Wave breaking resistance of ships

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Wave Breaking Resistance of Ships

1. Introduction

Free surface flow around a surface piercing body changes its form in various ways depending on the bluntness of the body and its speed. Blunt bodies usually are accompanied

with unsteady vortical free surface disturbance in both

front and rear regions of the bodies.

In the sixteenth century, Leonardo da Vinci made detailed drawings of the surface of a stream when an obstacle is placed in it. Fig. 1 shows one of his sketches, which shows well a characteristic of vortical free-surface flows around blunt bodies. A translation of his note on the drawings is given: 'Observe the movement of the surface of the water, which resembles that of hair, which has two


Fig. 1 Leonardo da Vinci's sketches of water flows around


Eiichi Baba*

The present paper deals with experimental and theoretical studies on breaking waves observed on the free surface around

conven-tional ships. Experimental studies on the structure of breaking of bow waves are reviewed first, and then some results of quantitative determination of a resistance component due to breaking waves are reviewed together with studies of the effect of the protruding bulb on the wave breaking resistance. Based on the above experimental studies, a flow model is introduced, in which a thin free surface layer is superposed on a nonuniform flow induced by the double model velocity potential. It is then suggested that shortening and steepening of the surface layer waves riding on the steep local waves are considered to be a cause of wave breaking. Finally, regarding the breaking of stern waves, a theoretical study is carried out. Then it is shown that stern waves at high Froude numbers behave like vortex motion near the free surface. This vortical fluid motion is considered as a cause of appearance of breaking stern waves.

motions, of which one depends on the weight of the hair and the other on the direction of the curls. Thus the water forms eddying whirlpools, of which one part depends on the force of the predominant current and the other on the incidental motion and return flow'. From his note we see that he seems to be thinking of ways of separating flow into steady and turbulent components.

The analytical description of such free surface flows presents a challenging task. However, it is a very difficult one, since such flows are essentially turbulent, and poten-tial theory is no longer applicable to the whole flow.

In 1969 three papers, which dealt with such unsteady, vortical free surface flows around ships, were published independently. Each paper is concerned equally with such flow phenomena around the bow of ships but the approach of each is different. The first one is Taneda and Amamoto's paper in which results of flow observation around a fishery training ship 'Nansei Maru' of 19.54 meters long are pre-sented together with the results of laboratory observations of the free-surface flow around her model'. They named the vortical flow around the bow 'necklace vortex'. The second paper is one by Dagan and Tulin, who carried out a theoretical study of the nonlinear free surface phenom-enon for the two-dimensional steady flow past a blunt body of semi-infinite length. Two asymptotic expansions of the governing equations of steady flow were developed. The small Froude number solution, representing the flow beneath an unbroken free surface before the body, was carried out to second order. The breaking condition on the free surface in the vicinity of the blunt bow was studied assuming that the breaking is related to a local Taylor instability. The high Frounde number solution was based on the jet flow model. Using this flow model, the bow drag was computed (2) The third paper is one by Baba, who regarded the unsteady vortical flows around the bow of a tanker model as the breaking of bow waves and measured * Dr. Eng., Resistance and Propulsion Research Laboratory, Nagasaki Technical Institute, Technical Headquarters.

The paper was presented at the International Seminar on Wave Resistance, Tokyo, February 1976.


MTB11O September 1976

the momentum loss due to wave breaking by means of a wake survey behind ship models. Energy loss due to the breaking waves was then calculated by the use of

hydraulic-jump flow model. Baba showed that a resistance com-ponent due to the. breaking of bow waves occupies. a considerable part of the total resistance of full forms.

Those. experimental and theoretical. studies stimulated further studies on the characteristics of free surface flows around blunt bodies piercing the free surface.

The objective of the present paper is to survey the studies related to the wave breaking phenomenon around ships. In the following section experimental studies on the structure of breaking waves around blunt bodies are de-scribed first. Then some results of quantitative determina-tion of wave breaking resistance are shown in Sec. 3, where effects of bow form on wave breaking resistance are also discussed. In Sec. 4 theoretical studies which take into consideration nonlinearity of free surface flows around blunt bodies are reviewed. Finally, a theoretiöal considera. tibn on the breaking waves observed near the stern of a ship is given in Sec. 5.


A sectional area of submerged part, of transom


B = ship breadth

C,, = block coefficient

Cw = R/4PU2L2 in three-dimensions, R/j-pU2 (ira) in two-dimensions


= disturbance induced by p,. on the free sur-face defined by (12)



Froude number



L = ship length

Oxyz = Cartesian axes fixed on a ship Oxy is a plane of free surface at rest, and Oz directs

upwards = wave resistance a, b, c d



g K0(x,y, 0) S a0 e

= speed Of uhiform stream directing positive x-axis

= radii of ellipsoid, 2aL, 2b=B, c=d

ship draft

= complex potential function for double model, where Zx +iz

= complex potential function for surface layer = acceleration of gravity

= wave number of surface layer waves defined by (16)

= B/2 in Secs. 5.1 and 52

= velocity potential of a whole flow = constant number defined by (28)

= c/a 2d/L = b/a=B/L

= wave height due to surface layer potential = apparent wave height due to double model.

potential Fig. 2 Vortices around a tanker


V = g/U2

p = density of water

0 = surface layer velocity potential In Secs. 4.6, 5.1 and 5.2, O=1 - Lix Or double model velocity potential

2. Characteristics of free surface flow around blunt bodies 2.1 Flow visualization

In 1968 Tàneda and others made measufements of viscous boundary layers on the hull surface by use of two fishery investigation vessels Wakasugi (11 7 meters long)' and 'Nansei Maru (19.5 meters long)',., At that

time the wake of the ships was also observed from the deck. Then it was found that the wake of a ship consists of five vortices which are visible on the water surface: one vortex originating from the center of the stern, due to a screw propeller of the ship; two separated vOrtice peeled off from the both sides near the stern; and the remaining two white streaks of air bubbles which, starting from the bow and after flowing downstream along the hull, finally appear

outside all the vortices mentioned before, the last two

vortices were supposed resulting from the bow waves. Through this experiment it was found that two vortices starting from the bow are completely different types of

vortices from those due to the separation of viscous bound-ary layer. SO the observation by use of a model of Nansei Maru was carried out by means of flow visualization tech-niques suôh as air bubble method and dye filament meth-od )(6),

It was then confirmed that the origin of two vortices starting from the bow is due to a phenomenon similar to the hydraulic jump. Itwas also observed that in the vicinity of the stern there is such a hydraulic jump, weaker than one

at the bow, resulting in the production of two vortices from the stern as observed from the deck of the boats. Fig. 2 is a typical photograph of such flows observed far above a tanker'. Since the vortices starting from the bow


look like a necklace, this vortical flow was named 'necklace vortex'.

The structure of the discontinuous flows observed in front of the bow is shown schematically by Taneda and Amamoto in Fig. 3 (a) and (b). It was observed that the dye filament injected into the discontinuous flow at the bow returns to the incoming flow containing air bubbles as

shown in Fig.3 (b) and comes to the boundary between the

incoming smooth flow and the disturbed flow and then

leaves aside forming necklace-like vortical flOws as shown in Fig. 3(a). It is noticed that there exists a main stream between the vortices and the hull side, i.e. the necklace vortex takes a path of about half a shipbreadth distance from the hull side

Taneda made further flow visualizations by the use of two dye filaments injected in the stream in front of the bow. One-was injected a little below the free surface and the other was injected on the free surface, It was then found that each dye filament takes a different path. The one injected on the free surface is involved in the vortical motion and the other injeôted below the free surface takes

a path just like a potential flow around a body in the infinite fluid. Fig. 4 shows an example of such flow visuali-zation for a vertical circular cylinder. It is noticed that there exist a strong shear flow nearthe free surface infrbnt of the blunt bodies piercing the free surface.

2.2 Flow measurement by use of pitot tube

In 1974 Takekumä, in conneôtiOn with his earlier

(a) -


Fig. 3 StructUre of discontinuous flows around the bow


work, conducted flow measurements in front of the bow of a full form by use of a 5-hole pitot tube(8). Takekuma used a simple hull form model of 6 meters long and 1 meter wide as shown in Fig. 5. Flow measurements around the bow were conducted for two draft- conditions. Fig 6 shows a flow pattern in the vicinity of the. bow. Fig. 7 shows one of the results of flow measurements by use of a 5-hole pitot tube of 7 mm diameter at Froude number F = U/../jT= 0.14. The location of the measurement is a little ahead of the front boundary of the disturbed flow

(350 mm ahead of F.P.). In the figure velocity components calculated from the double model velocity potential ør(X, y, z), which satisfies the rigid-wall free surface condi tion, are also -shown. Takekuma found that the measured velocity components u/U, v/U, w/U agree Well with the

calculated values except in the thin layer near the free sur-f ace, where measured vetoëity components change

depth-MTB 110 September 1976 u/U,.v/U,w/U


Free surfoce L00 lOad iIfljVfUW/u deep 0 £




:----0 0.5 E -100 U 3

Fig. 4 Flow around a vertical circular cylinder Fig. 7 Measured velocity components

ap FP

Fig. 5 Lines of a simple ship model

Pg.4 U=i.284 Weeo Pi.5 U=1.284 m/ec in deep ( tr//t=O.1674) in ehallow dmft



MTB11O September1976

wise rapidly. The flow property measured by Takekuma agrees well with Taneda's flow observation: except in the thin free surface layer, the water flow takes a path resembl-ing the potential flow around the body ifl the infinite fluid.

Takekuma observed further that free surface disturbance in the shallow draft condition is much greater than that in deep draft condition. For actual ships such flow phenom-ena are also observed. Figs. 8 and 9 show free surface dis-turbances around the bow of a large tanker. In the deep draft condition short waves with unstable crests are ob-served in front of the bow (Fig. 8), while in shallow draft condition those short waves are transformed into breaking waves (Fig. 9).

In summarizing the above experimental observations we may conclude that the unsteady vortiôal free Surface flows are confined Within a very thin free surface layer, and that. except, this thin layer, the flow can be expressed by double model potential. Further, free surface disturbance becomes larger with the decrease of the depth of immersion of the bodies.

Fig. 8 Short Waves around the bow in full load condition

Fig. 9 Breaking waves in ballast load condition

3. Measurement Of wave breaking resistance 3.1 Bow wave breaking

In connection With the finding of the wave breaking resistance Sharma, in a paper by Eckert and Sharma', 'fOund that the protruding bulb of full forms in ballast load condition is very effective in smoothing the free sur-face flOw around the bow and then contributing to the

reduction of the wave breaking resistance.

There have been variOus explanations about the effect of protruding bulbs of full forms in low Froude number, say F < 0.20, where one regards the wave resistance is negligibly small. From wave, pattern analyses of a tanker model with and without a protruding bulb Sharma showed that the magnitude of reduction of wave pattern resistance by the protruding bulb can not explain such a large reduc-tion of total resistance, such 'as a 20% reducreduc-tion of total resistance. Further, from the towing test results of sum-merged double models of ballast load condition with and without a protruding bulb, Sharma showed that the prot-ruding bulb does not work in reducing viscous resistance, although there has been such an explanation that. the

protruding bulb reduces vortices generated near the bottom part of the bow. Finally, from the results of wake survey behind the models, Sharma showed that the side peaks of head loss distribution are markedly reduced by the

protrud-ing bulb. At the bow remarkable smoothprotrud-ing of the free

surface flow was attained. Thus Sharma, by his systematic experiments, gave conclusively a clue to the mystery about

the' effectiveness of a protruding bulb of full forms: the protruding bulb in ballast load condition is effective 'in smoothing the disturbed free surface flow around the bow and then contributes to avoiding energy loss due to bow wave breaking

Taniguchi et al. conducted similar experiments and showed that protruding bow is also effective in reducing wave breaking resistance of full fbr'rns'°. Fig 10 showS a comparison of the contribution of each resistance

corn-awe btaking r.si$taflce Viscous resistance 14 Viscous resistance BaLlast condition

With protruded bow

Knot 20

Fig. 10 Decomposition of effective horsepower of a tanker of Cb = 0.80 in ballast load condition

WavI paus.rn 0 % 4 16 I I 0.16 0.16 I BaUast

With normaL bow

18 Knot2°

0:20 0.22 016 0.16 0.20 ct22 Fn


ponent to the total horsepower of a tanker of C,, 0.80 with and without the piotruding bow. The effective horse-power is estimated from the experiments by Taniguchi et


For a tanker without a protruding bow, wave break-ing resistance occupies about 25% of the total effective horsepower, while fOr a tanker with a protruding bow the wave breaking resistance is reduced to 10% of total ef fective power.

Recently Namimatsu and Kushimoto measured static pressure over the. hull surface of the fore body of a tanker of Gb = 0.83 with various bow forms including a bow form. with a protruding bulb. Measurements of head loss dis-tribution due to bow wave breaking are also conducted at the control surface 0.35L behind the model(h1). Fig. 11

shows a comparison of pressure resistance coeffiôient and wave breaking resistance coefficient for a tanker model with and without a protruding bulb The trend of resistance coefficient curves with respect to Froude number for both pressure and wave breaking resistance agrees well with each other. This shows that the wave breaking resistance is. a part of pressure resistance of ships.

Furthermore this study together with those of Sharma and Taniguchi et al. shows clearly that the change of bow 9eometry of full forms reflects mainly to the change of Wave breaking resistance.

Finally, it shOuld be noticed that a full form having a




f P.

Rusiduat reeletame (Towing teSt)

pu's "


2 Wave breaking resist(Wak survey)

I I.. l0 o 2-0 ModeL A --s-- Model D

P resist of fore body (Pressure

- Pu2V'/3 measurement)

5V DiplcemSnt voLume_

-4. _v___

Wave pattern resist (Wave pattern anaLysis)







7 I

Fig. 11 Measured pressure resistance and Waie breaking resistance

fore shoulder of large curvature is usually accompanied with breaking of shoulder waves. The head loss due to the breaking shoulder waves may also be included in the side peaks of head loss distribution.

3.2 Stern wave breaking

As pointed out by Taneda the breaking of stern waves is usually observed for any type of ship, though it is weaker than the breaking of bow waves for the case of full forms with blunt,bows. In order to obtain quantitative informa-tion about the stern wave breaking, Baba selected a fine form of C,, = 0 573 having a transom stern which is accom panied With rio breaking wavCs except near the stern. Then Baba made wake measurements at the control surface 0.5L behind the model(12),

Fig. 12 shows a comparison of head loss distributions measured for various depths. At both sides of the head loss distributions additional small peaks are observed. With an increase of the depth from the free surface those small peaks disappear rapidly, while the central peaks àf the head loss distribution decrease gradually. During the wake mea-surements it was observed that the transverse location of the small side peaks approximately correspond to the

transverse location of the disturbed free Surface flows

caused by breaking of stern waves.

Although it is considered that a part of the head loss due to breaking of stern waves is included in the central peak of the head loss distribution, a resistance component due to the stern wave breaking :isdetermined approximately by separating the small peaks from the central peak in a manner as shown in Fig. 12. Fig. 13 shOws a result of thefl decomposition of effective horsepower. It is found that

the effective horsepower due to stern wave breaking is corn parable to the wave pattern resistance determined by the wave pattern analysis and occupies about 13% of the total effective horsepower at the design speed.

In this experiment it Was possible to approximately determine a resistance component due to the stern wave breaking However if a large part of head loss due to stern wave breaking is included in the central wake peak, a part

03 ..50bSI,ind A. (Ho - H) Fn - 0.27

hd IOUdo.I.

USS br.iI,Inç ASs,

tress,,, iI.is MTB11O September 1976 IOO SUd L0 -05 0 i/B 05 :'Ud b,.edtII depTh 60 80 00 25 ISO 200 0 10


Fig. 12 Head loss distribution of a container ship form

with transom stern


020 v/mt



MTB11O SOptember 1976


Viscous resistance

Fig. 13 Decomposition of effective horsepower of a con tamer ship

Fig. 14 Form factor of geosims of a tanker determined

from the wake survey results

of the resistance component determined from the momen

turn loss of central wake peak may be Froude number dependent In fact, in the earlier study by Baba it was indicated that the form factor determined from the wake survey results was Froude number dependent, where the resistance component determined from the central wake peak Was regarded as the pure viscous resistance. Then the form factor k was determined by dividing this pure

viscous resistance by the frictional resistance of equivalent flat plate. Fig. 14 shows the values of fOrm factor for

geosims of a tanker of C'b = 0.80 determined by a manner mentioned' above. A gradual increase of form factor with Froude numbers is observed.. This figure may indicate in. completeness of separation of resistance components into

parts, one due to gravity effects and the other due to viscous effects.

3.3 Second wave breaking

Townsin discovered that at speeds when a tanker form of C,, 0.837 was being excessively overdriven, say at F,,>

0.25, two sets of breaking waves were evident: one is

breaking waves at the bow and the other is the second wave breaking further away from the model center line than the

0.3 02-Comb 0.1- ImmersiOn 0 0 0 53 MOdt beam (b) SPeed .74 m,4 t: Total Pmead measured/--PI.)

F: Static head/_pU2

Second wave breakinQ



Fig; 16 Examples of floW separatiOn. A spilling breaker.

The flow induced by an obstacle at the surface of a steady stream; 0.7 06 0,5 04 0.3 0.2 0.I 0 A 0 A o 4.2" 7 "' 10" A0 rradeL model model A Cuo C," 0 A 0 0.1 0.2 0.3 0.4 0.5 0.6 07 0.8 9 1.1 1.2 1.3 1.4 1.5

of Model Distance Metres

Fig. 15 The second wave breaking

breaking bow waves(13).

Fig. 15 shows one of' the results of Townsin's wake

measurements. The central peak is due to viscous boundary layer on the hull surfàcé and the other two peaks are due to the breaking of bow waves and the second breaking waves respectively. Townsin suggested that at speeds high enough for the second breaking waves to be significant, theoretical models concerned only with bow wave breaking

need reassesment. 4. Theoretical study

4.1 Introductory remarks

As we have considered in the previous sections, breaking phenomena of waves always exist around the conventional ships. Since suôh free surface flows are essentially

tUrbu-lent, the potential theory is no longer applicable to the whole flow. Longuet-Higgins remarks that "our knowledge of breaking waves is surprisingly scanty"'. Indeed, little work has been done on the detailed dynamics of breaking waves because of the difficulty of' analysis. In. 1973 LonguetHiggins suggested a possible theoretical model. which describes a gravity wave after it has broken. He

noticed a local feature, of such flOws, namely the flow near

the forward edge of a spilling breaker of ocean waves

(white-caps) and the flow upstreañi of an obstaOle abutting a steady free surface flow, where the turbulent flow meets the more tranquil water as illustrated in Fig. 16. He 'then considered that the separated region may be treated as a turbulent wedge with a certain eddy viscosity Whose 012 0.14 0.16 018 0.20 0. 0.245

- Fn



020 022 024 Fn 030 0.26 028


magnitude is determined by conditions outside the local flow. A local solution for the flow in the neighborhood of a discontinuity in the slope of the free surface has been derived and the inclination of the free surface is studied'. For designers of ship forms, another concern on wave breaking phenomenon is how to avoid energy loss due to wave breaking in order to minimize power consumption. Although we have already obtained a few practical solu tions such as protruding bulbs or bow forms, the develop-ment of analytical means to study the problem has been anxiously awaited for further refinement of ship forms. The only available tool in our hands may be the potential theory. In order to use the potential theory, we consider free surface flows before the occurrence of breaking waves

where we may assume that a flow around a ship is an irrotational one. However, we expect that the understand-ing of such free surface flows may help us in findunderstand-ing a way to reduce wave breaking resistance.

FOr this purpose we have to deal with a low speed

prob-lem. Usually, with a decrease of ship Speed, nonlinear properties dominate in governing equations of the free

surface problem. In the following section recent studies on low speed free surface problems are surveyed briefly and then a consideration of the mechanism of the occurrence of breaking waves is given.

4.2 Asymptotic:solutions in low speed limit

From the results of flow observations around blunt

bodies described in Sec. 2, we may introudce the following flow model as one of the possible models in tOw speed limit. The water flow around a body is divided into two parts i.e. one is a thin free surface layer and the other is the flow field represented by the flows which are produced when the free surface is replaced by a rigid-wall.

In 1968 Ogilvie first introduced such a flow model to study a free surface flow past a two dimensional submerged body in the low speed limit in which the velocity potential for the free surface problem is considered as a sum of two

parts(15 :

F(x,z) = cb(x,z)+4(x,z)

where x-axis is along with the direction of the uniform stream U and z axis directs upwards with the origin on the undisturbed free surface, and z) is the double model potential, and (x, z) is an additional one to z) so that the sum satisfies the free surface conditions.

In the studies by Dagan and Tulin, the double model potential is also used as the first approximation to the free surface flow before a two-dimensional blunt body of semi-inifinite length. The asymptotic solution in the low speed limit contains no term representing the dissipating gravity waves. Therefore the wave resistance of the body is

zero(21. On the other hand, Ogilvie's additional poten-tial q(x z) represents short waves riding on a nonuniform flow which is induced by the double model velocity poten-tiàl ør(X, z). Ogilvie assumed that the basic nonuniform flow varies slowly with space variables while in the thin

MTB11O September 1976 free surface layer physical variables such as velocity and wave height are assumed to vary rapidly.

The low speed problems, taking into account the non-uniformity of the basic flow, have been studied recently by oagan(17) and Hermans(18). FOr instance, in a paper by Hermans a wave resistance problem fOr bodies piercing the free surface is treated in the similar manner to Ogilvie's. By supposing a thin free surface layer a theory to analyze the free surface flow around a semisubmerged horizontal cylinder placed perpendicularly to the incoming flow has been developed.

Hermans expressed the total solution as a sum of two parts:

4(x,z) =°(x,.z) +g(x,z)

where (x, z) is the outer solution of the problem, and cb(x, z) the surface layer correction. °(x z) is not

identi-cal to Ogilvie's r(X, z), but is expressed as a perturbatiOn series considering U as a small parameter of the expansion. The first term of the series coincides with Ogilvie's Ø,(x, z). Since Hermans considers that the surface layer solution contributes to the outer solution as well, several terms of the perturbation series of '° (x,- z) are used so that the error of the rest of the terms is asymptotically small with respect to the first term of the surface layer expansion as U tends to zero.

In, order to solve the problem both Ogilvie and Hermans formulated a linear problem for the surface layer potential (x, z) while keeping nonlinear terms of the double model potential. For instance, Ogilvie's boundary value problem is written as: [L] O=Ø. + øzz (1) [F]

Ø(x.z) +-1-Ø,. (x. O)Ø(x,z)=-P(x),


(2) where P(x) = ørx(X. 0) r(X), =

- [U2-4(x, 0)]

apparent wave height due to double model potential, and

ix) = -

0)'(x, z)

wave height due to surface layer poteniial.

(3) In addition to these conditions, the radiation conditiOn, which insures that waves only follow the ship, has to be considered. In the above conditions the orgin of the z-axis is shifted up to the .free surface represented by ,(x), i.e. z = 0 in the above equations corresponds to the surface of apparent local waves induced by the double model poten-tial, and the value Of7 on the surface is approximated by the value at the original still water surface.


MTB11O September 1976

Using complex variables, Ogilvie obtained a solution in a form of the gradient of potential:




1.dk(Z) e_iK_




- dk + 0(U5).


we then find that the lowest order terms satisfy .the free surface' condition (2) in a complex notation (see Ogilvie')


iO)+ .jf(x - iO)] = P'(x).

Since there is a relation [see Wehausen and Laitone(19), p.481]



the asymptotic solution (5) is rewritten as:

f'(Z) =_




This expression is also derived from the Qgilvie's exact solution (4) when replacing K(a) and K(ü) by K(z). It is then considered that K(a) can be replaced approximately by the value at the pOint,a =z, sinceK(a)is a slowly vary-ing function.

In the far d7wn stream where K(z) tends to v= g/U2, we have from (5)


(7) Integrating by parts the wave height ix) superposed, on

x) is obtained by use of the relation (3):



g ,,


while Ogilvie's exact expression is



dsp"(s)sin [v(xs) i-Q(s)], (8)



[K(u) -

v] du.


Next, for a ôomparison of the present theory with a complete linearized theory, the following formal lineariza-tion top'(x) is introduced


f-(Øx(x O),(x)]


3-Ø,.(x, 0)

together with K(Z)=g/U2 = v. Then both the exact

solu-tion (4) and the asymptotic solusolu-tion (5) are expressed by



k dk


which is identical to the solution obtained by Guevel et al.(20), whose free surface condition is the one which is derived from (2) by a formal linearization, i.e.

(x,z)+vz(x,z)=ørxx(x,z), on z=0,

Guevel et al. showed that the solution (10) can be ap-plied to both submerged and surface piercing bodies. For a surface piercing body contributions from the singularities on the cross points between the body and the still water plane are taken into account in addition to the contribu tiOns frOm the singularities distributed on the surface of the submerged part of the body.

In the analogy of the linearized theory, it is expected that Ogilvie's solution (4) is used as a solution for bodies






K(u)du h +




order (4)

where the following definitions and assumptions of orders of magnitude are made.


z) ERe{ f,(Z) L ci(x, z) ERe1f(Z),

[UI EU5]






[U3] [U3] 2

aiaz = 0((T ) when operating onf(Z), a/aZ = 0(1) when operating onf,(Z).

Since K(Z) is expressed by the term of double model potential f(Z), we see that the operation aiaz is of 0(1) when operating on K(Z). Noticing this fact, we can deduce an asymptotic solution of (2) in a similar manner to the method employed by Baba and Takekuma(22) for the

three-dimensional problem:

K(z) r

I (Z)--j1jdsP(s)PvJ




dsp '(s)e 1K (Z) (Zs) (5)

It is shown, that this expression also satisfies Eq. (2). We observe first that



yf"(Z) =

fdssf e1dk

[U3] [U3]


piercing the free surface, too. Especially for blunt bodies, it is considered that Ogilvie's solution may be useful,

since it includes the nonlinearity of flows arOund the body. In fact, in a similar manner to Ogilvie's treatment, Hermans developed a two-dimensional theory for surface piercing bodies as mentiOned previously.

However, Ti'mman pointed out that for two-dimensional bodies piercing the free surface, such as a semi-submerged cylinder with its axis perpendicular to the incoming flow singular behavior appears at the frolit, part of the cylinder, i.e. at the line of stagnation points where K(x)=°o21. For instance, in Ogilvie's formula (8) such difficulty

ap-pears, since Q(s) defined by (9) becomes infinite at the line of stagnation points e g for a semi submerged circular cylinder of radius a, Q(s) is expressed as




as 3




+ log

---However, sUch difficulty is eliminated when we consider from the beginning the asymptotic solution (5) as the solution of the original boundary value problem, i.e. its far field solution (7) has no singular behavior as observed in (8), although at the stagnation points the potential function (5) does not exist. We may then calculate wave resistance of surface piercing twódimensional bodies as shown in 'the next Section.

Timman showed more explicitly that such difficulty

arising for two dimensional bodies piercing the free surface is eliminated by replacing the body by athree-dimensional one. Timman extended Hermans' theory to 'the case Of a semi-submerged ellipsoid with its middle axis horizontally on the still water plane so that the free surface meets the bow over a relatively broad part of the front Then the problem is transformed into a two-dimensional one in the plane perpendicular to the middle axis. In the pure two-dimensional case the whole equator is a line of the stagna-tiori points, but Timman says that for the ellipsoid this is only the case in the forward point. Along the equator a potential functiOn then exists producing a wave travelling from the ellipsoid.

Baba and Takekuma. developed a three-dimensional theory 'as' a direct extension of Ogilvies two-dimensional theory(22). 'The flow model considered by Ba.ba and Takekuma is shown schematically in Fig17. The boundary valUe problem 'for the surface layer potential is written as:




+Ø(x,y,z)=D(x,y), onzO




Eørx(X,Y, G),(x,y)]

4'(.y.z)Solution of thin surface Layer

rigid-watt Solution

FIg. 17 'Scheme of the modeled phenomenon


(12) .(x,.y)




ix,y) = --EØ,(x,y, O)(x,y, 0)










A three-dimensional solution which corresponds tO the asymptotic solUtion (5) in two-dimensions was obtained as










Xfko(xY. O)el o(x.0)z5flk0(XY O)j}dO,



= (x'-)cosO + (y-n)sinO



Note that a/ax = a/ay O(1) when operating on k0(x,y, 0). As observed in the two-dimensional problem, the veloc-ity potential (15) does not exist at the stagnation points, fOre and after ends. However, as Timman considered, except at these pOints the velocity potential exists. Thus, there is a wave travelling from the body.

The wave height ix, y) due to the surface layer potèn-tial is determined from (14) by taking the lowest order

terms of derivatives of (x, y, z): (x,y) = MTB 110 September 976' Ti2

x ':12


dO '9


MTB11O Septethbor 1976

0)cosO+ Ø,(x,y, 0)sinO

4.3 Wave resistance in low speed limit

In order to determine whether the theory gives wave resistance of practical orders, Baba and Tákekuma derived a wave resistance formula for slowly moving full forms as explained in the following.

In the far downstream where /,. - U,ctry 0, the wave

height is expressed as Illr/2


= j

dO[S(0)sin lvsec2o(xcose +ysinO}}


+C(0)cos{vsec2O(xcosO +ysiriO)] where

C(0) +iS(0)= sec3OJ,fdd,7D(77) x exp[ivsec2O(cosO


Then the wave resistance is expressed by



When taking a linearized form for D(x,y):

D(x,y) =-Ø,(xy, 0),

the amplitude function (18) is written as S(0) +iC(0)

-ecOH(O, vsec2O),




x expi


x h



This expression coincides with Guevel et al.'s amplitude function for the linearized problem in three.dimensions(2).

Guevel et al. called

H(O, k)

modified Kotchin function

which is expressed. as a sum of two parts: one is the usual

kochin functiOn due to singularities distributed on the body surface. and the other due to the singularities distri-buted on the intersection between the body and the still water plane.

Baba and Takekuma calculated wave resistance of a vertical infinite circular cylinder in low speed limit and

compared i with those obtained by linearized theories such as Guevel et als and the so called zero Froude number solution which does not ôontain the contribution from the line signularities. The following are the values of wave resistance coefficients of a vertical circular cylinder Of

radius a in low speed limit.

i) Zero Froude number solutiOn: -.




when sources are distributed on the surface Of the


ii) Guevel et,als solution(2:

= !?Fn6+

32F,sin(- +.j-) +OF,),


which is also derived from a theory developed by Brard who first showed conclusively the role of line singularities on wave resistance X22)

iii) Baba and Takekuma's solution(22):


W 315






Baba and Takekuma showed that the trend of C with respect to Froude number and the order of magnitude

derived by their theory are in agreement with those of con-ventiOnal full forms. As shown in Fig. 18 C-cUrve based on the linearized theory developed by Guevel et al., on the other hand, shows a rather oscillatory property, which is due to the linearization of the problem. However, both theories give the same order of magnitude for C-values,



while the zero Froude number solution

gives practically unacceptable high values.

Thus Baba and Tàkekuma considered that the asymp-totic solution (15) is applicable to a study of free surface flow around full forms in low seep range before the occur-rence of breaking waves. In a recent paper by Baba, appli-cations of the theory to the design problem Of full forms

are described.

Finally, for reference, wave resistance of a semi-sub-merged horizontal circular cylinder is calculated as. follows in the low speed limit:

i) Zero Froude number solution(20):



when sources a're distributed on the surface of circular cylin'der.

Guevel et als solution (10)(20):


F6cos2 (---r).


Based on the asymptotic solution (5):

-4 4'IO 3 2 Baba iTzIekuma -\ (CircuLor CyLinder) 8192 c6 Cw- n

Fig. 18 Comparison of C-vaIues x



Rw cLinearized theory

6 + 3?g

(Circular Cylinder)


which includes nonlinear effects of the basic flow. 4.4 A consideration on bow wave breaking

As mentioned in Sec 2 the wave breaking phenomenon

is stronger in shallow draft condition than in the deep draft conditiOn. In order to clarify this theoretically Baba calculated wave resistance of semi-submerged ellipsoids of various breadth/draft ratios based on the Baba and

Takekuma's theory.

Fig. 19 shows a comparison of C-vaIues for ellipsoids of variOus Bid ratios having a circular water plane, i.e. B/L

= 1 .0. It is shown that with an increase of Bid ratio

C-values increase drastically In order to explain the increase of wave resistance for a body of shallow draft, D(x,

y)-value which represents the intensity of free surface

disturb-ance due to the double model potential, the right hand side of (11) was calculated. D(x, y) on x-axis for an el-lipsoid is expressed as:







U-n where 14(X,0,0) 1







(l+t)3/2(l+t/E2)'/2(i+t/52)'h 2c6 1

(2an) IIV/2_(1_c2)V2(1_82)



0, 0) sgn() - 1_




4ijrzz(x. 0,0)


T (2a0) J2_

Cw =

j:F:cOs2 (.J)

0.07 0.09 0.09 0.10 0.11 0.12

Fig. 19 C-values for semisubmerged ellipsoids of various Bid ratios 2e& sgn() 1 (26)


2_(1_22_(1.2) 2_(1.2)

(27) =xia,ao=abcfdX(a2+Xrh[(a2+A)(b2+X)(c2+A)]_1h (28)

Fig. 20 showsD(x,0)-values for variousBid ratios. With an increase of B,/dratiO, while keeping LiB=1.0, a peak value of D(x, 0) becomes large and comes close to the

body. It is interesting to notice that the location of the peak value of D(x, 0) for a vertical circular cylinder i at

IxI/a1.5 äAd well coincides with the location of the

detached necklace vortex observed by Taneda which is shown in Fig. 4.

The positive D(x. y)-values indicate that in front of the bow upward velocities are induced by the double model potential,, while near the stern downward velocities are induced. Fig. 21 shows a comparison of apparent local waves represented by r(X,Y) defined by (13) on the xaxis. With an increase of Bid ratio the local wave becomes steeper. When comparing Figs. 20 and 21 it is found that steeper local waves give higher D(x, 0)values. Thus Baba considered that the steepening of local waves at the bow contributes to an increase of wave resistance.

For a body of extremely shallow draft the assumption of the present theory i e the condition of slowly varying basic flows may be violated since waves induced by the double model potential become steeper. Then it is antici

Fig. 20 D (x0) -Values for semisubmerged ellipsoids of

various Bid ratios

MTB11OSeptember 1976

Fig. 21 ,(x,0) -values for semisubmerged elliposids of

varioUs Bid ratibs


MTB11O September 1976

pated that the free surface becomes less and less stable so

that even a small perturbation induced by the surface

layer potential e(x,y, z) results in breaking waves.

From (17) the wave number of a superposed wave is known as

ko(x,y,O)=g/Ø(y,y, 0)cosO+Ø,(x,y,

0)sinO}2 which indicates that the wave length becomes shorter and shorter toward the bow, since the flow becomes stagnant. FurthermOre steep local waves enlarge the amplitude of the superposed waves, since the amplitude depends on the intensity of D(x, y)-values. Accordingly there may occur the breaking of waves.

Regarding the shortening and steepening Of waves super-posed on long waves, theoretical studies have, been done by

Longuet-Higgins and Stewart. They showed that short waves riding on long waves tend to be both shorter and steeper at the crest of the long wave troughs((26). If the

long waves become steeper, the steepening and shortening of the superposed short waves are drastic. This is considered as one of the reasons for the appearance of breaking waves

(white-caps) on the crest of waves in deep water.

Longuet-Higgins and Stewart explain further that the steepening and shortening of the superposed short waves are results of energy transfer from the longer waves to the shorter ones. It is also indicated that such a mechanism does not depend upon the sinusOidal character of the long waves but only upon their being progressive. Therefore, this flow mechanism occurs for short waves riding on a solitary wave, or any other kind of progressive disturbance, provided it is sufficiently lông. Since we can regard the local wave represented by r(X, y) as a progressive long wave in our low speed problem, the theory by Longuet-Higgins and Stewart may be applied to a ship wave prob-lem, too. Fortunately, supporting evidences of this con-sideration are Observed in Figs. 8 and 9 in Sec 2, i.e. short waves observed around the bow of a full ship have unstable crests 'just like white-caps on ocean waves (Fig. 8). With a decrease of the ship draft, in other words, with the steepen-ing of local waves, those short waves ridsteepen-ing on the steep local waves are transformed into breaking waves as shown in Fig. 9. In. the process energy of local waves is dissipated into water and no longer propagates in a form of travelling gravity waves. This flow phenomenon may exist not only

near the bow but also in the region where steep local waves exist. For instance, near the shoulder part having a large curvature this phenomenon appears.

In a recent paper by Dagan the importance of such an interference between a nonuniform basiô flow and a super-posed wave is also stressed regarding the stability of free surface fIows(27.

4.5 Effect' of protruding bulbs

As observed in Sec. 3. the protruding bulb is effective in reducing wave breaking resistance. TO explain this effect theoretically Baba replaced the bulb by a submerged sphere for the sake of simplicity. Then Baba calculated D(x,



Fig. 22 D(x, 0)-values for a submerged sphere.

values of the submerged sphere for various i,,mersions(24).

Fig. 22 shows D(x, y)-values on x-axis for different im-mersions. It is shown that a negative peak value appears in the front part of the sphere and it becomes larger with a decrease of immersion.

This simple calcuatiOn suggests that a submerged sphere or a protruding bulb is effective in cancellingD(x,y)-values

induced by a main body in front of the bow, i.e. the

protruding bulb is effective in reducing steepness of local bow waves. Further it is suggested that, the shallower im-mersion gives a larger effect in cancellation. Then we may consider that in ballast load condition the protruding bulb contributes to the reduction of wave breaking resistance which is attributed to the steepening of bow waves. In the full load condition, oh the other hand, it is considered that the effect of the protruding bulb is relatively small corn-pared with the effect in the ballast load condition. This

theoretical predication coincides with our experiences about the protruding bulbs which are mounted near the

bottom of bow.

4.6 Flat ship problem

Bodies of extremely shallow drafthave a sharp positive peak values of D(x, 0) near the bow and negative values near the stern as shown in Fig. 20. For this case it is con-sidered that an influence of' the body on the fluid disturb-ance is very small except near the body so that, within a small distance, fluid velocity .has to change from the uni-form stream velocity to 'zero at the stagnation point. This rapid change of velocity causes a large peak value of D(x, 0) near the edge of the body. Tb such flat bodies Baba and Takekuma's theory can not be applied, since basic flows are no longer slowly varying ones.

The sharp positive peak value of D(x, 0) in the vicinity of the bow suggests an existence of a jet flow, or a spray,


which is usually considered in the problem of planing

surfaces. Then the jet flow model can be considered as one

of the flow models representing turbulent vortical flows aroUnd the bow of f tat ships. So the theories for planing surfaces may be applied effectively to the bow flow prob-lems of full forms in shallow draft condition. Now the

problem belongs to the high Froude number problems.

Theories which take into account the nonlinearity of flow have been developed by Wu8), ogilvie(29), and pagan & Tulin(2) for the two-dimensional case. Within the framework of a linearized theory, Bäba studieS a three-dimensional free surface flow near the bow of 'flat ships as an application of a theory developed by Ogilvie in 1972 for the free surface flows around the The

theory is derived by modifying assumptions of the slender-body theory in the vicinity of the bow to allow for occur-rence of a longitudinal rate of change greater than normally assumed, viz. Ogilvie introduced the bow near field, which is an asymptotically defined region in which a/ax=o(e2) and a/ay, a/az =o(e'), while in the usUal slender-body theory a/ax=O(1) and a/aya/az=O(e').

It turned out that Baba s asymptotic solution of the bow near field of flat ships coincides with Maruo's solution of low-aspect-ratio planing surfaces32, which can be derived

directly as an expansion in the bow near field from the

three-dimensional velocity potential fOr a pressure dis-tribution on the steady flow.

The perturbation velocity potential for the pressure dis-tributiOn p (x y) on the free surface is written as'9 :

cb(x,y,z)= .Jw2+52z -e


2_ +132.

-x cos[V



dj3. (29)

Introducing the following stretched coordinates in the bow near field,

X'=xC4, YFyE',Z=zC'

together with

m=oh, n=13C,P(X,Y)=p(x,y)E',

which are variables introduced for the sake of çonveniènce in obtaining an asymptotic expansion, we have [see


Ø(x,y, z) 7?)

xf el3zcos[13(y

- )] cos['(x - )] dj3

at the limit E -+0.

Using this asymptotic solution Baba studied the rela-tiOnship between pressure distributiOn and bow geome-From numerical calculations for a flat ship having a blunt water line form Baba found that the gravity effect on pressure drag can be neglected when one considers the bow region less than about 30% of ship breadth B in the range of relatively high Froude numbers, Say, U/-..Jj> 05. The effectiveness of bow projection in reducing pres-sure drag is shown by the theory and then cOnfirmed ex-perimentally by towing tests and wake surveys in which momentum loss due to wave splashing at the bow is


5. Study on stern wave breaking 5.1 Stern near field problem

Until now we have considered mainly bow wave break-ing. In this section a consideratiOn on the stern wave

breaking is given by the present author. As shown in Sec. 3 a considerable amount of wave breaking from the transom stern of a high speed container ship is Observed. When immersion or width of immerged part of the, transom stern is chosen as a characteristic length, the free surface flow around the transom Stern is considered as a high FrOude number'f low.

For the sake of simplicity a ship having a transom stern is replaced by a pressure distribution Further we assume that the pressure is uniformly distributed in longitudinal direction so that the perturbation potential is simply ex-pressed as a sum of two parts:

ob(.r;y, z) øbow(x,Y,2) + østem(-,Y, z), (31) where øbow(x,Y,z) = MTB11O September 1976 x cos(wx) 0 0 (0




x cos[7(y)]sin('Jx)

(32) çbstem(X,Y,Z) ,Jj32+w2z 1'




x pv, dw i

d13 :.

cos[w(L x)]



(.0 p./w2+j32





is-2irpU J

dnP(1?)J d13e x (33)

where s is the half breadth of a ship,.

Now we consider a case s/L 1 so that the influence of

(30) a bow wave on a stern wave is negligibly small. We then

confine ourselves only on the stern wave.

In a similar manner to the boW near field' treatment(30),


MTB11O September 1976

we consider here 'stern near field', where the following

stretched coordinate System iS introduced


(xL)C'/2, Y=yE', ZzC',

where it,is assUmed that = s/L, and L is the ship length. Then (x-L) represents a distance from the stern.

Substitut-ing them in (33) together with


n=f3C P(Y)=p(y)C'/2

we have an asymptotic expression as e østern('L Y,z) =

r=. r=


x pv dm dn 2 cos(mX) 10


2irpU 0

x sinbJX].

Rewriting this. in terms ofusual variables, we have

øsim(X,Y, z) = r



(34) after carrying out the principal value integration.

The Wave height near the stern is then obtained:

ix,y) =





As a special case the wave height at the edge of transom stern(x=L) iswritten





In this case the water smoothly leaves from the stern in such a way that the stern geometry coincides with the wave height there. The relationship between stein geometry and pressure distributibn is then given explicitly by (36).

5.2 A flow characteristic near the transom stern

In order to be more specific an elliptic pressure distri-bution is considered:

p(y) (y/s)2 , (37)

where A is a constant value and corresponds to the sec-tional area of submerged part of transom stern. Substitut-ing (37) into (35), we have an expression of local waye pattern near the stern:



where J1 is the Bessel function.

Fig. 23 shows a wave pattern near the stern calculated by use of (38). Just at the back of the stern there exists a cavity-like dent which is followed by diverging waves.

14 (a) (b) 11 ...Fig. 26 .0 0.8 .4

Qi-Cavity liki dent Steep diver, in. wave

Fig. 23 Calculated wave heights of the stern waves

Longitudinal wave profiles are shown in Fig. 23 (a) From these figures we find that near the stern diverging waves are dominating. Figs. 24 and 25 show photographs of stern waves of a flat ship having .a transom stern of parabolic

section shape at U//"= 1.95 (U/.sJTJ = 3.37, where

H is the depth of the transom stern). The photographs were taken in Nagasaki Experimental Tank. Although breaking phenomena are observed on the diverging waves, the

observed wave pattern resembles' the calculated one (Fig. 23). From these figures it is considered that the steeper region of diverging waves relates breaking of waves. In

Fig. 24 Breaking stern waves (a)

Fig. 25 Breaking stern waves (b) 8

vL 1/S


order to achieve a better understanding of the breaking mechanism, stream lines at the cross plane right after the cavity-like dent (at s.,/. (x L)/s= 1.5) are calculated by use of the velocity potential (34) with a pressure distribu-tion defined by (37) As shown in Fig 26 calculated stream lines behave jUst like a pair of vortices. Since the vortex like fluid motions come close to the free surface, there may be expected such vortical, turbulent free surface flows as observed in Figs. 24 and 25.

The present theoretical interpretation coincides with Taneda's observation of a pair of vortices leaving from the


Fig. 26 Calculated stream lines in the transverse plane at

The author wishes to express his deep appreciation to Prof. Sadatoshi Taneda of Kyushu University for providing photographs of Figs 2 and 4 and Mr Masaaki Namimatsu of Research Institute of Ishikawajima-Harima Heavy

Industries Co., Ltd. for permission to use the experimental

Taneda, S., and Amamoto, H., On the necklace vortex, Bulletin

of Res Inst AppI Mech Kyushu Univ No 31 (Japanese) 1969.

Dagan, 0. and TUlin, M. P., Bow waves before blunt ships, Hydronautics, Inc., TechOical Report 117-14. 1969.

Baba, E., A New Component of Viscous Resistance of Ships; Journal of the Society of Naval Architects of Japan, Vol. 125, 1969, pp.23-34.

Kumat T and others Measurements of Boundary Layers of Ships (I), Selected papers from the Journal of the Society of Naval Architects of Japan, Vol.7, 1971, pp. 39-62.

Taneda, S., Observation of viscous fluid flows around bodies,

Synposium on Viscous Resistance of Ships, The SiCty of

Naval Architects of Japan, May 1973, pp.35-58.

Taneda, S., Necklace Vortices, Journal of the Physical Society of Japan, Vol.36, No. 1, 1974, pp. 298-303.

Takekuma, K., Study on the Nonlinear Free Surface Problem

around Bow, Journal of the Society of Naval Architects of

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Eckert, E. and Sharma, S. D, Bugwulste für langsame, vallige



stern of a ship as mentioned in Sec. -2. 6. Concluding remarks

In spite of the fact that a deep understanding of the

breaking phenomena of ship waves is important for design purposes of ship forms, especially for design of full forms, the theoretical study has been developed very little because

of the complexity of the phenomenon.

In the present paper a few pOssible theoretical models of the phenomenon are studied in order to understand the mechanism of breaking waves observed near the bow and the stern. A consideration of how to suppress the wave breaking is then given. However, further experimental and theoretical studies are necessary in the future in order to examine the validity of the present theoretical approaches.

In addition to the. present theoretical models other effective ways to explain breaking phenomena of ship

waves might be expected. In the present paper, truly non-linear analyses,. such as have been developed for the ship wave problem by, for instance, are not in-cluded. However, such studies are also expected to provide helpful information on the breaking -phenomenon of ship waves.

MTB11O September 1976

data shown in Fig. 11. Thanks are also due to Dr. Kyoji Watanabe and Mr Kinya Tamura Nagasaki Technical

Institute of Mitsubishi. Heavy Industries, Ltd. for their stimulating and encouraging discUssions in preparing the


-Schiffe, Jahrb. Schiffbautech. Ges. 64, 1970, pp. 129-171. (10) Taniguchi, K. Tamura, K.. and Baba, E, Reduction Of

Wave-Breaking Resistance by MHI Bow Mitsubishi Juko Giho Vol 8, No.1, 1971. pp.146-152 (Japanese), or Mitsubishi Tech-nical Review, Vol.9, No.1,1972, pp.62-69 (English).

(ii) Namimatsu, M. and Kushimoto, H.,. Some results of preure

measurement on the hull surface, unpublished report of

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Symposium on Viscous Resistance, of Ships, The Society of Naval Architects of Japan, May 1973, pp.157-168.

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of Engineers & Shipbullders, Vol.89, No.2, 1972, pp23-30. Longuet-Higins, M. S., A model of flow separation at a free

surface, Journal of Fluid Mechanics, Vol.57, part 1, 1973,


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Dagan, - G. and Tulin M. P., Two-dimensional free-surface 15


MTB11O Sepmber 1976

gravity flow past, blunt bodies. Journal of Fluid Mechanics.

Vol.51. part 3,1972, pp.529-543.

Dagan G Nonlinear ship wave theory 9th Symposium of Naval Hydro-dynarnics, August 1972.

Hermans, A. J., A matching principle in nonlinear ship wave theory at low Froude-number Deift Progress Reports, Series

F, Vohl, 1974.

Wèhaun J. V. and Laitone, E. V., Surface Waves, Handbush der Physik Vol 9 Springer Verlag 1960

Guevel P Vaussy P and Kobus J M The distribution of

singularities kinematically equivalent to a moving hull in the presence of a free surface International Shipbuilding Prog Vol.21. 1974, pp.311-324.

Timman, R. Small parameter expansives in ship hydro-dynamics 10th Symposium of Naval Hydrodynamics June


Baba, E. and Takekuma, K. A Study on Free-Surface FlOw

around Bow Of Slowly Moving Full Forms, Journal of the

Society of Naval Architests of Japan Vol 137 1975

Brard A The Representation of a Given Ship Form by Singu

larity DistributiOns When the Boundary ConditiOn On the

Free Surface is Linearized, Journal of Ship Research, Vol. 16, No.1, 1972, pp. 79-92.

Baba E Blunt Bow Forms and Wave Breaking The first

Symposium on Ship Technology and Research The Society of Naval Architects and Marine En9ineèr, Washington, D.C.,

Aug. 1975.

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form of short gravity waves on long waves and tidal currents, Journal of Fluid Mechanics, Vol.8, 1960, pp. 565-583

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tion of sea waves, Proc. Roy. Soc. A. 311, 1969, pp.371-389. Dagan G Taylor instability of a nonuniform free-surface

- flow Journal of Fluid Mechanics Vol 67 part 1 1975 pp 113


Wu, T. YaO-Tsu, A singular perturbation theory for nonliflear

free surface flow problems International Shipbuilding

Pro-9ress, Vol.4, 1967, pp. 88-97.

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problems The Journal of Engineering Mathematics Vol 1

1967, pp.215-235.

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the Society of Naval Architests of Japan, Vol. 135, 1974, pp.25-37.

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Planing Surfaces Schiffstechnik Bd 14 1967 pp 57-64 Baba, E., Analysis of Bow Near Field of Flat Ships, Mitsubishi Technical Bulletin, No.97, 1975.

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