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Ship Wave Ray Tracing Including Surface Tension D. B. Huang Harbin Shipbuilding Engineering Institute Harbin, China K. Eggers University of Hamburg Hamburg, Germany Abstract The aim of this work is to clarify the validity of slim wave ray theories at and near the ship's surface. As previous numerical investigations have led to ambigu- ities due to a breakdown of the ray analysis near the bow and stern stagnation points, we shall take care for the surface tension effect in order to Olden such defi- ciencies; then the wave length never surpasses a positive nimlun length which is attained at the boundary of a finite waveless zone around a stagnation point. It is found, however, that the ray equations degenerate at these boundaries, and that rays can be traced into the far field only if their starting point is selected outside a finite belt surrounding the waveless zone. For a class of bi-circula~ prismatic struts of infinite downward extent we investigate two alternative formu- lations of the free surface condition and their implica- tions for the ray geometry. For low speeds we find in both cases an increase of the Kelvin wave cusp angle clue to capillarity. We exte~cled tile ray tracing to capillary waves ahead of a blunt bow. Introduction The wave field at a point far away from a ship in stationary motion is well represented through Kelvin's pattern, found in a wedge-shaped region, with only a finite slumber of wave components, given through wave length, wave front angle and complex amplitude. The first two are constant; along straight lines (character- istics) through a hypothetical origin, conceived as the locus of a point disturbance. Observat;ions suggest that under local modifications such a wave model may be adequate even near a ship; Ursell [1; hence generalized this approach for waves due to a point disturbance in a slightly non-uniform flow. Mom a set of physical assumptions, he replaced the intensity aloud direction of the uniform flow by the lo- cal components to obtain an analoguous spatially vary- ing "dispersion relation" between wave angle and wave number; from a partial differential equation he obtained "rays" along the resultant of the local flow with a group 157 velocity vector. l'o simplify the problem, Ursell consid- ered only rays passing through the disturbance, thought he admitted that his assumptions are questionable there. Inui and Kajitani t2] used this approach for waves near a slip's bow, with the " double body flow " as the basic non-uniform flow. Keller [34 derived Ursell's results from a more formal approach, tacitly assuming pertinence and uniform va- lidity of his ray theory up to the ship's water line; he even concluded for certain ships that rays must origi- nate from the double body flow stagnation points only. Yimi44 evaluated this approach numerically, but due to zero wave length at these points he had to start ray trac- ing using values at some distance. For certain rays car- rying transverse waves he observed that they re-entered the hull; to avoid this, he introduced some mechanism of reflection. Brandsma t5] investigated a class of bi-circular forms with varying entrance angle. Even with "backshooting" from downstream, he failed to find rays due to trans- verse waves originating at the stagnation points; he therefore concluded that no transverse waves can em- anate from the bow.We shall demonstrate analytically that the rate of change of the wave front angle along the ray together with the change of rater direc- tan has a factor as inverse distance from the 3tagr~atior~ points. Thus it demands more detai- led analysis to Claire fy the validity of Xel- ler's ray theory near the stagnation points. Through our present investigation we want to clarify whether the inclusion of surface tension effects can im- prove the situation at least to the point that ray theory can give some qualitative information about the wave pattern geometry in accord with experimental observa- tions for not so slender ships. We selected the class of bi-circular struts and thus have even the case of a blunt bow included. - l Otherwise rays could be extended to the domain far ahead through backward tracing, at least in case of a submerged disturbance.

The underlying analysis was presented by Eggers [6], where two alternative appoaches were considered: (A)' based on the conventional surface condition (A) of slow ship theory, supplemented for surface tension following Maruot7] and (A+)' based on a modified free surface condition, derived by Eggerst8] from certain invariance requirements for wave resistance, again supplemented for capillarity eEects. In both cases we obtain zones around the stagnation points where no steady waves can exist; at their boundaries, only waves of minimum wave speed, with wave front orthogonal to the double body flow, can occur. If we start rays from these boundaries rather than from the stagnation points, we apparently have a well defined initial value problem, even for blunt bow forms. - In our computational investigations, we could con- firm Maruo's experimental finding that capillarity ef- fects can be significant even if the remodel speed exceeds the rnirumum wave speed considerably. However, we found ourselves confronted with some instability phenomenon. Due to the strong rate of change of the wave angle along the ray near its origin, flee wave length re-approached its minimum value after a short time and the analysis broke down. To find rays which can be continued into the far field, we lead to se- lect the starting point outside some "belt of sI~ort-livity" surrounding the waveless zone. The neglect of 9(r <'fizz (the second term of a formally divergent Taylor expansion) leads to the approach (A)' investigated by Maruo, which was developed from the "conventional" approach (A) underlying the ray analy- ses of Keller, Yim and Brandsma. Let us now consider u and v as slowly-varying (i.e.locally constant) quantities and let us disregard effects of phase and of amplitude, as they are of no concern for investigations on ray ge- ometry. A potential of the form `~' or e(kZ-is) with 5 = S(x,y) (3) represents a wave with wave number vector k IS{k:, k2 ~{k cos 8; k sin 8) (4) if h~ and k2 are also slowly varying. Here ~9 is the angle of k against tile x-direction, Let us define the speed ratio q and the flow angle ,0 through u _ Uq cos,l? and v Uq sin,l3. Then By _ ~ - ~ is the angle of k against the flow direction. In accord with Brandsma and Yim, we have selected the orientation of k such that cosy is non-negative,i.e. k is , opposite to the propagation of a wave stationary to the ship.- ': u , 1 Id 7rec lion Derivation Of Dispersion Relation And pow {ancient :) Ray Equations From Free Surface Conclitions. For simplicity, we shall restrict ourselves to a 2-D flow around prismatic struts of infinite vertical exten- sion. Let us consider a velocity potential of the town U¢r + Up, where U stands for the far field uniform flow in the +x direction, U¢r represents the "double body flow" (unbounded in tile upward z-direction) and US is the lowest order wavy potential. With u U¢rr, v USA, with (r _ (U2 _ u2 v2~/2g arid Dr(~,y)(u~r)2 + overly, ~ for z = 0 has to satisfy MU u2(p + 211v~7 + v2('o + 9~(~0 + :(~0 ~ = 29~(r3 <fir + Dry Say ~ J~ Dr (X ~ Y) ( 1~) (see Eggers t104~; for inclusion of surface tension, a term n~ozzz has to be added on the 1 h.s. (see Maruo t74~. Seeking for wave-type solutions, we concentrate or tire homogenous part of above d.e.; we further disre- gard the (amplitude modulating)3 terms with dot and say. Hence we consider the "modified approach" (A+ )' u S°e ~ +2UVSO:ry +v (pay 9(r ($°3::r + My ~ = /`y~zzz9(~°z ('2) 2Quite recently,this approach has been justified under new arguments by van Gemertt94. Dee Longuet-Higgins and Stewart (11] and t84. k = VS ~ Fig.1 Sketch for flow angle ,B and wave angles ~ Andy (both shown with negative values, typical for the starboard side of the bow). Inserting (3) into (2) leads to (ku cos ~ + kv sin §~2 = yk + y¢r k2 + ~k3 (5) Denote c as the phase velocity with d~rec- tion Amos ~ ,-sin of. If we consider station- ary slaves only, c must balance with the proje ction of fine ~~sic flow I've in the direction of c, i.ee c= ~ cos ~ ~ v sin 6 = Og cos I. qua ation <5~ thus leads to the "d' sper~ion rela- t ion" c = ~/k + 9<r + ok (6) Equation (5) may equivalently be expressed as F(`x, y, k1, k2 ) - 1 - (k2 + k2 ~c2 /(uk1 + vk2 )2 = 0 (~7) Noting that k1 = ~835 k2 = ails, we may consider this as a partial d.e. for the function S(x,y) for which 4 Note that the term 9(r appears only under (A+)', not under (A)'; it may regrind us of c = ~/;h for waves on shallow water of depth h. 158

"characteristic stripes" (i.e. characteristic curves in the (~, y, k~, k2 )-space) can be found from the equations dz F dy F dk~ dk2 dr ~k~; dr ,~k2; dr = -F~ ; d = -Fy (8) which define a curve parameter ~ Under multiple use of above relations, considering that bI(uk~ + vk2) b(`uk~ + vk2) {tk~ = u; = v (9) ,~k ~ = cg cos8; ,9k ~ = cg sin O (10) (where cg d~kc)/dk is in accord with the concept of group velocity related to Uq cos ~y as phase velocity), we find cg = We obtain dx d~r In a similar way Fiom (6) we find and thus Fy dk (kc) = dk>/kg + k2g~r + k3~ 2k (9 + 29(r + 3k2 ~) = 2 ( 1 + 9(' +2 ) ( 11 ) 63F kc ~ Lc = = 2 ~k~ uk~ + vk2 ~k~ uk~ + vk2 =~ k ,9k (kc-uk:)= k (u-cg cOs§) (12) dy = 2 (v - cg sin§) (13j ~Z 2C 61Z 9(r (14) d k1 kc ~ kc 1;, = - = - 2- 2: dT uk1 + vk2 ~X Uk1 -t Vk2 - 2 <~ (kc - Ukl - Vk2 ) ~9(rukl vk2 ) kc dx c = 2 ((uu2 +vv=)/2+c(u~ cos §+v~ sin}))~15) = 2 ((UUy +vvy)/2-Fctuy cos §+vy sin §)) (16) Then we find the rate of change of the wave angle ~ from kc2 dd kc2 d k2 = arctan 2 dr 2 dr k1 2Ck (ki d - k2 dT ) (17) = _c2 (Fy cos ~ - F~ sin §)/2 = ( (9 (r )y cO s ~ (9 (r )3' s in §) / 2 + cu2 sin 2} - cv~ cos 25 ~ 18 where we have used u~ = -vy, uy = v2 for our 2-D basic Ilow. In a similar way we obtain5 2 dT 2k (k: dT + k2 d 2 ) ((9(r )2, cOs8+ (9l~r )y sin~q)/2 - cu~ cos 2cqcv~ sin 2§ (19) From equs.~12) and (13) we can easily confirm the gen- eral result tan(3 + cx) dY = g ~ (20) (witI~ c~ defined as the ray angle against the double body flow) which contains the choice of approach only through the explicit expression for cg. We may thus re- call Ursell's observation that the ray direction is along tI~e resultant of the basic flow a~d the "group ve- locity" taken along the wave normal vector -k, and that this does not require asymptotic analysis such as tl~e principle of statio~ary phase (see discussion tot84~6 ~ Cat k 7~+)~ ~< // ~+~ Uq Fig.2 Ray direction as resultant of basic flow and cg along direction of -k (with angle c~ against flow direction). Restrictions For The Wave Parameters. From the dispersion relation (6) we have c2 = g/k +9(r + /ck (21) 5Tlle terms with 9(r (Iriissing under approach (A)') reflect the statemellt of Longuet-Higgins and Stewart t11] that short waves superposed a long wave shorten wileIl clirrlbiIlg, increasiIlg their length again when descend.illg. GFor later use, we have introduced in Fig.2 an "ac- tion transl~ort velocity" cat along tile tangent to the ray direction. 159

A minimum of c is found at k = >/577 giving C = Cm at +2~1 - 92 )/p2 = U:( 1 - q2 + 2p2 )/2 (22) where c'', = .,,~'4'cg is the minimum velocity of capillary- gravity waves and pcm/U is a dimensionless pararn- eter of surface tension. We introduce a dimensionless wave length Ag/(kU2~; then (5) is equivalent to q2 COS2 ~ = ~ + (1q2~/2 -F p4/4A (23) In a plane of the variables q2 and A, for p constant;, (22) represents a family of hyperbolas between the asymptotes A = 0 and ~ = ~(1 + 2cos2-yjq2 - 1/2. We are interested in the branch with ~ > 0; (other- wise,we would have an increase of the wave flow down- wards, seed. Solving for A, we obtain _ (1 +2cos27)q2 - 1 1 TSq (24) r Sq - 71 - (2p2 /(2 COS2 ~ + 1 )q2 _ 1 ~2 (25) Sq iS real only for q2 ~ (1 + 2p2~/(2cos27 + 1) > (1 + 2p2~/3; this implies that in the zones around the stagnation points where q2 < (1 + 2p2~/3 no steady waves can exist. The upper sign of the root corresponds to gravity dominated waves (A = Ag ); the capillarity dorrunaled waves (A = En, lower sign) are better described by7 4 (1 + 2cos2~)q2 - 1 i ~ Set (26) For any A, confluence of the two roots occurs for A = p2/2 corresponding to the minimum of c fouled earlier (21~. For the range (1 + 2p2~/3 < q2 < 1 + 2p2 the angle ~ is restricted through Act< 2 arccos ( 2P _ 2) (27) (under approach (A), fly I< arccosp, independent from q2~. Beyond this range, for q2 > 1 + 2p2, the minimum of Ag is no longer p2 /2 but the value corresponding to = ~2. Then we have O2 _ 1 1 + `/1 - (2p2/(q2 - 1))2 Q2 1 ,~_` 2 If we accept the argument that stationary waves can- not propagate into areas where Cg/C is negative, the do- main of admitted ~ values is further restricted (see(11~) through 2ccg = A + 1 _ q2 + 4PA > 0 (29) . 7The relevant analysis has been established and pro- foundly discussed by Crapper t13] for tile no-modified approach (A)' 160 ,.% . v ~ - P~Cnn/U~ O . ~J6 / ~ ,? (em o,7 An to 9,,,# ~ ; "' Ash ,,~ C._ ~ 2 or (; ~ ,~ 1 .G 1. ~ t 1 4 1.(~ 1 63 q 2 ~2 ~= 3 Do nain D calf Admitted Waves for p= 0. 46 ( i. e. U For gravity waves with q2 > 1 + ~p2 this implies A > Aid _ (q2-1) (1 -F >/1- 3p4/(1 _ q2)2) /2 (30) One may observe that this limitation is automatically met if ~ ~ ~ < ~r/4 with dA/dq2 > 1/2 + cos2 ~ > 1 then (see Fig.3~. Ogle inky wrote trial for any By we field from (22),(25) and (5) (C/U)2 = fig + ~ + (1 - q2)/2 (31) All the above restrictions can be visualized through a display of the dependence between the wave front an- gle 7 and the ray direction angle cat with p and q held constant. Mom a geometrical interpretation of (19), in- voking the sine theorem of elementary trigonometry (see Fig2.), we find sin a _sin(7 ~ ~) = _s~n) (32) and hence tang = singly = singe ` cos ~Uq/Cg 1 + cos 272C/Cg Setting p =°,~-l, we have Cg/C = 1/2 in accord with Kelvin's results; we find that act ~ will increase with ~~ ~ up to some maximum C:tk = arctan (1/~) and then fall off to zero with ~ = 7r/2. We may observe that for non- zero p, unless q2 > 1, ~ approaches zero only together with A, as cost will remain positive. Thus an outgoing ray act > 0) can turn inward again only if the wave front normal changes from inward (y ~ 0) to outward at the "caustic" (in the terminology of Yim [174) under a maximum of the wave length due to ~ = 0.

One may consider the range of negative car in order to leave By positive. If we exclude here those parts of curves where ~ turns positive due to cg < 0 (only for q2 > 1 under (A+)'),we may observe that ~ as an odd function will in general leave opposite sign to By. Let us refer to the range for which ~ cat ~ is increasing from zero with ~ ~ ~ as to that of transverse waves and define the maximum value attained for ~ in this range as the modified Melvin angle (Xk. Then we find treat for q2 < 1 -f 2p2 each curve for gravity clominated waves turns unpiler horizontal tangent ~ witI-r ~ = ~r/2 q- :) into one for capillarity dominated waves, so that after tan cat changed sign due to increase of Cg/C > 1 for such waves, falls oil: to zero again with the ray finally normal to tl~e wave crest again, but opposite now to the flow (li- rection. Note that our forrnulatio~s and considerations tI~rougl~out this paper are refered to the domain around the starboard side of the bow, where car is positive, hence By negative in general. To deal with the other ship side, statements remain valid if ~ and ~0, By and car are counted clockwise against the x-axis and flow direction there. Note that, observing 4kn/c2 = p4/~4q2 COS2 7) in (11), we have the deviations of Cg/C form 1/2 depending on p2 and on q2 even under approach (A)', where tile term (l q2~/'2 is disregarded. Cg/C arid Thence ~ c' ~ increases wills cle- creasing speed U (i.e. increasing p) and with decreasing distance from the stagnation point (i.e. decreasing q) for ~ held constant. This implies an increase of the mod- ified Kelvin angle (which is measured against the now lirectio~!) especially near the bow, in particular under approach (A+)'! This is well in accord with the exper- imental observations of Miyatat14] with wedge- shaped models with U = 0.5m/s, (p ~ 0.462) and U = lrn/s, (p~ 0.231) . Away from the bow, where q2 ~ 1, Claus ~ ~ 0, there is little difference between (A+)' and (A)'. But again we observe an increase of Cock with decreasing U in accord with Miyata's experiments with a rudder model (see Inui t15~) for the speeds U = 1.15m/s, 1.72m/s and U = 2.3m/s corresponding to p = 0.3, 0.14 and 0.1. Irt the domain where q2 > 1 aside of the ship, a re- duction of Elk iS predicted under (A+ )' including a ter- mination of rays with short gravity waves with cg ap- proaching negative values. We may retention that for a vertical circular cylinder, q2 increases up to 4.0, whereas for conventional forms q2 W]1 not exceed 1.2. It is only for not too small q and for not too large p that car is really stationary for ~ = Ark and talus marks a transition from transverse to divergent waves, so that with do/d: = 0 we may expect a wave cusp effect. Let us search for the most forward point along the water line of a ship where stationarity occurs (i.e. irnmedi- ately near the stagnation point for pointed bows) and trace a curve from there with dy/dx = tamed + Sky; tibia s~ cold ~efii~e so.-,e cusp line. ~ 9oo | q2 = 0.5 (near bow) | capillary wage! /;wave -90' MU = 0.5m/s,tU = 2.5m/s (both(A+)'),- 3 - (A)'. | q2 = 1 (far field) | capillary wav;s /gravity waves · , -180° ~~ 90° t · -r ~ U = 0.5m/s,- ?-u = 1.5m/s 3 U = 2.5m/s For q2 = 1 there is no difference between (A)' and (A+)'. ·/ cg/c negative! .2 1 q2 = 40 (aside of a thick body) l / ,.~,. / / ~ - it. . . capillary waves /gravity waves . / 180° 'am -90° (A)', .~.(A+)' (only negligible dependence on p) Fig.4Wave front angled versus ray angle cY 161 9oo t

or ALPHA . v ~ C:AMMA p=O.6 _ O 5A>1~dA ~ 2- - ALPHA . v 5 . GAMMA P=0.4 it/ ,' of / ~ Z Cl/ / ~ ;~ ~ ~ ~=~ GAt-1~1A Fig.4 (a) a. vs. Y (for(At )') curves over the dotted dine for capillary waves; curves under the dotted lisle for gravity waves. ALPHA . v 5 . GAMMA Fig.4 ( b ) a . vs. Y 1 1 -se. oo So. oo GAMMA 162 ~ rr ALPHA . v s . GAMMA p=0~1 rTr 2 .~, '/ ~ 1 Hi/./' r 67Ji I" ~,.~ C~^ 77 2 n ( for ( A) ') .. . Or ; O

The Si equation Of Ray Tracing Near A Corner And The Short Life Of Rays Near The Waveless Zone. Let us again consider 2-D potential flow as the basic flow, so that complex analysis can be used.We introduce Zcc+iy = r·ei~yu-iv = Uq epic, (34) Kkit - ik2 = k e-i (35) and P(9(r )2i(9(r by =(dV/dZ)V (36) The P stands for some gradient of the double body flow pressures We can write the first ray equations (12) and (13) as dZ 2 TV _ cs K' = 2 cat · e-i(+) (37) With ds as differential of the arclength along the ray,this implies that dr/ds = 1/ ~ dZ/dr I= kc/2ca`, so that we can write ( 19) and ( 18) as 1 dK {1 dk .d~ i\ dr K ds \<kdT-2d~J dS (38) = _e .dV (OV +ei8> Cat dZ ~ 2c ) Here o means 1 for (A+) and (A+)', it means 0 for (A) and (A)'.- The flow in the vicinity of a stagnation point due to a corner is basically the flow near a corner be- tween infinite planes as decribed by Milne- Thomson t12], we have V ~ Q . e(i'r~o /(~-~o ) , ale /(7r-~ where Q is a real constant; this means that the range To < ~ < 7r for the polar angle ~ is mapped on the range To > p(~) = hour - b)/~7r -,60) > 0 for the flow angle. In the special case of a bi-circular strut of opening angle 2po and length L, under parallel how of strength U. we have Q = U ~ . L - ¢° /(7r-~0 ) (40) so that q = qua) = 7r/~,rp0) (~/l)(~°/(~-¢',) and hence dV pa V _ = . _ dZ A- po Z (41) A=_ (°p zV V (42) 6Tulin t16] considered a quantity related to ~ P ~ as a (1isturbance parameter and came to the vexing conclu- sion that ray theory does not apply for bow entrance angles,B0 < 7r/3 as otherwise P is not bounded; on the other hand, Maruo (7] disclaimed the validity of ray theory for To ~ ~r/3 due to divergence of an integral representing the phase. 'tureen the rate of change of §,,L7, and k along the ray is found from 1 dK em . = _ . . K ds cat ~ -'Bo Z (°2c + e ) (43) Observing (34) and (5), separating real and imaginary part ilk (43), we find ds Or can 7rTo (O 2c~q bit + singe+ §_ it) (44) dip _ = ds and thus . . -Imdz V d = -Im [a 1 . ei(~+~) 1 . ~° . sin (a + ,B - 5) (45) d5 d6 = r 7r _ ~ x ... (46) (sin(a + ~b) ~ c (A (2+ ) + sing + ~t)) ) Ihe to the presence of the factor 1 j r, which tends to infinity at the stagnation popery, that might c use trouble when doing rater tracing, near the boy'. minis may explain the dile.ma of ~3ra~ds- m" ( ta7cin,~ 0 =0 ) In order that a ray should travel along the ship's wa- ter lisle (or more generally just coincide with a stream- line), we must have ax O i.e.: O hence da/as i.e. d:/dsO. It is not clear why this should be possible for general hull geometries, as for cat = by = 0, hence = l), we find with cat = c - cg ds Imp (c - c ) e2i¢] (~471) valid even away from stagnation points.- For the rate of change of the wave number k we find dk 1 Uq To (Ocos (~+ ~ 5) +cos 2+,- cis r cat 7rTO `< jocose (48) If the value of ~ along a ray should equal the critical value p2/2, this would correspond to the minimum for gravity waves; hence k then most decrease along the ray. However, due to the rapid increase of by near a stagna- tion point, the sum of cosine terms may change sign, so that k increases (in particular for (A)' where the first cosine term is deleted) and ~ approaches p2/2 again. Here the ray must terminate, as for A = p2/2, even off the waveless zone boundary, the partial derivatives of ~ both regarding q2 and ~ vanish simultaneously in conflict with the ray equations, q2 can not be varied independent tromp. This explains the previously men- tioned occurence of short life rays. Thus the choice of initial points for rays is moot, quite apart from the a~n- biguity of assigning initial values there for amplitude and phase. 153

Further Considerations About The Ray Approach - A "ray" i:~ the sense of our analysis is defined as a characteristic to a partial differential equ action F(~,y,S2,Sy) = 0 for a function S(x,y), seem. But we should ascer- tain that the essential features of the complex 3-D flow near the ship, including sensitive variations, can really be modeled adequately through functions S(x,y) with slowly varying gradient and associated complex a~npli- tude functions A(x,y). Note that until Yim's t17] re- cent investigations, no ejects of the Froude number on ray curvature could be modeled, and the variations of wave resistance only resulted from interference ejects in tlte far field computed through integration along the rays. Nevertheless, our in- vestigation showed that certain global characteristics of the wave pattern, such as the variation of Lark, and hence of the tangential direction of the wave domain bouncl- ary (visible in the rich stock of Miyata's experimental results) can be predicted even near flee bow with ap- proach (A+)'. An evaluation of merits for the competitive ap- proaches (A)' and (A+)' may be attempted. But it does not seem pertinent to discrim- inate between a "correct" and a "less consistent" ap- proach, although it is obvious that with inclusion of sur- face tension effects the rule of "automatic order change through differentiation", essential for (A), can not be maintained. Actually, the omission of terms with 9¢,r (and hence with 1 _ q2) under(A)' has no fundamental consequences for our analysis in general. Certainly, the extent of zones without steady waves ahead of a blunt bow is consider- ably larger under (A+~; well in accord with data from experiments with a vertical circular cylinder, for which we have evaluated both approaches. However, the nu- merous recent investigations on the flow ahead of a blunt bow (see the survey by Mori t183) make clear that be- tween the bow and the stationary capillary wave zone we have to expect a finite domain with either a stationary plateau, a turbulent free surface or instationary waves propagating forward (Osawa t193), and the flow visuali- sation experiments of Kayo et al t20] display a system of instationary "necklace vorticies" in this domain. And the decay of capillary waves through viscosity, as inves- tigated by Messick and NVu t21] should be considered. Numerical Calculations. Our calculations have been performed both for ap- proach (A+ )' and approach (A)'. We considered the class of bi-circular cylinders which had already been in- vestigated by Brandsma with conventional ray theory. The analytical expression for the velocity potential W is given in complex notation t12] ilk terms of bicircular coordinates ~ and ~ through Z = x + iy = L/2 cot(~/2) (49) W = ~ + ilk = UL i/ncot(~/n) (50) with ~ = (+ir~, ~ = b~ -52, rig = In(<r~/r2~), n = 2(7r7r/30)' where ,60 stands for half the entrance angle and L for the length of the strut. The symbols rI,5l,T2,{2 stand for polar coordinates regarding the strut end points. For economical reasons, we deduced explicit expressions in real mode for u, v and their derivatives. We evaluated the ray equations by the Runge-Kutta method. ,'5 r, ~ L ~= ~ Fig.5 Bi-circular coordinate system around a bi- circular strut. The aforementioned boundary of the short-life zone was determined numerically, assuming q2 = const there. It did not require much accuracy, we found that rays emanating outside such a border line were not sensitive to the choice of their origin. We selected ~ = 0 as initial value, securing a maximum for tl~e wave length and its rate of increase, hence minimum probability that it may become stationary and decrease again along the ray. Thus it is obviously adequate rather to operate with a continuous distribution of disturbances than with a concentration in the stagnation points. Due to inclusions of surface tension, steady compo- nents of bow capillary waves could be investigated as well. We traced rays with capillary waves from that part of the outer boundary of the short-life belt where the basic flow is incoming, and rays with gravity waves from the part with outgoing flow; the two domains have no common boundary, save the point where the basic flow is tangent to the belt. 164

Thc lays of gravity- find c.tpill`~rity- waves in the civics of different body advance speeds and entrance angles arc shown in F-ig.6, 8, 9, and Fig.10 To show the difference Hewn ours and the conventional ray Amy and ~ pan of char Run, fcwar~l- (dcw~nun) and backward- ( upstream from the far fickl ) tracing based can convcnticx~l ray theory are also pcrfonnod. For the latter, considering the uncertainty due to the stagnation point, rays arc backtr.tcod from the far fickl towards the bow, iterately, by changing initial conditions, the results in the far field by usual ( forward tracing ) method being used as the first set of which, till ray reaches a given- sized ~,cighborho~xl of bow stagnation point, and wave angle converges at the same time. the results are shown in Fig. 12. In all the figures of lays prcsentod, the pairs of members attached to each ray give ~ ,. arid ~ ~ the initial arid filial values of ~ ( in degree ) . Thc short segments on the rays show the local wave fronts. To get better accurracy, step widths in ray tracing with Runge- Kutta method were carefully chosen, and the reliability was checked by halving the step widths. Conclusions Ill the course of an assessement of the ray approach with regard to representing essential features of the wave pattern, we have incorporated capillarity ejects in our analysis to overcome obvious shortcomings near the hare stagnation point. ~~_..~,.e," ,&~A ~,`~ =w w Our formulation AL+ )' (eq. (2) ~ generalizes approach All', where certain terms related to the doubl~-tody flow pressure are disregarded. Although within our work we could neither provide a rational model for ray generation nor even a justification for extending the ray approach the hull surface vicinity, the following facts have been discovered or confirmed. Our numerical investigations have displayed several global effects on the wave pattern geometry result- ing from the inclusion of capillartity to our analytical model; they gain practical relevance for small speeds, say for U=2 m/s (if we consider a minimum capillary wave speed of 0.23 m/s.~: (1) Both the far-field Kelvin angle and the "modi- fied Kelvin angle" near the bow (i.e. the angle between tangents to the wave region boundary and to the hull water line) are found to increase with decreasing U. (2) With increasing bow entrance angle, both the zones of no steady waves and the surrounding short-life belt, from which no rays proceeding into the far field can be found, grow in size. (3) The outward extent of this belt is decreasing with increasing U. i.e. the stronger the capillari.Ly the lar~,er the snort life belt. The above findings are in qualitative accord with some tendencies one may observe from experimental vi- sualizations of flow and wave pattern as presented by Inuit15] (his Fig.2-2 is reproduced in our Fig.7), of Miy- atat14], of Maruot7] and of Osawat19; (see Fig.11~. It is true that we can not expect our analytical model to cover all features of the complex phenomena observced, ejects of viscosity and finite wave elevation in particu- lar, though the latter may be assumed to be less signif- icant considering the low speeds of the models. Thus it seems that in this regard ray theory displays a certain value for predicting ship wave phenomena, although the re-entrance of rays or their reflection at the water line must be considered an open problem, among others. In any case, the authors would like to emphasize the need to take account of surface tension at low speeds, well in accord with Maruot74. We hope that our work reported here can add some further weight on this aspect Acknowledgements The authors express appreciatiol! and gratitude to the 13eutsche Forscb ungs- und Versuchsanstalt fur Luft- und Raurnfahrt for sponsoring the first author's one year research fellowship at IfS Hamburg. Our thanks go to faculty and staff of IfS for all their kind assistance. The authors would like to dedicate this paper to Prof. T.Y. Wu to the occasion of his sixty fifth birthday. 165

References 1 Ursell,F.:"Steady wave patterns on non uniform fluid flow."J. Fluid Mech.Vol. 9 pp 337-364 (1960) 2 Inui T. and Kajitani,H.:"A study on local non-linear free surface effects in ship waves and wave resistance." Schiffstechnik Vol.24 pp.l78-213 (1979) 3 Keller,J.B.:"The ray theory of ship waves and the class of strea~rdined ships."J.Fluid Mech. Vol.91 pp.465-487 (1979) 4 Yim,B." A ray theory for non-linear ship waves and wave resistance."Proc.Third Intern. Conf. on Num. Ship Hydrodynamics Paris pp.55-70 (1981) 5 Brandsma,F.J.:"Low Cloudy number expansions for the wave pattern and the wave resistance of general ship forms." Thesis T.U. Delft lllpp.~1987) 6 Eggers,K.:"On stationary waves superposed to the flow around a body in a uniform stream." IUTAM Sympos. on nonlinear water waves, Tokyo. edt.by K.Horikawa and H.Maruo. Springer Verlag pp313-323 (1987) 7 Maruo,H.: and Ikehata,M.:"Some discussion on the free surface flow around the bow." Proc. 16th. Symp. On Naval Hydrodynamics Berkeley pp 65-77 (1986) 8 Eggers,K.:"Non-Kelvin dispersive waves around non- slender ships."Schiffstechnik Vol.28 pp.223-252 (1981) 9 van Gemert,P.H.:"A linearized surface condition on low speed hydrodynamics."Delft T.U. 26pp (1988) 10 Eggers,K:"A comment on free surface conditions for slow ship theory and ray tracing." Schiffstechnik32 pp42-47~1985) 11 Longuet-Higgins M.S. and Stewart R.W.:"Changes in the form of short gravity waves on long waves and tidal currents"J. Fluid Mech.8 pp566-588 (1960) 12 Milne Thomson:"Theoretical Hydrodynamics"MacMillan 13 Crapper,G.D.:"Surface waves generated by a travelling pressure point."Proc.Roy.Soc. A 282 pp.547 558 (1964) 14 Miyata,H.:"Characteristics of nonlinear waves in the near field of ships and their effect on resistance." Proc. 13th Symp. on Naval Hydrodynamics Tokyo pp.335-353 (1980) 15 Inui.T.:"Fiom bulbous bow to free surface shock waves- liends of 20 year's research at the Tokyo University." The Third Georg Weinblum Memorial Lecture. Journ.Ship Res, 25 147-180 (1981) 16 lulin,M.P."Surface waves from the ray point of view." The Seventh Georg Weinblum Memorial Lecture. Proc.14th. Symp.on Naval Hydrodynamics Hamburg pp.9-29 (1984) 17 Yim,B.:"Some recent developments in nonlinear ship wave theory."Proc.Int.Symp. on ship resistance and power performance, Shanghai. pp82-88 (1989) 18 Mori,K.":Necklace vortex and bow wave around blunt bodies." Proc.15th Symp. on Naval Hydro- dynamics Hamburg pp303-317 (1985) 19 Osawa,K .: "Aufmessung des Geschwindigkeitsfeldes a und unter der freien Wasseroberflache in der Bug- umstromung eines stumpfen Korpers Bericht Nr.476 Institut fur Schiffbau der Universitat Hamburg,125pp(1987) 20 Kayo,Y.,Takekuma,K.,Eggers,K.and Sharma S.D.: "Observation of free surface shear flow and its relation to bow wave-breaking on full forms."- Bericht Nr.420 Institut fur Schiffbau der Universitat Hamburg,33pp(1982) 21 Wu,T.Y. and Messick,R.E.: "Viscous effects on surface waves generated by steady disturbances. Rep. 85-8 Engg. Div. Cal.Inst.Techn. 31pp(1958) 166

9p, idAeF ENrRA^~£ ~E A. .~2. 5 o o (a) 8 S ~;' 0',60' 01,9~1.1''l'001 ,1,20' l ~ ~ ~_ ~ 0~ ( b ) -0 o o _ ~F ENrFA~= ~' 4. -22. 5 l - 1 _ 1 ~1 1 1 1~- (d' t o G Fig.6 Rays of gr~lvity W`IVCS for cl bicircLIlar cylin~icr with half a~trance angle Ae= 22. 5°, in cases p= 0. 6, O. 4 and 0. 1, corresponding to U= 0. 39, O. 58 and 2. 31m/s respectively, for Gn= 0. 231m/s. (a), (b), and (c) are based upon (iL+)'; (d) is ba~ed upon (A)'. ~ tracing i ~ s topped i f the ray ent ers the bo dy ~ · 60 2 c ~-W \ . ~ '~_ ~ ~a~ ~ ~ p.O.1 Fig.7 Wave pattern of a rudder model of length ().3II with U = 0.65' 0.5 and 0.34 m/s (by courtesy of Prof. T.1nUi from [15]) (Iilor this rudcler, Pn30.4' p=Oe33; I?n=0 e3 ~ P=(). 45; F.n-0.2' p=0~66) 167

pa D.37 6 0 0 In ~ p=o.25 rat to' ,,_--~~~~~--_,~ o - o p= o. 37 jp=o,25 Fig.8 Rays of bow capillary waves for different cntrancc angles and body speeds. ( based upon ( A+ ) ' ) ~ U= 0.62 and 0.92 mj~s ) D o o o c, 168 / ~ 16.],.= 3, te. 1. ~ ~ ·/ ~2~ ~ D1 _: ;F fi~ r ~N-~ r 3: ~. .J J ^ ~ ·~4~2~ ;~~~;;\> ~ ~ ~ = r~ '~ C. sa f. - o - o p(c-/U1~0.~00 ~ = `) 4 ~ (l b 29 5 · · ~ ]= ~ PtC-'Ul'0.100 ~ = 0 ~ Fig.9 Rays of gravity waves for diffcrcnt entrance angles and bc)dy speeds. ( based upon ( A+ ) ' ) ~ l,r= 0.58 and 2.31 mys)

~::? C j a:::` Fig.10 Rays of bow c~piLary awes in Wont of a cir- c~ar cylinder (moving to the hO) under approach (~' U = ~ ~ ~ ~ Fig.l] Sow waves ~ front of ~ lacing cylinder with dinette D= 0.~ m' moving to the left' at the speeds of U=O.61 Oe7 ~ 0~9 m/s (From Ee OBEY L191 169

o °_ HALF ENrRA~E A~f A, - 22.S o o~ srAQr ING P r . OF RAYS. I XO. YO I <0.0000)0,0050) (a) o O l 20 0 ~ ~ ~ -i: .sl O o · 5 2; ~ - ~ ~ / _~ A _ ~ \ -es.: - o 00 0. 20 0. ~0 0.60 0.60 ~ . 00 1 .20 ~ . ao ~ .60 ~ . SO 2. 00 X o ° - HALf ~NTRA~E A~ Ac-22.= ~ACKV - OS-TRACI~ ~ RAYS _ 67 ~~ , -8t .7 ~ ,_- O 66 t8 6'27 23 l - - ,' _ ~ '/ g o t0_ o >~ - o _ o o . _ o . 1 _t 01 o -o ~' ' ' ' 0 60 ' 0 80 ' 00 ' '.20 1.40 \.60 '.eo 2.00 'o' 26 ~-~2is . ~ P=0 .4 ~ --;~ I -1 1 ~_=: -------- X 1 . 23 1 ~ ~ 60 _° ~_ ~ i I I I I I - 1~.80 _` \ ~ ~ 209'S7~2;5~9 272 3 //~ 4 s; 34 '3 ¢.//// 3 ' .~ - t ~ // 1C a ;~7 O -' Fig.12 Rays traced with ap- proaches based upon conven- tional ray theory: (a) Forward~downstream~tra cing; (b) Backward (upstream) tracing from far field. Rays ~hat enter the body have been removed ) Fig.13 outer boundary of rayn of gravity waves, traced with dy/dx=tan (~+: ), based upon ~A+; . Il.B. it might not be the at- t ainable boundary of rays of gravi ty waves; and, it is not a boundary between ray families of gravity- and capi llary- waves, p=O.1 ~ 170

o . ~ r n ~ ~ I 1 nl,,il ~ 1,nl ~ l / GoO,~,2O,: 1C.O. 55 7 ~ t t . · -57 . 9 '/ o 1 o - o o 8 .~ og o og 24 ~ 2.'i=.'~'-i~ ~2~ '''''"-- . ~0.2O 0.40 0.60 o.6~L-l" lToo ,1 Ir20 ~ 1r40 ~ 1r60 ~ r 1 - -a ~~ PtC.~]'0 1nn _~' ''-a. , . . . . ~ . . . . . . . O ~ I I · , , . , , , ,.-, . . . O ~? O.bO 0.8~-' 1.00 1.20 ~ is shown negl gable. .~. - .. _O .0 '. ..; O' a/ Pt c. But 0. 100 Fig,15 Rays of bow capillary waves (based upon (A)' ~ Fig.16 Rays of gravity waves (based upon (A)' ~ . . .0 · .60 Fig.17 Precision check for the tracing technique, the upper curves are obta- , , , ined by halving the step width used 1 . SO 2 CC for the bellow curves, the difference 171

DISCUSSION by H. Kajitani 1) I suppose the ray tracing is a kind of low speed theory. I'm not sure that a pretty high Fn applied in Fig.7 is available or not. 2) Could you comment on what difference can be observed on the traced characteristic lines between with and without surface tension effects? 3) The wave length of capillary waves in front of a ship bow changes with its distance. Have you computed the capillary wave phase? Author's Reply Prof. Kajitani's worrying about applying ray theory for high Fn is certainly natural. We use Fig_ 7, the highest Fn is 0.4 there, (from Inui and Miyata) to show the qualitative confirmation with the test results We don't think that ray theories (at present) can predict strong non linear effect. Keller[3] claimed that ray theory may be useful for Fn <0.7. We are more conserva- tive in this regard. As to the differences between those with and without surface tension, they could be listed in Table Al. We have not yet calculated the capillary wave phase. It could be carried out through integration. DISCUSSION by H.S. Choi First of all, I would like to congratulate the authors that the surface-tension effect Table Al Conventional ray theory (no surface tension) 1 Point disturbance, all rays are from stagnation point; Stationry waves exist even near the stag- nation point. 2 Ray and wave patterns are inde- pendent of body steed U. has been successfully included in the ray theory to clarify the wave pattern around the bow more clearly. It may be more usefully applied to a small-scale models. If it is the case it is possible that the local phase velocity reaches to the minimum celerity of capillary gravity waves (=25 cm/see) and the capillary wave breaks. It implies that a new source of singularity has been invited to your method. I would be happy if you comment on it. Author's Reply Thank you for your comment. If the wave length of a gravity wave is decreasing when progressing, the wave might break before the local phase speed c reaches the local minimum phase speed cm. In our approach, we start rays of gravity waves from the short-life-belt boundary, where exist the shortest omitted gravity waves. The waves seem to become longer when propagating (cf.(48), which shows that near the bow, 1/k dk/ds < 0, that means increasing wave length along rays ). If c<cm does happen somewhere, our program will treat the corresponding ray as short life ray, terminate the tracing and enlarge the short life belt. We thus have not that kind of new source of singularity. You may be right to consider these new singularities. But I wonder if they would cause only secondary effect. DISCUSSION by A. Hermans I congratulate the authors with their interesting extension of ray theory. I agree with them that in the region of very short waves (near the stagnation points) surface tension is dominant and that its influence on the ray pattern is seen in the whole field. It Present ray theory ( include surface theory) 1 Region disturbance, rays that can reach far field are from the boud- ary of that region; No stationary wares can exist inside of it. ** 2 Patterns are dependent on U. Change tendencies of size of short life belt, local and far field Kelvin Angles etc. are in accord- ance with experiments. In Fig.7, the rudder being small, Fn=0.4 corresponds to a "slow" speed in the scale of minimum phase velocity Cm (in uniform flow) Be) Observing carefully the region right in front of slowly advancing body of blunt bow, say a circular cylinder as shown in Fig.11, one could find (in some region of body speed) in stationary waves between the stationary capillary region and a turbulent region close to the bow. What we considered was only the stationary waves. 172

makes the mode for the ray pattern more accurate than the one described by Brandsma and myself. It is a pity that the authors do not say one word on the influence of the wave excitation coefficients and the corresponding wave amplitude. It is my philosophy that one must try to balance all components of the building. To my opinion one approach has such a balance at its own level. Do the authors expect that our approach to the amplitude problem is applicable in this case? If so, do they expect that the influence of surface tension is noticeable there just as well. Author's Reply Thank you for your congratulations and comments. The aim of this paper is to find out if surface tension is taken into con- sideration, the ambiguity and difficulties of the conventional Ray theory, as found by many others, can be overcome. We would not blame an existing theory for not being perfect. On the contrary, we appreciate every effort made by previous authors who developed ray theory and made it possible to apply it in practice. In view of that some important features of the real world can not be predicted with conventional ray theory, we think that some improvements may be necessary. We have not yet calculated amplitude. Our concern in this paper is on ray pattern. Our results show that surface tension may not be disregarded for slow ship problems, at least in small scale cases. Even if it turns out to have no significant effect on the final results in some cases, it can still be used as a way to circumvent the difficulties in ray theory. From the viewpoint of validation, the assumption of infinitesimal wave length at the stagnation point is always an unpleasant thing. We tried to get rid of it. 173