. hus ab out synchror.ous width are slackened and the hollows gradually
approach the
value for an infinite width. Wé can hardly expect,however, a tank width being close to infinite.
50 to conduct
rolling experiments at rather low speeds of advance is practically impossible as long as suitable wave breakers are not
ai1able along the tánk'! walls.
The above discussions are on the rolling that has continued
uniformly for a long period of time. In the càse.of experiments, .
where rolling ceases after betweenseveral and oie fifteen
swings, some nodificat ions
are required..
If the period, between the ti.te when the first waves are
shot by a ship and th time when the reflectéd waves come back
to her, equals or exceeds the tire required
for one rollig.tcoiet.e., the oonditioñ is ecuivalent to
that on an unbohded
water surface.. This condition is
.fq
group y ocity
)riuverf rolls (N)xrolling period(T).
owe ver.,
half wave length
1
5. Discussions of the Above kesults.
T
k0L.
If k0L 5. ¿
This is hardly possible
to prattce in experiment.
n the other hand if assumed q/L 1.5 is a mean, is
nearly unity, that is,
only the first two swings are of use.
Further, since the uirst. swing contains transitional
extraordinary phenomena (9), only
the record for half the
period
uccecdir3g the first swing is
reliable.
Th ?igs. 2  II, the range of qjk in existing experimental tanks is sho hatched. The range lies in the width of water where wavemaking resistance i very sxrall. This makes the
record of
wavemking resistance
ccnservatiVe.Again, when the speed of
advance exce&ls U
correspondingto V
1/4, the form of resistance curves
suddenly changeS.
Thotgh the first 'hump
correpond'thg to q/
/4 for U0 ramainS
for larger values of q a curve
gets considerably gradual
and
approach that representing the value for ininite width. Thisof course'
iS de tQ the fact that
the major part of
waveresistance consists of waves
other than progressive waves. Consequently, it is difficultto avoid sidewall effects unìes
a speed is made to exceed V 1/4or speedlength
ratio 0.8.
In Fig. 12 is plotted the comparison between wavemaking resistance affected by walls 1.5L.apart and that'in the water with an unbounded surface.
The relation in (22) holds'
approxintély true for the
finite depth h of 'water.
Wavelength J
of progrSSiVe waves
produced by the rolling in the same period is given by the equation
2CL'
2V
2.cotn7
Tg _{L}
L' is shorter than. j for infinite depth, while
the group velocity of.waves increases by the ratioSfly
Hence, instead' of (22),we have for k0L
4
tL).
However, the rihthand side
is nearly
/2'N for h up to
approximatelyL/l5.
It is clear from the above that special
conSiderations need to be rrade with
respect to side wall effects whñ
rolling experiments are carried out with actual ships.
6.
Rolling experiments with
selfpropelling model ships.
In order to learn resistance
to
the ro1lLg of ships innotiofl, it is necessary to examine
the
variations of component resistances, due to bilge keels, wavemaking, etc., togetherwith the speed of advance.
t
9A.W. Johns (2) stated that the resistances .dtce to surface
friction and bilge keels would be ir1creaec becave' these vere
attacked by a vater flow with a resultant velocity of both
rolling and advance.
However, the author cannot irudiately
agree with him.
So far as the author is thÍoriid tbre are no
other invastigaions treating the relatiân betwe n the. oiponente
of resistaice to rólling an
the sed of advance,.
Almost all works have dealt with total resiatane,
regardless whether a ship has bilge keels or uot. Ts makes it ail
the more difficult to grasp the general properties of ship's
resistance.
With this
vtew the author has carr'iéd out experilTnts
usirg ship models with and wit1out bilge keels and under the
same conditions throughout.
Table 1.
smalipassenger ship, G of a cargo ship, B of a tz'aiÉ1x and I,
t ae Sijyijiar' models of 1oenge.ahape.
The particnlrs of
these' models are shown in Tablè 1, amI the lines 'of hull toxins 0f 1
re indicated in Figs. 13,
»».Sh.iç
FH.I,
3cale cf niodel
1/25
1/50
1/25
1 2L (m)
1.60
1.86
1,52
100
2.00
!(m)
.300'
.274
.288
.30
6OQci (cm)
).0
10.0
11.6
rrn (cm)
o
o
4
0
' by stern
0
Dsplacernent (kg)2.9
37.3
2L$
25.0
200.<>.
3M (cri)
3.48
2.00,.:
1,'.O
_{2,45}
CG (cm).
..1.72 O.O
252
5..2
10.44
(tckX1eth)51
5x54
,
510x60.
Bar keel
»(thick.x1eight
"

.2.SX6.4.
. . .T (sec,)
1.20
140
1.10
122
,l.74
.42
,2
48
k,L
¿.5
3.8
5.0
2.7.
2.7
The experinents were performed ï twopon;, for t.h use
of an experinntaI tank is not prcper for the euttor's purpose as is clear from the above discusions. The an depths of water
are
2 n and 1.8 m respective1y while th shortéstdie. tarces frOErn
the ship's courses to thé 1orders re 30 rn and 15 n
repective1y The model ships are prope1ed byscrew propellers driven bj small electric motors As electric sources
batterte
,are used.
'4 Rolling is starte4 by the rieans ás shown in Fig. .
Weights are attached to the arms of frames using
electrômagnets.
After à model ship i .startéd at anestimated time
when its speed has become constant,
a
eléct'ic current in. a
magnet on one side is cut off b7 a chÑnometer switch and a
weight on the same, side falls down along a guide wire to a shock
absorber situated at the
módel's center.Plling
velocity of
the weight is absorbed by the friction Of
the wire ans ¿ spring.
The ship coiences to heel. When thebèeling angle is above
a ce"tain amount, a small weight. inan electric
relay tti1es, thereby making a switch off for another magnet. The heeling mônent 7anishes, but rolling remains. At the time, when rolling damps out, thechronometer switch' cut.s
of? all current. Then the &hip comes to a stop.To record rolling, a gyrorecorder is used. This is adjusted before thé model is started. The
speed of advance is
measured by the same method as in a speed tria1., except that itis done on ].and,in the period ftï'required to
run out theco e which is indicated by buOys placed 5 n apart.. The
st _{Ang point is at a distance about ¿ m tò the first buoy.} Some 'items in the process hd to be modified for the model
H becaise of damage to te
chronometer switch. Ro]Jing wasstarted by pushing
the frame when the ship was alose to thefirst buoy. The recorder was set in action at the
ame time
when the ship was started. And another boat was sent tO stop
the ship. _{The rolling curves, bòwever. are in good coincidere} with those ot the rolling automatically start.ed.
'5
Fig. and _{show a ciibration curve' Of the} _{} gyrorecorder and a curve of extinction of the rolling of a model ship obtained
'from the recorder in çomparion wth those
_{of an}
optical recording. _{Fig.'729 is an exale of records made by} _{the}gyrcreorder, which are affected by the
precession of the gyro.A middle line
between thboth aplitudé curves, theréfore,
_{needs}to be taken as a neutral liÀe.
¡8 2Z
The experimental results are shown in Figs,  . They
are the data measured on a perfectly smooth water .'ón midsummer
days.
23
In Figs.
6
9, the data are reaÑriged
_{on abscissae of}
the. speed of advance, taking the roIling ¿litude as a.
parameter.
.It is found from these 'figures that the differene in
resistance between ships with and without . bilge keels is
approximately constant, alt'hough some of such diiTerenòes. decreasesslightly in degree at higher speeds. Te rame end ay is found
an opinion that a bilge keel is affected by a 1ft because a streir atta* tts faces with an ancle of incidence and the
rcr'
ispor.ent of the lift cojtributes to antirolling. rtis supposed, however, that the effect of a. lift seen on such a long and slender plate as a bilge keel wld be localized
within a rather short part abtft the leading edge. This tendency is endorsed by the fact that bilge fins öf a comb form give
strong dátTlping to a ship in motiofl head.
Next, the ratios of resistance to the rolling of a bare ship Ln notion ahead and that of a ship in tandstill are shown in Figs. 12in coarison with the theoretical
curves f ro Fig. 2 above.
These experimantal ratios containing
fríctto:al resistances of sa1Ï anount, ágree rather well withthose represented by the theoretical curves of an ellipsoid
of rvo1ution. That L to say, they agree with the residuary resi,'ance (the
niaor part of which is waveiaking resistance)
ratío, arid probably these ratios roughly equal the residuaryresistance cf an eflipoid regardless of ship fs. In. this
connection the experintal values of the model ship R8(a)
after
3aker (7) are found to agree approximately with those
represented by the theoretical curve Shown in Ftg 94.
33
It ry be said that an Increase of resistance to rolling
with
speed of advance is mainly due to wavemaking resistance.
Summary.
Wave.reaistance to the
rclUngòf aneliipsoid of
revñution in motion ahead has been calculated. A motion ahead always causes an increase of resistance. The rate f the increase i coiaratjvely snail at rather low speed, while it shows arìpid increa8e at a speed about =. Ujg =. 1/4.. Añd the sn1ler k01, the larger is the rate of
resistance increase in. the sana
speedlength ratio.rapidincease in the vicinity of ))= 1/4 is observed also in the experintal results in foreign countries..
i= 1/4
corresponds to
the Speed of advance which equals the propaga tier.velecity of
ve group.
With this, speed as a boundary, the
ooxçonents of 'ave resistance cbaneu: for at lawer
eeda
resistancE is of progressive waves, ìiil
at higher abeeds
resistane is conssiciicais
du
t.
..
,water
as it contacted by a ship advang. The larger k0L, the
løweris a boundary line in the speidlength ratio.
The ratio between the
cononenta o1 wave.making
resistanc (which may contain fricttona.l resistance, if it i? sr1l in quar.tity) of a model ship is app xitely euqal to tht of wavemaking resistance of an el poi.d which has the Same length, speed and rolling period a the former.The wave re.sistamce to rolling on a re5tricted water surface with a constant Width q varies frequently with q or
the
rolling period T.
it greatly increaeee if q is near odd times
of ha1
the length of a progressive. waVe with the periód T
being in two dr.sios. Fut if q .les between such lengths,
resistance js smaller than that on n unrestricted surace.
o
/
In the seedength ratio apprcite1y above O., however,
the said variation is considrab1y mitinated arid resistance
ztproaches th ' zn an unrest icted surface because the ship
i
lit.t)e affected by r'
ed waves corfling
froisidewalls.
At a low
speed of advance, the record inonly the
second
ewing
of roll is attributable, if the tank has anordinary
width. And thesucceeding swing record will
givesinal].er
res5tane than in a water with
an infinite width.
It is not awisable to conduct an
experinrit
onthe
r2Jing of ships unless it is done at roderately
high
speeda of.dvance.
It is necessary to choose a sea surface
aswide
aspossible for rolling exeriints on an actual ship,
too.The resistance due to bilge keels is affected little.
by the speed of advance.
It may be said, therefore, that an
increase in resistance th
rolling
due to an increase in speed risistsrin1y
of wavea1cing resistance.(p7)
_{Forn}
the ahoye, resistance to the rolling of ain
notiûn
ahead is approximately qual to the sun of the twoparts;
the resist.ance due .o bilge keels and a bar kee. ofthe
shi: in
a standstiU and the resistance of the bare hull ina standstil) irultiplied by the wave reistanoe ratio
of
an
e1lipsc6 which hà.s the arrie length, speed and. rolling period
asthe.'ip.
AP?ENIX
T.
The velocity potential due to doublets in an unbounded
fluid, whi±
t.and in a row between x= ae and x= as
onthe
xaxis and bave axes parallel to the sarte xaxis, is given by the
ì?wirg
',,if the strngth per unit length is proprtionai
to
a2ezh
n the vicinity of x=h.
=
(xh)(a'th )dh.
1(x_h)hs;7+zJ4a '
The inter..ticn turns easily to

/(ae.:x2+y2+z_J(ae+x)2
i_)
(ii)
Puiting x=ae,iÇ , y1sz* a1e
=
2aelogaP
_{2aei.}
aeaeì+ae(ç+.)
If
M=.
e
1_{14e}
log113
(H: a C3nSt.).
(i)
Eq. (iii) represents evidently the velocity potential (10) of
the fluid in which an ellipsoid of revolution d.th the focal
length 2ae advances with a constant velocity U in the direction
of the major axis.
Herve the tstribution cf the doubletts
strength fcr the ellipsoid is to be a2e2.h'.
If E. (i) is rewritten by the reiation

Ji
cos8d&fk
.(y)
'e bave the first term in the righthand
mber of (3). in the
text.
5mi1arly the second term in the &are equation is.
obtained whex doutlets with axes in the ydirection are
distributed.
ssurTIing the third and fourth terw.s in the sara equatioi
to satisfy the free surface coMition,
..
(3) is cox1eted.
II.
Extracting the ínt1egration I with. respect to k ar.d 9
from (13)
:
j4eJ
k3 2kf
28.X
s('
)s()dk
(vi)
Rf
= (gk o +2iJkcos612k2 soso)z +frUkcos
6)zuZ _{.}_{cvii)}
If th
roots of a hiqadrtic 'quaion with respect to k, the
righthand side of (vii) = O, are k,,k,, k3 and k,,
X
crUkcos8
_{1} _{1} _{1}r+s2
2piU2cose
J Tk3 k
$ìn.e
i
cc(khcosO)coa(ktIlcos8)=.rLe
dfk
g+2tJrcos8 +i$.Uco58+J(g+i Tos9)2 34Uagcosß
I.
2U2cs
k3 =expression with.
instead of 1zi. in the above,
k2, k
_{expressio with f insteaã of J}
in k,, 1(3 respectively.
)I
d
 r
(r
_{(0= ±(h*os8.}
a compLex constarrt,
(Ç=k.iu,
(fl; poeitive integr,
(5=1,2,3,4,).
14
_k(hm)o58
th
integra tian with reapect to k is eontatned. in the citur
integration
.Co!ing respectiv
contours a
shown in Fig. 3+ according to
the sin of the rea]. or the imaginary par
oX r5 and that Q.f
wehave
J,.
K e
dkJ, +i2i
exp(2r;+ir,)
,
( w > O)
J2,
,
(w<O)

(i>O)
(xi)
=
i2xr
exp (ar; +1i"x)
,
(w
o)
JI3,4,
(u40)
(w< O)
In the above equations
I.
)t ...2fU.k)&L.,.
(iu) e
_{j}
_{.(iu}
_{e}
idi.
(xii)
j
inr5
s'j
We trnsforr (ix) in order to examine the sgn of the real
and the iirary parts of k1
k. Name1y
j (g±iutJcos9Jz +h.Ugcoe
+6iLzUzgcos2J4
I
i
,
2gUcosB
x exp±
_{ga_}
_{l2tOS}
+4UrgçosT
J.
I
UcoaB
H
(g+4Ugcosa)2
_{.}
eXP±
_{e+4UosoJ}
Therefore, when g44Uacos B > O,
!, ,k2 ,k3 ,k4
1k1 ,k2
_{jJos.(l±g//g2 +4Urgc'osej/2U2co2O}
_{,}
1k3 ,k
=
_{;}5.f y+4tYucos
O,
_{(xiv)}
2k, ,k, ,k1 ,k
n., ,k3
=
/jz_t7o.gcoso
_{/2Uc8}
Ik
,k4
=
In the above eqtatiçns, take pIus for k1
_{an k3}
_{from the double}
signs and R, I specify the real and the iginarj
_{parts respec.}
tivel.
The crresponIence between k, w k and
_{is, therefore, as fo1iows}
15
in 4hiCh
9= 7L.cûs1 (g/Ur),
tCOS1 (g/2U)
iid only the range of
> ô
is cnsidered because the integration
iitI
respect to 6
iS
1. 1
Thefefire, since k1 
k3 arid k2 
k4 or 4:
4her.O,
arid J,
vanish in ail cases and the term t2xir,"
.exp(2fr, i'l'r
al ne reiains.
Further, this term appears only for the
root
with a plus real part and the
double sigri
of the term are
chosen in accordance with the
sign ofthe irrginary part o
the
root.
These combinations are shown in
F1.g.and Table 2.
35
Table 2.
O Vf
gfItJc.S8 > O,
g+zUrcoS&K O,
0=0 ,
z
I,
1e knoi fror the above
are indep
sent of the
sign for
, if
integration is redaced
k1= r1
r2= r1
= r3
k,
r1r,
r.. r4k 
_{3} _{2} r1 r1r3
o
o.
.
o oo
taule that (a.) the integrated values
sign of w., (b) the values change..the
they contain c, and (o) the rang
of
to 9 O6,
I. 4b .'t,
.,0
o
O
>0
o
'O Qo
or,
r
.2r,
r4Comeq'ueìt1y I beeces
, 2 _{51T1&} ¡A_{)}
_{+/4ThrgcoS8} xs(k,hco8)co5(k,rC3S&)6
_,(_1Jk2â)k exp(2fk1)
Cx
ccs(k2cs&)coS(kiO8)1J
k,
g+2'Jico5
±/g2+4Tir3S
2L13co5'& K2next., the
with respect to h or m turns to
ae
J ae
a3e3 J
(k, aecos
ing;.\and ?
we have (14) and (15).
III.
The velocity potent4al when an
infinite
er of
1lip!jdS 3tand abrvast, with
distance
betweon bath othei.
N
((a2eh2)dh
h)
zjr
(xix)
velcity on the surface of
he ellipsoid (m=O)
to .
?.,67at xeO, z=f and y=O.
(
ae
a2e_h%)dh(_1)"Çh_2t
(rrt' = ,2,..)'.
)
ac
If q is at least as large as 2ae, it is reduced ap:oxite1y to
LÍ
ae3h1)dh=
(_l)m<2 L3
(tq)3)
3
q3'
m' 3'
'raking the ratio of this to tte
aiopitudè of V in
2)
2
i
The amount of the (
) is very eI1 sine e
' 1 for such an
e1lipsid which is as slender as a snip.
The disturbance,
t.herefoÑ,S very sa11 unless Lfq is very
large.
T!EL
disturbance
is nearly equal
e
 17
(xvit:
(xz
s ç
REFENCES
WHITE,. W.:
_{Notes on further ex eriHr1t8}
_{with firstclass}
battleshjp, T.I.N.A., 1896.
JOHNS, A..:
_{The effects of rotiot ahead on the rolling}
of ships, T.I.N.A,, 1905.
BERTIN, L.:
_{The azrlitude of rolling}
_{on a nonsynchro.}
wave, T.I.N.A., 1896.
(4)
PAYflE, 14.?. :
iesi1ts cf sáe roilitg
_{experiments on ship}
rwdels, T.I.NA., 1924.
LE3ENflR., R.:
_{The influence of speed on ro11ng.}
Shtpbi1der, April, 1933.
6)
SANTIS, R. AND RUSSO,
_{Ro.1ingof the S.S. "Conte di}
Saroia
_{in tank experimer.t and}
_{at sea, T.LJ;.A.LE., 1936.}
13AKLR, (LS.:
_{Roliing of ships under}
_{way.}
_{Shipb. and}
shipping record, 1ov. 9, 1939, Nov. 23, 1939.
KtTO, H.:
_{On the stability' ofsmall}
_{vessels,(jn Japanese).}
J..K., Vol. '79, 1948.
ISHIDA, T.;
_{TranSitional phenomena in}
_{wave resstanc}
_{to}
the roll ing of. s hips.
_{Jour}
_{of Kansai Soc. N.A., Vol. 67,}
150.
(lo)
LAMB, iL:_{"Hydrodynarni:s", 6th edition, p. 142.}
18
(7)
5/
Fig. ¡
Fg.Z
I I 1,2/4
1.6 .46
(.6.6
.8
1.0Sp..d MngM ra'o
F
9:11
1620
9 (deg.)
..."
Ioo....

X/0.
20
U(kno)
OI I 6 2
i,
oo.
F19.8
Fig.1/
.2E,
k0L=1
_{}
t'L0L
&j.
t)ocrL
L=L'X
7_= 4 32 / oSyuh1Q +, COt with reipect te mseçtoyi
I
qI
_{6 s.}F.I2
0/
DZ 03F9.I3
I1. ModeL SCm20
FQ. 14
Fig.
7,ostren9th..
e8 volti.
o.I0
_{vo1hged} motorx
Bvolts u /OVolt 100 /40 200/2
16 _{20}60(deq.)
With iI5e )eeIs
e
Without bilge
keeLs
22
Fi9.
23
v .
bilge kee1/
Wi$1ouf'F9. 24
6
"o
o
Fia.
25
D I I 2t
Fi9.
26
'o
 wiTh
b*r keeL 
Wliho"t qo'
F
Fig. 27
.1
_{/}
32 /
/
V
(iup.soid
.4 ,..volw#.'
S?eed U,,g rio
e
(IiJ_= 3.9)
Fag. 2'?
.6
Speed ¡Qn9M ra*tO
Speed ¡esiØ& rgto.
o. 4s°
V