TUtIE. Ci' T1FVE RESISTkN
TO THE }L
(Part II - EffFcts c' ?-oticr
aheae on Wave Re
.stacce to Rdllingt
in the journal of th Eocity
f .iap. Vol.
This is th
:f the criinal paper, reviied
anì en1ared by the author foi'
cirf.fl if th çoirS isussd
ft with a
sist.: parer "flolkr
blihi in the journal of thKan
Arcflitects cf Japar.. Vol.
'ave-Makir.g 1esi'tace cf ar. ¿llipsoid
Disctssions of the above results.
f Lirited width of Water..
xperthentS witl. self-Propelling Model
well knowr, fact that wave resistance
increases '.i.ih ìncreang her forwaro
bn examined both thec.retìally an
x:erimentally by ïny
among t.hn, W. White
(1), A.W. Johns (2), E. Brtin
.F. Fayne (4)
lì. Legendre (5), R. 5ar.tis and N. Russe (b,.
S. Baker (7), N. Kato (8) and so on.
explara-tion .of the rrchanis cf augmentaexplara-tion of resistance to rolling
still recainS to he done.
The author disuseS Ìre the resLlts öf
'sing sef-propelling model ships with and without bilge kc'is
as well as the theoretical
cf wave resitr.ce t.
rolling of an ellipsoid of re.îc1ution in
Ware-t'aKing t&istance of an ELlipsd
e tae the x-, y- and z-axes as in F'i. 1. Ari ellipsoid
of revolutin wt.h a major axis 2a and ai equatorial radtus h
water carI1eì to the fr
surface with a
con-tant:peed U in the drtr of x posìtve.
rolls about. a longitudinal axisabove
ajor axis, the
trans-vere velocity V may be expressed by Eq. (2), provided
tw rolling angle is defined by
& 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
real part cf (rfer to Appendix T)
x os 9dB dk
kG(k)8 )exp-k(f-z )+ik t(x-h)co&
'o -. .
f2.\ I v1
and G are functions to be dterniind by the condition of the
free surÇce given belo.
In Ihese eqationS
is the elevation o.f the free
p- Rayleigh's internal resistance.
It is evident that the first
and third teriis or the right-hand 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
=i_ae e -h1 )dh
keSinO,.(_XR+SY)cosrt-(SX+RY)Sir3f t X
°_(_xR+SY)sinrt-(SX+RY)cosrt; R= gk_+2Urkcos8-U1k1coS & S=,u.(r-UkcosB), X (r-Ukcos8)2 ,
Energy dissipated per unit time by wave-making3s given by the formula P-fl
(9g)dxdydt', (1.0) .
where T is a rolling perio&
ws now haveF1 G1 4F2
fdk( dhÇ dm(a2 e1_hl
3 i ).-ee
Sifl26h ) (iou a) R1i-S2 ijae f ae -211f
e2 -h2 )(a e2
Transforming by a contour integratthn (refer to
xexp(- 2f .l+2+4)cOs8)+il
,'. il+2'CosD-/1+vcOs6 )exp(-
L in (14) to (16) approxivteìv represente the length of
th ellipsoid if it is long andslender 1ike a ship.
rtio of thspeed of ahead motion to the propagaticn velocity of a wave ripple and
Xa Fróude 's number.
Although is the function of 'and)., it may be considered as that o',\ and k0L, for
And kL/2means the ratio of Lto the
length of tw
dimensionalwaves 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, verygradually at first
but rapidly at Froude'snumbers 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, wave-making resistance is due to
The energy per unit tire
- by these waves is of an order c, A1 L, provided the mean hèight
of- the waves is 2A and e, a proportional constant.
ellipsoid is in motion ahead, it moves through undisturbed ter constantly iv-ing
Some of the energy is left behindthe e11ipojd, and some reriains in a conser'r-tive system of staniiñg waves around the ellipsoid. Let us suppose the
-' 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.
rateof 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 iscomparatively
great onboth sides of the ellipsoid and it is small in the fore- and aft- body. But when the ellipsoid r'ioves ahead, the ditribution
of wave heights relative to it
i.e.,dense at the bow .;hile thin at the stern, for
6y allparts of the ellipsoid which compose a standing wave, are crowded at the bow.
Therefore, the increint of lost energr changes gradually up to a certain Speed because the height of the standing
wavesleft behind the ellipsoid is coiarative1y low, although its length increases with U.
When speed reaches themagnitude corresponding to V = 1/4, the whole system of standing waves is left behind the ellisoid. A
then becomes constant.
Theincrernt of the resistance is linear'y proportional to U.
The physical meaning of y= 1/4 may be
explainedas follows: For small U, although the ellipsoid is in
a wave systeiigenerated
by itself; with incréasing U, the wave shifts
back relative to
At )=U/g=l/4, wave speed becomes
equal to U.
the stem of the ellipsoid is at A in
Fig. 4, the principal partof the waves generate.d by the stem
is expressed as
w-re 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
And we get
This value of 's"
is independent of the dimension and form
the moving obstacle. For speeds U.over
g/4a, the obstacleoutruns ripples, as shown
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 rollingo
"Conte di Savoia IIincludes slight resistance due
srmll bU-ge keels.
The same tendency nay.be found in the
experiments on A-model ("Royal Sovereign without bulges nor
bilge keels) afterPayne (4), although we can
not make conparative
study because the particulars of the ship are not
Effect of Limited Width of Wter.
A motion ahead always increases wave
eSitèLnce to the
rolling as discussed above.
ut there. are soi
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
-The disturbance on the surface of thè ellj.psoid coming from the images, will be small if q is modertely
with the transverse sectional length of the. 'ellipsoid andwe
may explain general. tendencies (réfer to Appeñdix III).
Velocity potential for each image is
y replaced by y+i
(rn=±l, *2,. ...) on theight-hand side of Eq. (3) and
multi-piled by (-l)'..
Rewriting similarly in .(7).and (8) and suflning
with respect to m (including rn=O), we have
1,2,3 . ..
À similar forti way be had for-..
Eqs. (12) and l3) change therefore to :thosewith
integrandsmultiplied by [1+2
and made tcnd to n-o°
As this multiplier inc1ues.only circularfunctions, if we
consider it together with cos(khcbs&)(csknIós8) in (13), it is
transformed Into a formconsisting only of circular
like the righthandmember of Eq (x) in Appendix II Accordingly with respect to the integration of.E.,1
also, the first and secondgroups in (xvi) in Appendix II must be multiplied by the same multiplier
within which k is replaced by k1or k2 . Consequently we have . corresponding tó of Eq. (25):
where f, and f are the first and second terms01
the irite.grar.d in
Eq. (15) respectively.
Ising a well-known surcation forrala of a trigonometrical
This is very s11 for every value of 9
â11-which make cok,qsin&/2)zero 1ik kjqsin4= (2r+1)-7L
2rt (r..O,l,2,.,.) wider liiitations
¿9, >6,,.>o -a-nd
When k2 replaces k, taking
f( I. i -C
)2n-i-I 8sr 2fr.,)
'qsin& =o<.2 j n
For such r, that for whjh
coincides with either
or ò, the values under
in the above
equation i-iust be ha1ve.
2C2n + ,1 2
-Resúlts of nwr.erical coutatior1
for k0L=4 are shown above
the abscissa J=2qk in Figs. 8
When the speed U is-ciiparativèly
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 wave-making resistance. decreases
rapidly as q cones off these widths decrease to less than
-The widths correspond to wave lengths for free transverse
oscillations oí' thetank water. For snail U, wave creats are
The lower is U th narrower
are the widths.
Itis con-sidered from
wavessent by the ellipsoid's
entrance 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 ifthe
length 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
reulting in resistance to some degre.
satie time the energy giver. b
motion is absorbed
to create standing ives
in the water that comes in contactwit-h 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 thevalue 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 practi-cally 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 ionsare 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 requiredfor one rollig.--t
coiet.e., the oonditioñ is ecuivalent to
that on an unbohded
water surface.. This condition is
group y ocity
)riuverf rolls (N)xrolling period(T).
half wave length
5. Discussions of the Above kesults.
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
extra-ordinary phenomena (9), only
the record for half the
uccecdir3g the first swing is
Th ?igs. 2 - II, the range of qjk in existing experimental tanks is sho hatched. The range lies in the width of water where wave-making resistance i very sxrall. This makes the
-Again, when the speed of
advance exce&ls Ucorresponding
1/4, the form of resistance curves
Thotgh the first 'hump
correpond'thg to q/
/4 for U0 ramainS
for larger values of q a curve
gets considerably gradual
andapproach that representing the value for ininite width. This
iS de tQ the fact that
the major part ofwave
resistance consists of wavesother than progressive waves. Consequently, it is difficult
to avoid side-wall effects unìesa speed is made to exceed V 1/4
In Fig. 12 is plotted the comparison between wave-making 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.
of progrSSiVe waves
produced by the rolling in the same period is given by the equation
L' is shorter than. j for infinite depth, whilethe group velocity of.waves increases by the ratio
Hence, instead' of (22),we have for k0L
However, the riht-hand side
/2'N for h up to
It is clear from the above that special
conSidera-tions need to be rrade with
respect to side wall effects whñ
rolling experiments are carried out with actual ships.
Rolling experiments with
self-propelling model ships.
In order to learn resistance
tothe ro1lLg of ships in
notiofl, it is necessary to examine
thevariations of component resistances, due to bilge keels, wave-making, etc., together
with the speed of advance.
-9-A.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,
regard-less whether a ship has bilge keels or uot. Ts makes it ail
the more difficult to grasp the general properties of ship's
vtew the author has carr'iéd out experilTnts
usirg ship models with and wit1out bilge keels and under the
same conditions throughout.
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,»».
3cale cf niodel
-' by stern
.2.SX6.4.. -. .
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
are2 n and 1.8 m respective1y while th shortést
die-. 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
'4 Rolling is starte4 by the rieans ás shown in Fig. .
Weights are attached to the arms of frames using
After à model ship i .startéd at an-estimated time
when its speed has become constant,
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 themódel's center.
the weight is absorbed by the friction Of
the wire ans ¿ spring.The ship coiences to heel. When the
bèeling angle is above
a ce"tain amount, a small weight. in-an electricrelay tti1es, thereby making a switch off for another magnet. The heeling mônent 7anishes, but rolling remains. At the time, when rolling damps out, the
chronometer switch' cut.sof? all current. Then the &hip comes to a stop.
To record rolling, a gyro-recorder is used. This is adjusted before thé model is started. The
speed of advance ismeasured by the same method as in a speed tria1., except that it
is done on ].and,in the period ftï'required torun out the
co 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 techronometer switch. Ro]Jing was
started by pushingthe frame when the ship was alose to the
first buoy. The recorder was set in action at the
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.
Fig. and show a ciibration curve' Of the gyro-recorder and a curve of extinction of the rolling of a model ship obtained
'from the recorder in çomparion wth those
of anoptical recording. Fig.'729 is an exale of records made by the
gyrc-reorder, which are affected by theprecession of the gyro.
A middle linebetween th
both aplitudé curves, theréfore,needs
to be taken as a neutral liÀe-.
The experimental results are shown in Figs, - . They
are the data measured on a perfectly smooth water .'ón midsummer
9, the data are reaÑriged
on abscissae of
the. speed of advance, -taking the roIling ¿litude as a.
-It is found from these 'figures that the differene in
resistance between ships with and without . bilge keels isapproxi-mately constant, alt'hough some of such diiTerenòes. decreases
slightly 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 anti-rolling. rt
is 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 tand-still are shown in Figs. 12in coarison with the theoretical
curves f ro Fig. 2 above.
These experimantal ratios containingfríctto:al resistances of sa1Ï anount, ágree rather well with
those 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 wave-iaking resistance)ratío, arid probably these ratios roughly equal the residuary
resistance cf an eflipoid regardless of ship fs. In. this
connection the experintal values of the model ship R8(a)
3aker (7) are found to agree approximately with those
represented by the theoretical curve Shown in Ftg 94.
It ry be said that an Increase of resistance to rolling
speed of advance is mainly due to wave-making resistance.
Wave.reaistance to the
rclUngòf aneliipsoid ofrevñution 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 a
rìpid increa8e at a speed about =. Ujg =. 1/4.. Añd the sn1ler k01, the larger is the rate of
resistance increase in. the sanaspeed-length ratio.
rapidincease in the vicinity of ))= 1/4 is observed also in the experintal results in foreign countries..
corresponds tothe Speed of advance which equals the propaga tier.
With this, speed as a boundary, the
ooxçonents of 'ave resistance cbaneu: for at lawer
resistancE is of progressive waves, ìiil
at higher abeeds
resistane is conssiciicais
as it contacted by a ship advang. The larger k0L, the
løwer-is a boundary line in the speidlength ratio.
The ratio between the
cononenta o1 wave-.makingresist-anc (which may contain fricttona.l resistance, if it i? sr1l in quar.tity) of a model ship is app xitely- euqal to tht of wave-making 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
rolling period T.
it greatly increaeee if q is near odd times
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.
In the seed-ength ratio apprcite1y above O., however,
the said variation is considrab1y mitinated arid resistance
ztproaches th ' zn an unrest icted surface because the ship
lit.t)e affected by r'
ed waves corflingfroi
At a lowspeed of advance, the record in
ewingof roll is attributable, if the tank has an
ordinarywidth. And the
succeeding swing record willgive
res5tane than in a water with
an infinite width.
It is not awisable to conduct an
r2J-ing of ships unless it is done at roderately
It is necessary to choose a sea surfaceas
possible 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
rollingdue to an increase in speed risists
rin1yof wave-a1cing resistance.
Fornthe ahoye, resistance to the rolling of a
notiûnahead is approximately qual to the sun of the two
parts;the resist.ance due .o bilge keels and a bar kee. of
shi: ina standstiU and the resistance of the bare hull in
a standstil) irultiplied by the wave reistanoe ratio
e1lipsc6 which hà.s the arrie length, speed and. rolling period
The velocity potential due to doublets in an unbounded
t.and in a row between x= -ae and x= ason
x-axis and bave axes parallel to the sarte x-axis, is given by the
if the strngth per unit length is proprtionai
n the vicinity of x=h.
The inter..ticn turns easily to
Puiting x=ae,iÇ , y1sz* a1e
-(H: a C3nSt.).
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
'e bave the first term in the righthand
mber of (3). in the
5mi1arly the second term in the &are equation is.
obtained whex doutlets with axes in the y-direction are
ssurTIing the third and fourth terw.s in the sara equatioi
to satisfy the free surface coMition,
(3) is cox1eted.
Extracting the ínt1egration I with. respect to k ar.d 9
= (gk- o +2iJkcos6-12k2 soso)z +fr-Ukcos6)zuZ .
roots of a hiqadrtic 'quaion with respect to k, the
righthand side of (vii) = O, are k,,k,, k3 and k,,
cr-Ukcos81 1 1
g+2tJrcos8 +i$.Uco58+J(g+i Tos9)2 34Uagcosß
k3 =expression with.-
instead of 1zi. in the above,
expressio with -f- insteaã of J
in k,, 1(3 respectively.
a compLex constarrt,
(fl; poeitive integr,
integra tian with reapect to k is eontatned. in the citur
shown in Fig. 3+ according to
the sin of the rea]. or the imaginary par
oX r5 and that Q.f
( w > O)
exp (-ar; +1i"x)
In the above equations
We trnsforr (ix) in order to examine the sgn of the real
and the iirary parts of k1
j (g±iutJcos9Jz +h.Ugcoe
Therefore, when g44Ua-cos B > O,
!, ,k2 ,k3 ,k4
2k, ,k, ,k1 ,k
In the above eqtatiçns, take pIus for k1
from the double
signs and R, I specify the real and the iginarj
The crresponIence between k, -w k and
is, therefore, as fo1iows
9= 7L.-cûs1 (g/Ur),
iid only the range of
is cnsidered because the integration
respect to 6
Thefefire, since k1 -
k3 arid k2 -
k4 or 4:4her.
vanish in ail cases and the term t2xir,"
al -ne re-iains.
Further, this term appears only for the
with a plus real part and the
of the term are
chosen in accordance with the
sign ofthe irrginary part o
These combinations are shown inF1.g.
and Table 2.
Table 2.O V
gfItJc.S8 > O,
1e knoi fror the above
sent of the
integration is redacedk1
k -3 2 r1 r1
taule that (a.) the integrated values
sign of w., (b) the values change..the
they contain c, and (o) the rang
to 9- O-6,I. 4
Comeq'ueìt1y I beeces, -2 51T1& ¡A
with respect to h or m turns to
we have (14) and (15).
The velocity potent4al when an
1lip!jdS 3tand abrvast, with
betweon bath othei.
velcity on the surface of
he ellipsoid (m=O)
?.,67at xeO, z=-f and y=O.
(rrt' = ,2,..)'.
If q is at least as large as 2ae, it is reduced ap:oxite1y to
'raking the ratio of this to tte
aiopitudè of V in
The amount of the (
) is very eI1 sine e
' 1 for such an
e1lipsid which is as slender as a snip.
t.herefoÑ,S very sa11 unless Lfq is very
is nearly equal
Notes on further ex eriHr1t8
battle-shjp, T.I.N.A., 1896.
The effects of rotiot ahead on the rolling
of ships, T.I.N.A,, 1905.
The azrlitude of rolling
on a nonsynchro.
wave, T.I.N.A., 1896.
PAYflE, 14.?. :
iesi1ts cf sáe roilitg
experiments on ship
rwdels, T.I.NA., 1924.
The influence of speed on ro11ng.
Shtpbi1der, April, 1933.
SANTIS, R. AND RUSSO,
Ro.1ingof the S.S. "Conte di
in tank experimer.t and
at sea, T.LJ;.A.LE., 1936.
Roliing of ships under
shipping record, 1ov. 9, 1939, Nov. 23, 1939.
On the stability' ofsmall
J..K., Vol. '79, 1948.
TranSitional phenomena in
the roll ing of. s hips.
of Kansai Soc. N.A., Vol. 67,
"Hydrodynarni:s", 6th edition, p. 142.
Fg.ZI I 1,2
Sp..d MngM ra'o
OI I 6 2
L=L'---X7_= 4 3-2 / o
Syuh1Q +, COt with reipect te m-seçtoyi
F9.I3I1. ModeL SCm
x--Bvolts u /OVolt 100 /4-0 200
With iI5e )eeIs
v -.bilge kee1
25D I I 2
---- wiThb*r keeL
- -Wliho"t q
S?eed U,,g rio
Speed ¡Qn9M ra*tO
Speed ¡esiØ& rgto.