bTlNSRDC-^^g

GCi

3)3no . * l/oi5

DAVID W. TAYLOR NAVAL SHIPRESEARCH AND DEVELOPMENT CENTER

Bethesda, Maryland 20084

A RAY THEORY FOR NONLINEAR SHIPWAVES AND WAVE RESISTANCE

DOCUMENTCOLLECTION

by

Bohyun Yim

APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

Presented at theThird International Conference

on Numerical Ship HydrodynamicsParis, 16-19 June 1981

SHIP PERFORMANCE DEPARTMENTRESEARCH AND DEVELOPMENT REPORT

December 1981 DTNSRDC-81/095

MAJOR DTNSRDC ORGANIZATIONAL COMPONENTS

DTNSRDC

COMMANDER00

TECHNICAL DIRECTOR01

OFFICER-IN-CHARGECARDEROCK

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SYSTEMSDEVELOPMENTDEPARTMENT

SHIP PERFORMANCEDEPARTMENT

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STRUCTURESDEPARTMENT

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OFFICER-IN-CHARGEANNAPOLIS

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AVIATION ANDSURFACE EFFECTSDEPARTMENT

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COMPUTATION,MATHEMATICS AND

LOGISTICS DEPARTMENT18

PROPULSION ANDAUXILIARY SYSTEMS

DEPARTMENT27

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G P 866 993 NDW-DTIMSRDC 3960/43 (Rev. 2-I

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2. GOVT ACCESSION NO EClPVENT'S catalog number

4. TITLE (and Subtitle)

A RAY THEORY FOR NONLINEAR SHIPWAVES AND WAVE RESISTANCE

5. "TYPE Of REPORT 4 PERIOD COVERED

Final6. PERFORMING ORG. REPORT NUMBER

7. AUTHORfsJ

Bohyun Yim

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9. PERFORMING ORGANIZATION NAME AND ADDRESSDavid W. Taylor Naval Ship Researchand Development CenterBethesda, Maryland 20084

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18. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverse aide t

RayWave NumberCausticFree Surface Shock Waves

assary and Identify by block number)

Ship WavesWave PhaseSecond CausticNonlinear Ship Waves

Wave ReflectionWave ResistanceBulbous Bow

20. ABSTRACT (Continue on reverse aide It necessary and Identity by b/ocfc number)Analytical and numerical methods for application of ray theory in

computing ship waves are investigated. The potentially important role ofray theory in analyses of nonlinear waves and wave resistance is demon-strated. The reflection of ship waves from the hull boundary is analyzedhere for the first time.

(Continued on reverse side)

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(Block 20 continued)

When a wave crest touches the ship surface, the ray exactly followsthe ship surface. When the wave crest is nearly perpendicular to theship surface the ray is reflected many times as it propagates along theship surface. Many rays of reflected elementary waves intersect eachother. The envelope to the first reflected rays forms a line like ashock front which borders the area of large waves or breaking waves nearthe ship.

For the Wigley hull, ray paths, wave phases, and directions ofelementary waves are computed by the ray theory and a method of computingwave resistance is developed. The wave phase is compared with that oflinear theory as a function of ship-beam length ratio to identify theadvancement of the bow wave phase which influences the design of bowbulbs. The wave resistance of the Wigley hull is computed using theamplitude function from Michell's then ship theory and compared withvalues of Michell's wave resistance. The total wave resistance has thephase of hump and hollow shifted considerably.

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TABLE OF CONTENTS

Page

LIST OF FIGURES iii

ABSTRACT 1

ADMINISTRATIVE INFORMATION 1

INTRODUCTION 1

RAY EQUATIONS 2

RAYS OF SHIP WAVES AND LINEAR THEORY 3

RAYS OF SHIP WAVES AND SHIP BOUNDARIES 5

RAY REFLECTION 7

NUMERICAL EXPERIMENTS OF RAY PATHS AND FREESURFACE SHOCK WAVES 7

WAVE AMPLITUDE AND PHASE 9

WAVE RESISTANCE 10

DISCUSSION 13

ACKNOWLEDGMENTS 13

APPENDIX A 13

REFERENCES i5

LIST OF FIGURES

1 - Ray Paths for a Wigley Hull (b = 0.1, h = 0.625) ' 4

2 - Ray Paths for a Wigley Hull (b = 0.2, h = 0.0625) 6

3 - Ray Paths for a Wigley Hull (b = 0.2, h = 0. 03) 6

4 - Ray Paths for a Wigley Hull (b = 0.5, h = 0.0625) 6

5 - Angles of Waves Reflecting on the Ship Hull

6 - Width of the Second Caustic, a/x , for Wigley Hulls 8

7 - Rays of the Wigley Ship (b = 0.2, h = 0.0625) with Bulb(radius, r

b= 0.0214h, depth, 2.^ = 0.5h)

iii

Page

- The y Coordinate of Ray at x = 2, Phase Function S (0)

and the Starting Point of Elementary Wave, x of aWigley Hull (b = 0.2) 7 11

9 - y (x = 2), S 2 (6 00),and -x of a Wigley Hull

(b = 0.1, h = 0.0625) 11

10 - S (6^), and -x

of a Wigley Hull (b = 0.2,

h = 0.0625) with and without Bulb(rv = 0.0285h, z. = 0.7h) 12b

11 - Ray Paths of Stern Waves of a Wigley Hull(b = 0.2, h = 0.0625) 12

12 - Wave Resistance of the Wigley Ship (b = 0.2,h = 0.0625) 13

A RAY THEORY FOR NONLINEAR SHIP WAVESAND WAVE RESISTANCE

Bohyun YimTaylor Naval Ship Research and Development Center

Bethesda, Maryland 20084, U.S.A.

Abstract

Analytical and numerical methods forapplication of ray theory in computing shipwaves are investigated. The potentiallyimportant role of ray theory in analyses ofnonlinear waves and wave resistance is demon-strated. The reflection of ship waves fromthe hull boundary is analyzed here for thefirst time.

When a wave crest touches the ship sur-face, the ray exactly follows the ship surface.When the wave crest is nearly perpendicular tothe ship surface the ray is reflected manytimes as it propagates along the ship surface.Many rays of reflected elementary wavesintersect each other. The envelope to the firstreflected rays forms a line like a shockfront which borders the area of large wavesor breaking waves near the ship.

For the Wigley hull, ray paths, wavephases, and directions of elementary wavesare computed by the ray theory and a method ofcomputing wave resistance is developed. Thewave phase is compared with that of lineartheory as a function of ship-beam lengthratio to Identify the advancement of the bowwave phase which influences the design of bowbulbs. The wave resistance of the Wigley hullis computed using the amplitude function fromMichell's thin ship theory and compared withvalues of Michell's wave resistance. The totalwave resistance has the phase of hump and hollowshifted considerably.

Introduction

Significant developments in the ship wavetheory have been made in recent years. Theseinclude the application of ray theoryl>2* an(jthe experimental discovery of a phenomenoncalled a free surface shock wave.

3

Because a ship is not thin enough to applyboth the thin ship theory and the complicatedfree-surface effect, theoretical developmentof an accurate ship wave theory has been slow.The problem should be evaluated differentlyfrom the conventional means. Ray theory hasbeen used in geometrical optics or for waveshaving small wave lengths. Ursell used the

*A complete listing of references is givenon page 1 5

.

theory to consider wave propagation in non-uniform flow, and Shen, Meyer, and Keller-* usedit to investigate water waves in channels andaround islands. Recently Keller developed aray theory for ship waves and pointed out thatthe theory could supply useful informationabout the waves of thick ships at relativelyslow speeds. However, he had difficulty ob-taining the excitation function for wave ampli-tude and solved only a thin-ship problem. Inuiand Kajitani^ applied the Ursell ray theory4 toship waves, using the amplitude function fromlinear theory.

Because the ray is the path of wave energy,it is not supposed to penetrate the ship surface;and this was emphasized by Keller. However,neither Keller nor Inui and Kajitani considernonpenetration of the ray seriously. RecentlyYim" found the existence of rays which emanatedfrom the ship bow and reflected from the shipsurface.

In the present paper, further study hasrevealed many bow rays reflecting from theship surface, creating an area in which theserays intersect each other. The envelope tothe first reflected rays forms a line like ashock front which borders the area of largewaves or breaking waves near the ship. Thiswill be referred to as the second caustic. Thisphenomenon was observed by Inui et al.' in thetowing tank and called a free surface shockwave (FSSW) . Much has been done concerning thetheoretical investigation of FSSW by Introduc-ing a fictitious depth for a shallow water non-dispersive wave.

The ray equation and the ship boundarycondition are analyzed further to show theexistence of reflecting rays, except in thecase of a flat wedgelike ship surface. Theray path is very sensitive to the initial bound-ary condition near the bow or the stern, andshould be identified at downstream infinitynot by the initial condition. The conservationlaw of wave energy in the nonuniform flow isdifferent from that of the uniform flow due tothe exchange of energy with the local flow.

8

Therefore, the linear wave amplitude as afunction of initial wave angle near the bowor stern is meaningless and cannot be used inthe ray theory. It is shown that the amplitudefunction for the ray theory matched with the

linear theory far downstream is reasonable andlikely to produce a reasonable result as inthe two dimensional theory.'

The wave phase in the ray theory is ob-tained, together with the ray path, indepen-dent from the amplitude function. Both thephase and the ray path are quite different fromthe prediction of linear theory near the ship.The rays far downstream are straight as in thelinear theory yet have phases different fromthe linear theory; they are parallel to thelinear ray with the same wave angle but donot coincide. The difference in wave phasesis the main factor that makes the wave resis-tance different from the linear theory. Raypaths and phase difference are computed forvarious parameters of the Wigley ship, with andwithout a bulbous bow. The ray paths fordifferent drafts and different beam-lengthratio of the Wigley hull are slightly different,widening the wave area near the hull for thewider beam and /or larger draft. However, thephase difference is more sensitive to the beam-length ratio and/or draft-length ratio, byalways advancing the linear wave phase.

The wave resistance of the Wigley hull iscomputed and shown to have a considerable shiftof phase of hump and hollow.

By eliminating S z from these two equations heobtained a dispersion relation

The most interesting phenomenon of a shipwith a bulbous bow is the reduction of slopesof rays and the second caustic, i.e., thelarger the bulb, the greater the reduction.This fact was observed in the towing tank.-'There exists a bulb size which totally elimi-nates the reflecting rays. However, the phasedifference due to the bulbil ^s very smallshowing that the phase difference, which hasbeen observed in the towing tank, is the effectof nonuniform flow caused by the main hull.

Ray Equations

The concept of ray theory in ship waves isanalogous to the concepts used in geometricaloptics and in geometric acoustics. A ship isconsidered advancing with a constant velocity-U which is the direction of the negative xaxis of a right handed rectangular coordinatesystem O-xyz with the origin at the ship bowon the mean free surface, z0, z is positiveupwards

.

First the phase function s(x,y,z) is de-fined so that the equation (s constant) re-represents the wave front where the value of sis the optical distance from the wave source,e.g., the ship bow. When Keller developedhis ray theory of ships by expanding boundaryconditions and a solution, which should satisfyboth the Laplace equation and the boundaryconditions, in a series of Froude number squaresF , he obtained:

(s..2+ s V

(2)

where t is the double model potential. Whensteady state motion is assumed with respect tothe moving coordinate system, - xyz,

When the angle between the normal n to thephase curve s and the 'axis is denoted by 6,

n - icosB + jsinS (3)

Then the wave number vector is defined by

k = k.T + k,j = si + sj1

(4)- nk = ik cos6 + jk sine at z =

From equations (2) and (4)

k"|u cos6 + v sin9J

where

- = u and -4 = vx y

These results are an approximation withinthe order of f4, and the phase function and itsrelated equations are all limited to theirvalues at z = 0. Thus, from now on, unlessotherwise mentioned, all the physical valuesare at z 0. The ray equation of ship wavesis obtained from the irrotationality of thewave number vector

9k2

3k

j

3x 3y

From Equations (4) through (6)

,

{2 sine (u sine - v cosB)

+ cosB (u cosB + v sine)}

+ {2 cosB (-u sine + v cos6)

+ sine (u cose + v sine)} |

- 2 sine (cose |S + sine |^)

(6)

(7)

(Vs) -

-i (S + V4 7s) 2 at z -

(1)

This is the ray equation which can besolved by the method of characteristics, andis equivalent to simultaneous ordinary differ-ential equations.

dy {v- *s sine (u cos6 + v sine)} ..

dx {u- h cos6 (u cosB + v sinS)}

7- -=.{2 sine (cose |H + sine |^-)

3u iV>- 2 cose (cose

-T- + sine ?r-)}/{2 sine (u sine (9)dy dy

- v cose) + cos6 (u cose + v sin6)}

Here u cos6 + v sin 6 is the velocity componentnormal to the phase curve of the flow relativeto the ship. Because the wave is stationaryrelative to the ship, the phase velocitythrough the water surface should be

- u cosB - v sine

The group velocity is one-half of the phasevelocity. Thus the ray direction in Equation(8) is along the resultant of the group velo-city taken normal to the phase curve and of thevelocity of the basic flow as Ursell haB shown.*Because the wave energy is propagated at thegroup velocity, the ray path is interpreted asthe path of energy relative to the ship. Thiscan be obtained by solving Equations (8) and(9) with the proper initial condition. Thephase s can be obtained from equations (4) by

s - f k dr

as in potential theory; s(x,y) is a function of(x,y) but is unrelated to the integration path.However

,

ds k cose dx + k sine dy (10)

can be solved together with the ray equationsalong the ray path.

Rays of Ship Waves and Linear Theory

To investigate the path of a ray of aship wave, a ship, represented by a doublemodel source distribution m (x,y) ony o, h>z>-h, is considered. Although thelinear relation between the source strengthm and the ship surface

y - f(x.z)

1 dym- 2^ at

(11)

(12)

the actual double model ship body streamlineshould be obtained by solving

through the stagnation point where u and v arethe velocity components of the total velocitycaused by the double model source distributionm and the uniform flow relative to the ship.

In the linear ship wave theory a smoothsource distribution produces two systems ofregular ship waves: the bow and the sternwaves starting from the bow and stern,respectively, represented!' as

tt/2

;(x,y) = j A(e) exp {i Sj (x,y,e)} d6 (14)-tt/2

s. = k sec 8 {(x-x.) cose + y sine} (15)

x = the x coordinate of the bow or stern

g acceleration due to gravity

A(6) =* amplitude function which is afunction of source distribution m

The regular wave c is the solution oflinear ship wave theory far from the ship. 1

Actually, it is easy to see. that the exponentialfunction satisfies both the Laplace equationand the linear free-surface boundary conditionfor any value of xi. Havelock interpretedEquation (14) by discussing the integrand aselementary waves, ,13 i.e., the regular wavesare aggregates of elementary waves starting fromthe bow and stern of a ship. The normaldirection of each elementary wave crest isn - i cosB + J sine.

Because the local disturbance of the doublemodel decays rapidly away from the ship, evenin nonlinear theory in the far field, theregular wave should be of the form of Equation(14) with possibly different values of x^ andA(6). When the integral of 5 is evaluated bythe method of the stationary phase 14 by takingroots of

If (x.y.6) = (16)

2 tan 8 + - tane + 1-0 (17)y

in general two values of e are obtained for agiven value of each x and y, satisfying,

A > 8^ (18)

dx u(13)

Furthermore, when

>-h

because only the transformation of coordinates(x,y) to (x - Ut , y) is needed to get the un-steady flow. Strictly speaking the relativefrequency is

s = k u cos6 + k v sine = k U cose

6 = 35 deg

and the value of the integral of Equation (14)for any x,y in x/|y|-8^ can be evaluated bythe stationary phase method as the sum of twowaves: transverse waves (0 deg < |e| < 35 deg)and divergent waves (35 deg < |e| < 90 deg).

Except for the amplitude function exactlythe same result for the relation of x,y and 6as in the linear theory can be obtained fromEquation (8) by substituting u = 1 and v = 0.Thus, this means that the energy of eachelementary wave propagates on the uniform flowalong a straight line ray. The rays are con-fined in x/|y| > 8^, both of the elementarywaves at 6 = and 9 = 90 deg correspond to theray y /x = 0, and 8 = 35 deg corresponds withthe ray x/|y| 8"*. One ray between these tworays corresponds to two elementary waves whereone is transversel and the other is divergent.

When the velocity is not uniform due to theflow perturbation caused by a ship, the ray isnot straight but curved near the ship asshown in Figure 1. It is known that wave energyflux divided by the relative frequency withrespect to the coordinates for which the fluidvelocity is zero, is conserved along thecurved ray tube. If the coordinates are fixedin space, then the ship waves are unsteadyrelative to the fixed coordinates and thefrequency is

= U k cose (19)

0.20

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.0

/0.2265/

0 = -0.66892/ ./ (Geo; 0j) =-0.1175:0.08

-

///el"JL1 SECOND

-^CAUSTIC

^n2^^5p=2==

Rays of Ship Waves and Ship Boundaries

Ship waves created by a smooth ship hullpropagate as regular bow and stern waves fromthe bow and stern stagnation points to infinityalong rays. Because a ray carries the waveenergy it cannot penetrate the ship surface. Ifthe linear free-surface condition is consideredwith the exact hull boundary condition, as hasbeen popular in recent ship wave analysis ,the straight rays which pass through the hullboundary have to be considered. For a raytheory with the exact hull boundary condition,the ray is not allowed to penetrate the ship.In fact, when the initial wave crest touchesthe ship boundary it can be proved that theray of such a wave grazes along the shipboundary without penetrating the ship boundary.

In Equation (8) when the wave cresttouches the hull

u cose + v sine (20)

because e is the angle between the normal tothe crest and to the*axis, and the velocitynormal to the ship hull is zero on the hull.Equation (8) then becomes

dy_

v

showing that the ray touches the ship hullstreamline from Equation (13).

When the wave crest is perpendicular tothe ship hull

= tan6 (26)

If Equation (26) Is inserted into Equation (8)

tanti - (27)dy_

The ray also touches the ship initially.However, Equation (9) is not compatible withEquation (26) on the hull. This can be provedin a similar way as follows:

Differentiating Equation (26) with respectto x along the ship hull

dx f + 1x

Inserting Equation (26) into Equation (9) gives

2tan9 (|H +taneg)_ 2(g +tane|x)u (1 + tan6)

From Equations (25), (27), and (29), noting

3u 3v

When the wave crest touches the hull,from Equations (11) and (13) (30)

and from Equations (20) and (21)

tane = ^

(21)

(22)

Differentiating Equation (22) with respect tox along the ship hull

(23)

However, inserting Equation (20) into Equation(9) yields

u v 2. 3u 3vcote -r- + t cot e - cote d6 3x 3x 3y 3y (24)

u(l cote)

Differentiating Equation (21) with respect tox along the hull yields

v 3u 3y 3v 3yu 3y 3x 3y 3x (25)3 v u 3x

3x u u

From Equations (20), (21), (22), and (25)it can be shown that Equations (23) and (24)are equivalent. That is, when the wave cresttouches the ship boundary, the ray equationsand the ship hull streamline equations areequivalent.

Equations (28) and (30) are compatible onlywhen f - , or fx - constant

.

That is, only when the ship is a flat platedoes the ray of the wave, whose crest is perpen-dicular to the ship, follow the ship boundary.This means that when the ship bow is like awedge, the ray of the bow wave, whose crest isperpendicular to the wedge surface, initiallyfollows the wedge surface.

When the ray equations are solved numer-ically by the Runge-Kutta method, with initialvalues near the bow stagnation point, the raytouches and follows the ship boundary at

e

-

-j + a (31)

where a is the half entrance angle of the shipbow and e^ denotes the initial value of 6.

When 6^ increases from this value the raymoves gradually away from the ship as shows inFigures 1 through 4. The rays are curved nearthe ship but at far downstream they are straightand the ray direction becomes exactly the samefunction of 6 as the linear theory. Thus atinfinity the rays are inside |dx/dy| = 8*5.However, when 6^ approaches zero, the curvedray near the ship gradually approaches theship, and eventually crosses the ship boundary,as in Figures 1 through 4. Here the wavereflection should be considered to prevent theray penetration of the ship hull.

0?0

P.18! (6a.; Oj)

ft ^ 1-0.36145; 0.265// 0.3015- 27

0.14

0.12

0.10

008

0.06

0.04

0.02

0.00

0.20

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00.

. 0. = -0.6514 /.-0.8515/X (60=; 9i) =

///^0.1761; 0.2415

/// ^ -0.07132; 0.2480.25&~\0.249^

- /yU2S4B> -*"^^X^SEC0ND^O\CAUSTIC NT-

0.0 0.2 0.4 0.6 0.8 1.0

Figure 2 - Ray Paths for a Wigley Hull(b = 0.2, h = 0.0625)

Figure 3 - Ray Paths for a Wigley Hull(b = 0.2, h = 0.03)

0.4

0.3r

y 0.2 -

0.1

0.0

I

0.545

^^^0558

0.55^-

i

= 0.54

i

0oo

1

; 6; = -0.141; 0.535

i 1 ii

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Figure 4 - Ray Paths for a Wigley Hull (b = 0.5, h = 0.0625)

Ray Reflection

There are no waves coming from placesother than the ship bow or the ship stern. Inthe ray theory, the flow field perturbed by theship deflects the ray path starting from theship bow toward the ship. Thus, some raysof bow waves impinge on the ship hull.Because the ray theory is for small Froudenumbers and the wave phenomenon is consideredonly on the free surface, it is reasonable toconsider only reflected waves, as in geometricaloptics, neglecting transmitted waves when theoncoming waves impinge on the ship hull.

Let the ray at the wave angle 8 on theship boundary (x,y) be reflected to 8r andfx = tan8 as in Figure 5.

(32)

Figure 5 Angles of Waves Reflecting onthe Ship Hull

Whenever the ray intersects the shipboundary at (x,y) with angle 8, then the valueof 6 at (x,y) will be changed to 8r = 28 - 8,and x,y,8 are continuously calculated by theRunge-Kutta's method. Then the ray will reflectas in Figures 1 through 4. Here if 8 is zero,it is easily seen in Equation (8) that theimpinging ray angle (9) changes to - for thereflected ray angle. This fact can be easilyshown to be true even for 8 p by just therotation of the coordinate system.

Numerical Experiments of Ray Paths andFree Surface Shock Waves

Because the ray equations can be solvedonly numerically, careful numerical experimentswith the ray equations may give valuable in-formation. For simplicity, the Wigley hullsource distribution with

2ir dx

f. - 2rt (l - x2

) {l - (h 2) (33)

is considered for various numbers of b and hwhich are related to the hull beam and draft,respectively. The actual hull shape corre-

sponding to the source Equation (33), is ob-tained by plotting the body streamline passingthrough stagnation points as the solution of

dy_

v

dx u

where u and v corresponding to Equation (32)

,

are shown in Appendix A together with u , u ,and v for the ray equation. Equations (8){(9), and (10), together with the streamlineequation, are solved by the Runge-Kutta methodwith initial conditions (x,y,8) near the stag-nation points with various values of 8.

Many ray paths both reflecting and non-reflecting from the surfaces of various Wigleyhulls are shown in Figures 1 through A. Thesepaths were computed by a high speed Burroughscomputer at David Taylor Naval Ship Researchand Development Center. The reflection condi-tion is incorporated in the high-speed computa-tion with a routine to find the intersection ofthe ray and the ship boundary which is pre-calculated and saved in the memory. The stepsizes of integrations and interpolation weredetermined after many numerical tests, and theshown results are considered to be reasonablyaccurate. For a given initial condition, thesolution is stable and converges well.

In Figures 1 through 4, some common fea-tures of rays can be drawn as follows. Therays in - ir/2 < 8 < far behind the shipbehave like rays of linear theory except wavephases are advanced in the ray theory and thosenear 8oo are rays propagating from the shipbow and reflecting from those of ship hull.The rays near the ship are very different fromthe linear theory as Inui and Kajitani^ pointedout

. The curved rays from the bow have a farlarger slope than those of linear theory andthe phase of each ray is considerably advanced.The magnitude of the phase difference is moresensitive to the beam-length ratio and thedraft-length ratio than the magnitude of theray slope.

When 6 = 35 deg, the ray will be theoutermost ray and the ray angle at > will beapproximately tan" 8-^ as in the linear theoryand there is the corresponding initial value of8 or 8i near the bow, or the origin. However,the 8i which is corresponding to a single valueof 8o> is very sensitive to small changes of xand y near the origin. At a fixed point nearthe origin there exists a unique correspondencebetween 6^ and 8,.

When from the 8^ which is corresponding to8,,, = 35 deg, the initial value of 8^ Increases,8oo also Increases and the ray angle decreases.In general, when 8i = 0, 8

twice from the ship hull. In this way, furtherincrement of 0^ makes the ray reflect from theship hull three, four ... times. However, at$1 near the value of a, the ray tries to pene-trate the ship hull at the starting point ofthe bow. This cannot be allowed because thiskind of ray should come from outside of theship. Let the border point of 6i be 6iu .This means that rays of initial value of 6between 6i < 6i < 9iu reflect from the shiphull. As is clear in Figures 1 through 4,in general, all the rays before reflection donot intersect each other. However, reflectedrays intersect other rays. The once reflectedrays intersect not only with each ray oncereflected from ship boundary points close toeach other, but also with at least one raybefore reflection.

In the stationary phase, each ray has anamplitude. Likewise in the ray theory, eachray carries it's energy. The reflected ray mayhave approximately the same energy as the rayat Sj = Oio or 8 =. o where the amplitudefunction of the linear theory is in generalmore significant than amplitudes of the othervalues of 6=>. Because the phase must beapproximately close to each other for the wavesnear 6= = 0, the wave height of the once re-flected ray may be close to three times thatof the transversal wave for 9, = 0. When theenvelope of the once reflected waves is drawn,the domain bounded by the ship surface andthe envelope, denoted by Dm, must be distinctlydifferent from other domains because in Dmthere are not only once reflected rays but alsotwic or multi-reflected rays on which more thanthree reflected rays intersect by an argumentsimilar to that used for the once reflectedrays. The envelope of the once reflected raysbehaves like the shock front which was observedin Japan. ->

In general, a line is called a caustic whenon one side of the line one can find a continuousdistribution of rays but not on the other side,and along the caustic the wave slope is foundto be large. The wave near the ray angle 8, =

35 deg is the caustic, and the wave height nearthe caustic far downstream can be obtained byan application of the Airy function in thelinear theory. The envelope of the once re-flected rays may be a kind of caustic formedby the refracted bow wave rays due to the non-uniform flow perturbed by the ship. Shen,Meyer, and Keller^ studied such caustics causedby the sloping beach of channels and aroundislands. Thus, the additional caustic of shipwaves may be called the second caustic of shipwaves, and should not be confused with the firstcaustic which is the known caustic at 8 35deg.

Second Caustic of Ship Waves of Various Ships

For the Wigley hull, several differentvalues of parameters b and h were taken tofind their effect on the second caustic. Inaddition, the effect of a bulbous bow on thesecond caustic was considered. The mostdistinguishable physical characteristic of thesecond caustic is its distance from the shiphull. This is related to the distance of

reflected rays from the ship hull. If thenumber of reflection rays increases, or if 6iincreases from 6j. , the maximum distance be-tween the ray and the ship hull decreases.The distance, before or after the ray reflection,approximately behaves like a sine curve. Themaximum distance between the ray before thefirst reflection and the ship hull dividedby the x coordinate of the point of the firstreflection a/xj is plotted in Figure 6 forvarious ships. The value of a/xi for differentvalues of 9i are approximately the same for agiven hull and are related to the area betweenthe second caustic and the ship hull where theremay be breaking waves or turbulent waves. Thus,if the area is large, viscous dissipation ofenergy becomes large. Accordingly, the measuredmomentum loss behind the ship for the breakingwaves' becomes large.

The values of a/xj , increase with increas-ing beam-length ratio. However, the mostinteresting part is the effect of the bulbousbow. 10 When the bulb size is increased or thedoublet strength is increased the curvature ofthe ray near the bow becomes less, although thestreamline near the bow is such that the en-trance angle is slightly large. The values ofa/xj decrease with increasing bulb size, andeventually the ray for S -. cannot propagatewithout penetrating the ship hull at the begin-ning. That is, there is no reflecting raycoming out of the bow with a bulb of the propersize, as shown in Figure 7.

0.2 0.4

(h = 0.0625)

Figure 6 - Width of the Second Caustica/xj for Wigley Hulls

Figure 7 - Rays of the Wigley Ship (b = 0.2, h = 0.0625) with Bulb(radius, r = 0.0214h, depth, Zj = 0.5h)

These phenomena associated with the secondcaustic are exactly the same as those of the"free surface shock wave" which was observedin the towing tank.

Finally it should be pointed out that theray of the wave whose crest is perpendicularto the hull surface follows exactly along theflat surface as was proven by Equations (28)and (30) and the following paragraphs. Thismeans that rays originating from the vertexof a wedge very likely never reflect on thewedge surface. This is because rays near theray which follows the ship surface did notintersect each other near 6^ = -tt/2 + o, or6 - - tt/2. Rays near 6,,, '=. impinge on theship surface due to the effect of the water-line curvature of the ship. Therefore, if aship has a wedge bow, the second caustic mustbe found near or behind the shoulder. Be-cause the first caustic near the bow may bevery prominent and both waves near the firstcaustic and waves near the stagnation pointbreak, careful experimental analysis and moretheoretical study of the bow near field may beneeded.

Wave Amplitude and Phase

Because the perturbation due to a ship,or both regular waves and local disturbances,decays at far field, the linear theory musthold in the far field. In particular, thewave resistance can be calculated from theenergy passing through a vertical plane x=constant far downstream; the linear theorywhich is properly matched to the near field ofthe ship will be used for calculating the waveresistance. As explained in Equation (14) andthe following paragraphs, the expression forregular waves far downstream is known to be ofthe form of Equation (14) where the amplitudefunction

A(6) = P(8) + i Q(6) (34)

may be taken as an approximation from the lineartheory but the phase difference xi must be ob-tained from matching with the near field. Ifthe near field is also expressed by the lineartheory

Q = "k / /(35)

dxdz f e sec 9

where fx is the derivative of Equation (11) withrespect to x. When the inner integrand ofEquation (35) is integrated with respect to x,the value with the limit x = 1 will form sternwaves and the value with the limit x = willform bow waves . 1 1 Then the bow waves can berepresented by

tt/2

?b " / *b (8) 6XP f 1 S l (9) ) d9 (36)-tt/2

where

(6) Pv + i q. (37)

When the phase s is computed from Equation(10) along with the ray path from Equations(8) and (9) considering that s - at the bownear the origin, there are two results differentfrom those of linear theory: (1) the ray pathis deflected as if the elementary wave of thelinear theory started from (x^.O) not from theorigin, and (2) the phase change denoted byAs should be considered. That is, the

equivalent linear elementary wave may bewritten as

A, (6) exp |"i k sec 6^ (38)

|(x-x ) cos8 + y sine| + i kQ As]

where xi is obtained as an intersection of thetangent to the ray at and the x axis and

k As = k s - k sec 8 |(x-x,) cos8 + y sinSJo o o ' 1

'

(39)

The value of

Wave Resistance

If all the elementary waves are assumed tobe propagated without reflection, the amplitudefunction and the phase difference studied inthe previous sections will supply enough infor-mation for the calculation of wave resistance.Because at far downstream, the wave heightmay be considered linear, the Havelock waveresistance formula^ may be used for the wavesrepresented by Equations (36) through (40),considering that the wave with the changedphase s_, (8) has the amplitude=

2b

(?, i QK ) >ik s 2b (6)

s, (8) = As - yl

sec6 (40;

can be obtained at any point along the ray.In general, XJ is negative and S2 (8) ispositive meaning that the bow wave phase in theray theory is larger or more advanced than thephase of the linear theory. This fact has longbeen observed in experiments in towing tanks.

The advancement of wave phase is computedfor various ships and the values of S2 and xiat x = 2 are shown in Figures 8 through 10.When the beam-length ratio increases and/orthe draft-length ratio increases, the valuesof S2 increase and the values of x decreasefor all values of 8. As compared with theincrement of the slopes of rays near the ship,the increment of the phase angle is moresensitive to the beam- and/or draft-to-length

The most interesting phenomenon about thephase difference is in regard to the bulbousbow. 10 That is, the phase differences for hullswith and without bulbs are almost the same evenwith a considerably larger bulb. In the past,because of the observed phase difference ofship waves, the bulb was located far forwardto obtain good bow wave cancellation. *

According to the present analysis, if there isno other reason, the bulb position need not befar forward. Because the nonuniform flow cre-ated by the ship is much more significant thanthat of the bulb, as far as phase change isconcerned, both the ship bow waves (in general,positive sine waves) and the bulb waves(negative sine waves) propagate through thesame region and cancel each other.

As for the amplitude function, althoughit was shown by Doctors and Dagan9 that raytheory produced the best result for a two-dimensional submerged body even though theyused a linear amplitude function, the surfacepiercing three-dimensional case may be quitedifferent. The amplitude function is mainlyrelated to the singularity strength which sat-isfies the ship hull boundary condition andsome improvement might result by considering thesheltering effect. However, in the presentstudy, the linear amplitude function is used tosimplify the problem, showing the effect ofcurved rays.

Ill U L

IT/,

C = k IJ

,

-tt/2

ik s nl_ (6)

'b ' "V '

or in Sretten6ky's formula"

2,

16

c =16tt k

j=0(42)

v*m\ ik.s,, (dj)fl^J. |A, (dj) e^4ttj\2,k w

(dj)ik s,. (6j)

f yJ j dxdz m exp j k

d.

(zd.

+ ix) + ik s2 |

=

2rr^

(43)

h-\h [^ + H 1+ TTw-

2

! 1=sec6

w = width of the towing tank nondimension-alized by ship length L

e . = 2 f or j S 1

|(Pb

+ iQ) eik s,(6) 2

10

1.0

h = 0.0625 h = 0.03

J L J 1_0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Figure 8 - The y Coordinate of Ray at x = 2, Phase Function S2(9) and the Starting Point ofElementary Wave, x of a Wigley Hull (b " 0.2)

1.0

0.8 COORDINATE OF RAYYr

0.6

0.4

S2(0co)-X1

0.2

0.00.0

NO BULB

WITH BULB

Figure 10 - S, of a Wigley Hull (bBulb (r, = 0.0285h, z

the phase difference of each elementary wavedoes not change bow wave resistance from thelinear theory.

The stern wave amplitude is exactly thesame as the linear value

0.2, h = 0.0625) with and without= 0.7h)

considered to be totally missing in the waveresistance, j in the Srettensky formula for bowaves would start not from zero but from acertain number minimum J = Jj such that

A (8) = P + iQs s s

but the phase difference S2 S (6) should be com-puted by the ray theory together with the raypath which is shown in Figure 11. Then it isalso obvious that the stern wave resistance isthe same as the linear stern wave resistance.The total wave resistance may be obtained simi-larly by considering that the total wave ampli-tude is

where 6i is the value of 8twaves start to reflect. If

from which bow

and e = 2 for Jfor J i J

The result is shown in Figure 12 whereconsiderable shift of the phase of hump andhollow of the wave resistance due to bowstern wave interaction is noticeable.

iks (8)(P + iQ, ) e -

D+ (P + iQ )

ik (s, (8) - sec9)(44)

Here, the bow and stern wave interaction ap-pears in the wave resistance. That is, onlythe interaction term changes due to the phasechange caused by the nonuniform flow. Thisfact is exactly the same as in two-dimensionaltheory.

The actual computation of wave resistanceis performed by the Srettensky formula usingthe relation

(d. - 1) (45)

and the corresponding values of S2 are obtainedby interpolation. When a portion of elementarywaves near 8c = is reflected from the shiphull, the larger part of the energy in thisportion of elementary waves will be dissipatedby breaking waves, and the wave resistance willdecrease, but the momentum loss due to breakingwaves will increase. If such energy was con-

y 0.2

1.0 1-2 1.4 1.6 1.8 2.0 2.2

Figure 11 - Ray Paths of Stern Waves ofa Wigley Hull (b = 0.2, h = 0.0625)

12

MICHELL'S THEORY

"RAY THEORY

0.15 0.20 0.25

Figure 12 - Wave Resistance of CheWigley Ship (b = 0.2, h = 0.0625)

Discussion

Recently Eggers obtained a dispersionrelation from a low speed free-surface boun-dary condition which was a slightly modifiedversion of the one Babal8 used. Eggers showedthat there was a small region near the stag-nation point where the wave number becamenegative; since this was not permissible for awave and it could be interpreted to mean thatthere was no wave in the region. He suggestedthat his form might help alleviate the sen-sitivity of the initial condition to the raypaths. According to the energy conservationlaw8 of the wave propagating through non-uniform flow the wave energy flux is proportion-al to the wave number along the ray tube.When the wave number along a ray is considered,although the Keller dispersion relation has aninfinite wave number only at the stagnationpoint, the Eggers dispersion relation has aninfinite wave number in the flow near the bow.Thus, near such a singular point or a singularline, waves may break and the present theorycannot be applied. When Eggers 1 equation wasincorporated into the ray equation in thepresent ray computer program it was found thatthe ray path was still very sensitive to asmall change of initial value (x,y,tt) and thecurved ray still reflected from the ship hull.

In the experiments^ conducted at TokyoUniversity, the second caustic can be noticed,and near the second caustic the flow field isviolently different from the linear theory.However, it seems that extreme caution isneeded to distinguish the second caustic fromthe first caustic. The case of experiments-1

with wedges is interesting because, accordingto the present theory, there cannot be a secondcaustic on the wedge although the first causticshould be there. However, the wave phases ofthe wedge should be advanced and quite sensitiveto the draft- and beam-length ratios due to thenonuniform flow caused by the wedge. On theother hand, the present theory is still notexact although it takes into account the effecton the propagation of water waves of nonuniform

flow due to the double model. Thus, the non-linear effect of water waves is not totallyanalyzed here. Nevertheless, the presenttheory supplies a great deal of hope to anentirely different approach to the ship wavetheory - the ray theory.

Although the strictly linear amplitudefunction is used for simplicity in the presentstudy, a slightly improved amplitude functionmay be easily incorporated by adding thesheltering effect or other effects. However,it should be noted here that, in any case, thebow wave resistance and the stern wave resis-tance are the same as the results without theray theory and the effect of the ray theorywould appear as a shift of the phase of humpand hollow in the total wave resistance. There-fore, the computational results would not matchthe results of towing tank experiments unlessthe viscous boundary layer effect on the. sternwave or some other effect is considered.

Acknowledgements

This work was supported by the NumericalNaval Hydrodynamics Program at the David W.Taylor Naval Ship Research and DevelopmentCenter. This Program is jointly sponsored byDTNSRDC and the Office of Naval Research.

Appendix A

For the computation of ray paths of aship, the flow velocity and its derivatives on(x,y,o), u,v,ux , v ,Uy are needed. A Wigleyhull has the double model source distribution

b (-2x. + 1) il - fci)

= 0, h

-u (x,y,o) =2 j | m |^ (^) dXl dZj - 1o o

fh 1

=- 2 7 / h (7> d*i dz i - 1

= 2b j f ; i^T dz,I J r (x 1

= 1) 1

h i .

J r (x, = o)o

+ 4b

+ r(z, = h)

= 1 " 1

-j h log j x1

- x

13

( (xj - x) y^ 2 2o (z^ + y') I

h(x. - x)

r rj_i

it 2.

2 S(z, + y ) r

(xl

- X) ( *1 ' AdzJ

dx. dz

h -r f 2x - 3x + 1

yJ Li . 2. .

-

oUz

i+ y ) r(x

i "

d+Zj) - 2x - 3x + ^h r (x, = 1) h

2zt

r(x1=l) h '1

s2-x + 2z, 2 -I

^ dz(x^O) ti r(Xj =0)J

2x'-x + 2zf 2

r(:i

r(x,=l, z.=0)-u = -4b log

rCx^O, Zj=6)

+ =* [- Ms z. r(x.-l)

log I z, - r(x=l)(x-1)

1l^l "7 2

2 (te + ^-h

--^z^Cxj-O)ao

2

-j log |zj + r(Xj-0)|

-2J(i + !)log|

r(x =0) - r(x,-0)

2 2,2a = x + y

real hi,v (x.v.O) = 2 I I m-f- (

)

r r 32

/ 1 \J J ^ r -u (x.y.O) = 2 btV - dx, dz.on y 'J 3y3x \ r/ 1 1

-I + -I

--i-r(x =1)

=-2by

1

{(1-x)2+ y

2} r(l,h)

1

r(l'.h) h r(O.h) h

1,

(h + r(l,h){h + r(o,h)}5 log __i >h r(l,0) r(0,0)

+4by{ |

(1-x) 1 + 72J^ 1 +2 2\

z' + y* r(x -1)

i fz 2 +z, + y r (x,=0)

i r i-x ,"-2 J

dzi

hZ

r(Xl =l)

h

- J jnxTo)dz

il

,-K*?>x

14

(x.-x) y

o (z

i+y

)r

2 h3

|(Xl -x)2+ y

2| r(z x=h)

h + (z -h)

2 2 2(li + y ) (x^x) y

Z

1

^2y2+ (l-2x) (x.-x) h

6 by'3h" \ (x -x) + y r(z =h)

h +r(z =h) 2h5r(z =0) r(a =h)

2(x,-x) (l-2x)f

d2l

-

1 (z2r + y

2)r

2. Keller, J., "The Ray Theory of Ship Wavesand the Class of Streamlined Ships," J. ofFluid Mechanics, Vol. 91, Part 3, pp. 465-488 (1978).

3. Miyata, H. , "Characteristics of NonlinearWaves in the Near-Field of Ships and TheirEffects on Resistance," Proc . of 13th ONRSymposium on Naval Hydrodynamics (1980).

4. Ursell, F. , "Steady Wave Patterns on a Non-Uniform Steady Fluid Flow," J. of FluidMechanics, Vol 9, pp. 333-346 (1960).

5. Shen, M.C., R.E. Meyer and J.B. Keller,"Spectra of Water Waves in Channels andAround Islands," The Physics of Fluids,Vol 11. No. 11, pp. 2289-2304 (1968).

6. Yim, B., "Notes Concerning the Ray Theoryof Ship Waves," Proc. of the ContinuedWorkshop on Ship Wave-Resistance Comput-tations, pp. 139-153 (1980).

7. Inui, T., H. Kajitani, H. Miyata, M.Tsuruoka, and A. Suzuki, "Nonlinear Proper-ties of Wave Making Resistance of Wide-BeamShips," J. of the Society of Naval Arch, ofJapan, Vol. 146, pp. 18-26-' (Dec 1979).

8. Lighthill, J., "Waves in Fluids," CambridgeUniversity Press, Cambridge, London pp. 332-337 (1978).

9. Doctors, L. and C. Dagan , "Comparison ofNonlinear Wave-Resistance Theories for aTwo-Dimensional Pressure Distribution,"J. of Fluid Mechanics, Vol. 98, Part. 3,pp. 647-672 (1980).

10. Yim, B., "Design of Bulbous Bow by the RayTheory of Ship Waves," to be published.

v (x.y.O)y

wnere r(a,b) = r(x.=a, a =b)

In these expressions the integrals

are available in closed forms.

References

1. Inui, T. and H. Kajitani, "Study on LocalNon-Linear Free Surface Effects in ShipWaves and Wave Resistance," Schif f stechnik,Band 24, Heft 118 (Nov 1977).

11. Yim, B., "A Simple Design Theory and Methodfor Bulbous Bows of Ships," J. of ShipResearch, Vol. 18, No. 3, pp. 141-152(Sep 1974).

12. Havelock, T.H., "Wave Patterns and WaveResistance," Tran . of the Inst, of NavalArchitects, Vol. 76, pp. 430-443 (1934).

13. Inui, T., "Wave-Making Resistance ofShips," Trans, of SNAME, Vol. 70, pp. 283-313 (1962).

14. Lamb, H., "Hydrodynamics," Dover Pub.,New York, N.Y., p. 395, (1945).

15. Havelock, T.H., "The Calculation of WaveResistance," Proc. of the Royal Society(London), Vol. A144, pp. 514-521 (1934).

16. Srettensky, L.N., "On The Wave MakingResistance of a Ship Moving Along in aCanal," Philosophical Magazine, Vol. 22,7th Series (1936).

17. Eggers, K. , "On the Dispersion Relationand Exponential Variation of Wave Components

15

Satisfying the Slow Ship DifferentialLquation on the Undisturbed Free Surface,"International Joint Research on Study onLocal Nonlinear Effect in Ship Waves,Research Report 1979, Edited by T. Inui,pp. 43-62 (Apr 1980).

18. Baba, E. and M. Hara , "Numerical Evaluationof Wave Resistance Theory for Slow Ships,"Proc. 2nd International Conference onNumerical Ship Hydrodynamics, Universityof California, Berkeley, pp. 17-29 (1977).

i9. Moreno, M., L. Perez-Rojas and L, Landweber,"Effect of Wake on Wave Resistance of aShip Model," Iowa Institute of HydraulicResearch, Report 180 (Aug 1975).

20. Yim, B., "Theory of Ventilating onCavitating Flows about Symmetric SurfacePiercing Struts," David W. Taylor NavalShip Research and Development Center,Report 4616 (Sep 1976)

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