THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING ON SECOND-ORDER WAVE THEORY FOR SUBMERGED TWO-DIMENSIONAL BODIES Nils Salvesen A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Naval Architecture and Marine Engineering 1966 April, 1966 IP-736

Doctoral Committee: Professor Finn Co Michelsen, Co-chairman Professor Chia-Shun Yih, Co-chairman Associate Professor Walter Ro Debler Assistant Professor Jack L. Goldberg Assistant Professor Horst Nowacki Professor Raymond Ao Yagle

ACKNOWLEDGEMENTS To the members of the doctoral committee I wish to express my deep gratitude for their guidance. I give special thanks to Professor Finn C. Michelsen and Professor Chia-Shun Yih; not only for their unending inspiration, encouragement, and assistance in the course of this work, but also for their dedication to teaching and their personal interest in me throughout my nine years at The University of Michigan. My appreciation is also extended to Professor R. B. Couch and other staff members of the Ship Hydrodynamics Laboratory at the University for their assistance and cooperation. Furthermore, I am grateful to the University Industry Program for the fine preparation of the material for publication. Thanks must also be extended to the following institutions for granting me fellowships: the Horace H. Rackham School of Graduate Studies, The Society of Naval Architects and Marine Engineers, and the Norwegian Research Institute for Science and Technology. ii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS................................................. ii LIST OF FIGURES.................................................. iv LIST OF APPENDICES............................................... v LIST OF SYMBOLS.................................................. vi CHAPTER I INTRODUCTION.......................................... 1 II MATHEMATICAL FORMULATION............................... 5 A. Approximations...................................... 5 B. Boundary Conditions................................ 6 C. Perturbation Procedure............................. 7 III MATHEMATICAL SOLUTION.................................. 11 A. Complex Potential and Wave Elevation................ 11 B. Wave Resistance..................................... 21 C. Accuracy of Mathematical Representation of Body....... 25 IV EXPERIMENTAL RESULTS................................... 28 V CONCLUSIONS......................................... 41 APPENDICES....................................................... 43 REFERENCES....................................... 62 iii

LIST OF FIGURES Figure Page 1. Singularity Representation and Cross Section of Body.... 11 2. Plot of Function, Re I{iv(x-s+iO)]...................... 19 3. Plot of "Pressure" Function, f(x)....................... 19 4. Plot of Second-Order Wave Function, ri)(x)............. 19 5. First- and Second-Order Wave Elevation Curves for 1.25-Foot Submergence............................. 20 6. First- and Second-Order Wave Resistance Curves for 1.25-Foot Submergence............................... 24 7. Effect of Linearized Free Surface on Flow Around Assumed Singularities........................... 26 8. Picture of Foil with Supports........................... 29 9. Instrumentation for Measuring Horizontal Drag Force and Free-Surface Elevation............................. 29 10. Test Section of Foil with Strain Gauges................ 30 11. Cart and Ten-Foot Rail with Capacitance Wire............ 30 12. Wave Elevation Curves for 1.25-Foot Submergence......... 32 13. Wave Resistance Curves for 1.25-Foot Submergence......... 34 14. Wave Elevation Curves for 1.50-Foot Submergence......... 36 15. Wave Resistance Curves for 1.50-Foot Submergence........ 37 16. Wave Elevation Curves for One-Foot Submergence.......... 38 17. Pictures of Breaking Waves (Submergence = 1.0 ft)....... 39 18. Breaking Wave (Submergence = 0.75 ft and Speed = 3.50 ft/sec)............................... 40 iv

LIST OF APPENDICES Appendix Page A. Stokes Waves and Their Application to Wave Resistance Problems............................ 43 B. Derivation of "Exact" Wave Resistance Formula and Note on Variable Resistance Paradox........... 48 C. Expansion of Wave Integral......................... 52 D. Derivation of Pressure Function.................... 54 E. Computer Program for First- and Second-Order Wave Profiles....................................... 56 F. Wave Resistance Computations....................... 60 v

LIST OF SYMBOLS B body surface C1 Bernoulli constant C2 constant E total energy of fluid H wave height I defined complex function Im imaginary part M,N vertical control planes R wave resistance Re real part S surface of control volume U constant uniform velocity V velocity of control surface _f control -volume W complex velocity potential a,c x-coordinates of control planes b submergence of body f(x) "pressure" function g gravitational acceleration h water depth i imaginary unit j summation index i, j unit vectors vi

LIST OF SYMBOLS (CONT'D) k variable of integration m source strength n summation index p pressure s real variable of integration t time variable w complex velocity potential x,y rectangular coordinates z complex variable Oa first-order wave amplitude 3 ~ wave amplitude for Stokes waves Y Euler's constant 6 wave amplitude of order o C small parameter complex variable l ~free- surface elevation X\ variable of integration v wave number t Hx-coordinate of singularity p mass density T phase shift do velocity potential velocity potential Eft stream function stream function ~2 a2 a2 V 2, + 52 = Laplacian Ox2 2yv vii

I. INTRODUCTION The determination of the wave resistance of ship hulls is one of the most important and challenging problems the naval architect has to contend witho In the last century, many outstanding scholars in the field have been engaged in this subject; moreover, the last several years have brought a renewed interest in the theory of ship waves. Due to the complexity of the problem, however, we are by no means able to predict theoretically the wave resistance of a ship with sufficient accuracy. The three methods in use for obtaining the wave resistance are as follows: l. Model testing, subtracting the estimated viscous drag from the measured total drag force. 2, Wave survey behind the model, assuming small waves and using linear wave theory. 3. Theoretical analysis, using linear theory and representing the hull by some singularity distribution. The correlations between these three techniques have been far from satisfactory. In many cases, the differences are as large as 40 to 50 percent. [See S.D. Sharma (1963)] Fairly good agreements have been achieved only by introducing artificial correction factors; however, very little is known in general about how these factors are rerelated to physical parameters. The author believes that the main part of the discrepancies between analytical and experimental results could be due to the neglect of nonlinear effects at the free surface and thatthe viscous effect is -1

-2probably not as important as often stated. This nonlinearity is investigated here, and in particular its effect on the wave resistance. The first nonlinear wave theory for irrotational flow of an ideal fluid was derived by G. G. Stokes (1847). Applying this theory to the problem of flow past submerged two-dimensional bodies, one can easily relate the wave resistance of the body to the far downstream wave height. On the other hand, no relationship between the shape of the body and the wave elevation or the wave resistance can be obtained from the Stokes wave. Appendix A gives a detailed discussion of "Stokes Waves and their Application to Wave Resistance Problems." Only two investigators have applied a consistent second-order wave theory to the problem of free-surface effects on the flow past bodies: M. Bessho (1957) and E. O. Tuck (1965). Both restricted themselves to the simplified two-dimensional case of a submerged circular cylinder, due to the great mathematical difficulties of the more general problem. Bessho in his fine work derived correctly the complex potential, but unfortunately applied Blasius theorem incorrectly such that the most important higherorder term was not included. His final result and many of his conclusions are therefore incorrect. Tuck, on the other hand, correctly obtained the wave resistance and the lift for the circular cylinder, and he also correctly stated the very opposite conclusion of Bessho, namely that "it is more important to correct for nonlinearity at the free surface than for the fact that the boundary condition is not satisfied exactly by the first approximation on the body surface." This excellent paper by Tuck appears to be to the effect of nonlinearity at the free surface has been treated correctly to the second order.

-3Certain features of Tuck's results, however, indicate that the validity of this second-order theory may be questionable. The three following points will be mentioned hereo 1. At a submergence equal twice the diameter and at Froude number, U/fgb =.55, the second-order theory gives twice the wave resistance obtained by linear theory, and at even smaller Froude number, U,/gb =.47, his work predicts the resistance three times the linear theory. Results of this nature are not in agreement with the assumed converging perturbation series w = w(l) + 2w(2) +... and in this speed range at least, it is therefore doubtful if his theory is applicable. 2. The fact that the difference between the linear and secondorder theory increases as the Froude number decreases may indicate that the second-order wave resistance approaches infinity as the Froude number approaches zero. Unfortunately, no data are shown by Tuck for Froude numbers smaller than 0.47. 3. The largest submergence investigated by Tuck was twice the diameter of the cylinder. When the body is as close as that to the free surface, the waves created by the cylinder will break, however, resulting in a highly nonlinear phenomenon which cannot be treated by second-order perturbation theory. In addition to these points, it should be mentioned that the circular cylinder lends itself nicely to mathematical treatment and is therefre very useful for a preliminary mathematical investigation of nonlinear free-surface effects. But we must keep in mind that when assuming inviscid fluid flow past a circular cylinder we should not expect the

-4mathematical solution to be "a good approximation to the true wave resistance" (as hoped by Tuck). The object of this work has been to investigate the validity of second-order wave theory by a comparison of analytical and experimental results, and also to clear up some of the uncertainties in previous work. In order to perform such a comparison the author has applied second-order wave theory to a streamlined two-dimensional body and conducted experiments on an 11-foot long strut of 13 inch chord length. Wave resistance data have been obtained by the three techniques: 1) drag measurements, 2) wave survey, accounting for secondorder effects, and 3) second-order theory. In addition to these data, the wave elevations have also been measured and computed for the selected body shape. This is the first time wave profiles correct to the second order in wave amplitude have been computed for flow past a body, and it is shown that a comparison between these computed wave elevations and the measured wave profiles is more useful for the purpose of checking the validity of the theory, than the use of wave resistance data. The results show in general amazingly good agreement between theory and experiments for low and moderate speeds. Especially encouraging is the excellent agreement at low speed, where the difference between linear and second-order theory is very large. Useful information and data on wave breaking for small submergences are also included.

II. MATHEMATICAL FORMULATION A. Approximations An infinitely long cylinder is supposed to move with a constant velocity U in a direction perpendicular to its axis and at a ixed distance below the free surface. The problem is to determine the surface waves and the wave resistance. The flow will be treated as steady in a coordinate system moving with the cylinder. A two-dimensional coordinate system will be used, with the y-axis pointing to the direction of the vertical, and the x-axis located a distance b below the undisturbed free surface. The direction of decreasing x coincides with the direction of motion of the cylinder. This two-dimensional, steady-state problem is too difficult to be solved in its exact form. It will therefore be necessary to assume the fluid to be inviscid, incompressible, and without surface tension. In addition it will be necessary to introduce two approximations of a more mathematical nature: 1) irrotational flow, and 2) small disturbances. Let us look at the simplifications resulting from the irrotationality. In the first place, it ensures the existence of a singlevalued velocity potential T(x,y) from which the velocity field can be derived, i.e. v = grad 0 = Tx i + M j (2.1) y In view of the equation of continuity, the velocity potential 0 is a solution of the Laplace equation -5

-6V2 = 0, (2.2) and $ is thus a harmonic function. Another significant consequence of the irrotational character is obtained by integrating the equation of motion which leads to the Bernoulli equation: 1 grad |2 + P + gy = C (2.3) 2 p where p is the pressure, g is the gravitational acceleration, and p is the density. Hence, under these assumptions the Laplace equation (2.2) together with Bernoulli equation take the place of the equation of motion and the continuity equation. While it is true that the Laplace equation is a linear differential equation, it does not follow that we shall be able to avoid all difficulties arising from the nonlinear character of the basic differential equation of motion. The problem of interest here remains essentially nonlinear because the boundary condition at the free surface is p = 0, and according to (2.3) it is nonlinear in. B. Boundary Conditions Two conditions are to be satisfied on the free surface. The kinematic condition states that any particle which is on the surface remains there. Thus if y = b + ~(x) is the equation of the free surface and T(x,y) the stream function, then f(x,b + n(x)) = Constant = b (2.4) on the free surface. The dynamical condition requires that the pressure

-7above the free surface is constant, here set equal to zero. By the Bernoulli equation (2.3) it therefore follows that 2 | grad 0 |2 + g(x) = C1 (2.5) 12 5) The nonlinear conditions above are often referred to as the "exact" free-surface conditions. The boundary condition to be satisfied at the wall of the cylinder is clearly a = 0, (2.6) on in which 6/6n represents differentiation along the normal to the surface wall. The depth of the water is assumed to be infinite which results in the condition lim (grad 0) = Ui. (2.7) y-e.to One additional condition is necessary to ensure uniqueness, namely the absence of waves far upstream. In order to make the list of boundary conditions complete, we should also mention the absence of'wave reflection far downstream, often referred to as the Sommerfeld radiation condition. C. Perturbation Procedure In order to treat the nonlinear free-surface conditions it will be assumed that the disturbance velocity of the water particles, In this work, "exact" in quotation marks refers to exact within the potential flow theory~

-8thle free- surface elevation and their derivatives, are all small quantities, Suchl that a perturbation about the uniform flow can be carried out. We assumne, in fact, the following expansions in the parameter c D(x,y) - Ux + 0(x,y) = Ux + c(L)(xy) + C x2(2xy) +.'(x,y) - Uy + (x,y() = + c2) + e )( ) +. (2.8) (X) -= (1)(x) + C2E(2)(x) +. where c is some physical parameter which vanishes with the disturbance. It follows that each of the functions 0 (x,y) is a solution of the Laplace equation, i.e. 2(i) 0. (2.9) Substitution of the expansions (2.8) in the "exact" free-surface conditions (2.4) and (2.5), gives the first-order condition to be satisfied at y = b to be (1) _ 2 (1) = 0, (2.10) provided only first-order terms are included. Including terms of the second order, one obtains after some manipulation, that the second-order free-surface condition, also to be satisfied at y = b, is (2) (2) 1 (1) 2(1) 1 (1)2 (1) g - y - u - - x It also follows from these substitutions that the corresponding approximations to the wave height are ^(1)(x) (1= (x,b) (2.12) (2)(x) I (2) (x,b) + 2 t( )(x,b) ()(x,b). (2.13) U Uy

-9By introducing the complex variable z = x + iy and an expansion of the complex velocity potential W(z) = D(x,y) + iY(x,y) in the form W(z) = Uz + w(z) = Uz + W(1)(z)+ e (z)+..., (2.14) the free-surface conditions (2.10) and (2.11) can be written in a more convenient form. From (2.10) it follows that the first-order complex potential w(l)(z) must satisfy Re {wl) + ivw(l)} = at z = x + ib, (2.15) where v = g/U2 is the wave number. The second-order complex potential w(2)(z) must satisfy Re {lw2) + ivw()} 1 f(x) at z = x + ib, (2.16) pU where f(x) = J () 12 - Im(w(l)). Re(iw() - vw. P 2 z zz z In addition to these free-surface conditions each of the complex potentials w(i)(z) must satisfy the following four conditions: the rigid-boundary condition (2.6), the condition (2.7) at infinite depth, and the conditions specifying the absence of upstream waves and of wave reflection far downstream. A first- and second-order complex potential, w(l)(z) and (2) w (z), satisfying all these conditions would give us the desired "exact" second-order solution. It is rather difficult, however, to satisfy the cylinder-wall condition to this order of accuracy. Let us therefore look at this condition more closely. Clearly if the submergence of the cylinder is large compared with its own dimensions it will produce small waves simply because it is a relatively distant disturbance to the

-10free surface. Therefore the body can for the first-order approximation, be represented by a singularity distribution which generates the desired cylinder in an infinite fluid without a free surface; while for a secondorder theory, a modification of this singularity will be necessary in order to generate the same body in a fluid with a linearized free surface. E. O. Tuck (1965) has shown, however, that in obtaining the wave resistance for a circular cylinder it is more important to take account of second-order approximations at the free surface, than to include this modification of the singularity. Therefore, in this work, the body will simply be assumed to be represented by its singularity distribution in an infinite fluid. This implies that the cylinder-wall condition is satisfied only to the first order of approximation. It can certainly be argued that the second-order theory presented here is not consistent; however, it is believed that the method used will give a solution far more realistic than the first-order theory and therefore believed to be an improvement over the linear theory.

1i. MATHEMATI ICAL SOLUTION A. Complex Plotential arld Wave Elevation Thle two-dimensional submerged body will be mathematically represented by a singularity distribution of the following strength and location: Eleven sources equally spaced along the x-axis between x = 0 and x = 1.0, and with strengths (mj) given by the relationship m. - +.04 when 0 < j < 2, 2 jn U (3.1) -.- (65-3j) when 3 < j 10, 168 where j = 0, 1, 2, etc. refer to the sources at x = 0,.1,.2, etc. y FREE SURFACE b I ) ) ~ ) I I,.0.2 0.1 2.3.4.5.6.07. ^_____________ ~1.09 _. Figure 1. Singularity Representation and Cross Section of Body. -11

-12The stream function for this singularity distribution in a uniform flow (velocity = U) with no free surface is 10 m-. V(x,y) = Uy -. E - tan-. (3.2) j=o 2j n x- (j10( The body is given by the streamline E = 0. Hence the equation of the body is 10 m.. y= z (iL)'.tr ( 10 (5.5) -j=o (2 U x-(j/10) (3) The cross section of the body is seen in Figure 1. In terms of the selected unit length, the chord length is 1.09 and the thickness is.37i-. The unit length used here is, for numerical examples as well as for the experimental work, set equal to 1 foot. The complex potential and the wave elevation for a single source will first be obtained, then the solution to the given singularity distribution can be obtained by summation. Let the stationary source of constant strength m be located in the complex plane at z = ~ + i 0, let the undisturbed free surface be at y = b, and let the undisturbed fluid be moving in the positive x-direction with a constant velocity U. Then by Chapter II it follows that the complex potential for the source is of the form w(z) = Uz + w(l)(z) + 2W(2)(z) + where the first-order potential w(1) must satisfy the conditions

-131. V2 w(1) = 0 except at z = ~ + i, 2. Re {(') + ivw()} = at z = x + ib, (3.4) 3. lim | wl) = 0 as Im z - -c 4. lim w = as Re z - -o The solution to this differential system (3.4) is given by Wehausen and Laitone (1960, p. 489) in the form w(')(Z)= i n (z-E) + 2m n (z - 5 -2ib) m P.V,. -ik(z —2ib) dk - im e-iv( —2ib) (3.5) o k- v The two last terms in this solution can be more conveniently written in terms of a series expansion. In order to do so, let us define the complex function I(:) - P.V. / e-u du - is e. (3.6) O 1-u This function has been shown in Appendix C to have the expansion -~ { + in + ix + n3 -nn7l I(:) = - e { + n + i + nn (7) n=l where the Euler's constant, 7 =.5772.... Then by the substitution G = iv(z-~-2ib) we can rewrite the solution (3.5) in terms of the expansion (3.7) as w(l)(z) = In (z- ) + 2- In (z-e-2ib) + I {iv(z-e-2ib)}. (3.8) The next step is to find the second-order potential w(2)(z), which must satisfy the conditions

-141. V2W(2) = 0 except at z = ~ + i0, 2. Re w + iv 2} = f(x) at z = x + ib, z - pU where 1 f(x) = |w |2 - I m (w(l)) Re (iw!1) vwl)), P 2 Z zz z (3 35 lim (w 2) =0 as Im z -- m, 4. lim |w() = as Re z - - oo This is exactly the same differential system as for the linear problem of a fixed pressure distribution f(x) on the free surface of a uniform stream of velocity U. Thus f(x) may be interpreted as a pressure distribution due to the first-order wave system. The solution to the differential system (359) is therefore the same as the well known solution to the moving pressure distribution problem, given by Wehausen and Laitone (1960, p. 601) in the form (2. +0 m 0 -i\(z-ib-s) w()(z) = f ds f(s) P.V. f e - dX jTU -o o - v _eiv0z-ib) ivs + 1 e (z-ib) f(s) e ds (3.10) U _ 00 This potential can also be expressed in terms of the expansion (35.7) as (2)~ ~ ~a00 w()() = - f ds f(s) * I {iv(z-ib-s)}, (5.11) 7tU -o where the function f(s) is defined in (2.16) and, as shown in Appendix D can be written by virtue of (3.8) as 2 2 [m I (iv(s1-ib))]12 b 2 f(s) = pU {2v [7g Im I (iv(s-i)) + 2 [2U ( + v Re I(iv(sl-ib))] [ m l iv(s-2b))] + [I1 (5. + )}, 12) +[U Imu(iv(sl-ib))] [U(si+b)+ (3.1b

-15where sl = s-S Real and imaginary parts of the expansion I {iv(s1-ib)} are given by Equations (C-4) and (C-5) in Appendix C as Im I {iv(s 1-ib)} = e V{(y+ln vR + nl n- cos no) sin vs1 0 vR)n - (r + Q + n nn sin n@) cos vs} (5.15) n=l n-n 3~3 and {iv~s1ib)} ~ vb { (i + (vR)n Re I iv(s -ib)= -e+ + vR n cos n~) sin vs oo (v'R )n + (7 + In vR + v sin nO) cos vsl},(5.l4) where R = Ni /S^+ b21 and 0 = tan1 b Hence, the set of Equations (5.11) through (3.14) gives the second-order complex potential of a single two-dimensional source in a uniform streamo The wave height can now easily be obtained. The first-order wave elevation is by (2,12) and by the first-order potential (3.8) 1)(x) =- 1 ) ^ (xb) = Im I i -{iv(x —ib)}. (5.15) The second-order wave elevation is by (2.15) 2 -)(x) 1 (2) + (1) (2)(x) X()(x~b) + (2 r)(xb) ( x,b), (5.16) where, by virtue of (D-5) in Appendix D, (1) x,b) = Re {w(') ( x+ib)} = v Im I {iv(x-)-ib) ~y Therefore by (.5.15) it is seen that the last term in (5.16) becomes ^ r(L)(x~b) - ^4 (xb) =. (5.17)

-16By applying the second-order potential (3511) and the relationship (3.17), the second-order wave elevation (3.16) is (2)(x) = 1- L ds f(s) Re I {iv(x - s + iO)} + v(r(1)), (3.18) where Re I {iv(x-s+iO)} is given by Equation (C-6) in Appendix C as Re I {iv(x-s+iO)} = - cos v( x-s){y + in v(x-s) +, v2n (x-s) (-l)} n=l 2non' + sin vx {i(1 + sign x) + - 2 + oo v2n-l s)2n-ll)n (19) in=l (2n-l)>(2n-l1) Note that (Pi) o Re I {iv(x-s+i0)} (rgp is the wave height created by an integral pressure P concentrated on an infinitely narrow band of the free surface at x = s. A detailed discussion of this is given by Lamb (1945) in sections 243 and 244. A plot of Re I {iv(x-s+iO)} is seen in Figure 2 on page 19. The next step is to obtain the complex potential and the wave height for the singularity distribution (351)o This can be done by adding the contribution of each individual source. The first-order complex potential then becomes by (3.8) )(Z) l 10 mm. w(l)(z) = jo in (z- j + o -- In (z - a - 2ib) j=n 10 )j=0 2it 10 + Z - I {iv(z - - ib)} (.20) j=Te 10l The second-order complex potential is, by (3.11)

-17w(2)(z) = V f ds f(s) ~ I{iv(z-ib-s)}, (3.21) where the function f(s) now becomes 10 m. 2 10 mj b 2 f(s) = pU2I{2v2 Im I] + 2[. U (b + v Re Ij)] j=o j=o 2-tU s7+bJ 1 1mi m 2bs vs. + j4 o [ U Im Ij] [jEo 2IU ( (.22) 3 J with I. = I{iv(s - - - ib)} and s. s- J JI 10 10 Real and imaginary parts of the expansion I{iv(s - -- - ib) are given 10 by Equations (3.13) and (3.14). The first-order wave height for the given singularity distribution becomes r(l)(x) = 10j Im i (3523) j=0o *IU — 10 and the second-order wave is ~(2)(x) = 1- ds f(s) Re I{iv(x-s+iO)} +v(r(l)) (3.24) where f(s) is given by (3.22) and Re I{iv(x-s+iO)} by (3.19). For later reference we shall write the two terms in (3.24) as 00 (2)(x) -= 1 ds f(s) Re I{iv(x-s+iO)} niI (X) V(~ (3.25) rW)(x) = v(rtl))2. (5.25) We now turn to the numerical computation of the first- and second-order waves. A MAD computer program, shown in Appendix E, was prepared for this purpose. Before we discuss the final results of these

-18computations, let us look at some of the intermediate steps. It is interesting to note that the "pressure" function f:(s), defined by Equation (3.22), is not sinusoidal far downstream but tends to the con1 stant value 2 pgva-, as can be seen from Figure 3. We also note that due to this constant "pressure" value, the first part of the secondorder wave I(2)(x) far downstream is sinusoidal, but with the mean 1 line a distance - va below the undisturbed free-surface level as is clearly seen in Figure 4. The important fact is that this constant dislocation of the mean line is of exactly the right magnitude to balance the constant part of the term (x) = v(l), leaving only harmonic wave components far downstream. Plots of the final results, the surface elevation according to the first- and second-order theory, are given in Figure 5 for the case of 1.25-foot body submergence and for four different speeds. This is, to the author's knowledge, the first time that waves created by a body have been computed to the second order. Looking at these wave profiles, we immediately notice the extremely large difference between first- and second-order waves for the lowest speed of 2.5 ft/ sec. This seems to violate the original assumption of expansions of the form r(x) = (l)(x) + 2 (2) + *.. (3.26) where the second-order term c2P(2)(x) was assumed to be one order smaller than the first term e i!)(x). The expansion (3.26) may, therefore be divergent for the lower speeds, which could result in the first-order theory giving better results than the second-order in this speed range.

-19Re I{(x-s+io)} X=S Figure 2. Plot of Function, Re I{iv(x-s+iO)} U=4.0 ft/sec a b=.O ft f(x) v22 Figure 3. Plot of "Pressure" Function, f(x) 4(X~ I I 2 U=4.O ft/sec a b=1.O ft Figure 4. Plot of Second-Order Wave Function, rl2)(x).

-20U 2.5 ft/sec.211.5' I. OND/ I.0 2.5T Submergec,.0 -*2"^, ^2/ 7.. U = 4.5 ft/sec 7 /~,, \ /" \ o0 /// 1.5'.5' 7.5' 7"- 3/ U 5.5 ft/sec 1.5 2 4~~~~~~/\ 1I,.5 Figure 5. First- and Second-Order Wave Elevation Curves for 1.25-Foot Submergence.

-21The experimental results discussed in Chapter IV will, however, show that the second-order waves agree surprisingly well with the measured waves for these lower speeds. It should also be recognized that the shape of the second-order waves downstream are quite the same as given by second-order Stokes waves, as was expected. B. Wave Resistance The wave resistance will be obtained from the asymptotic form of the velocity potential and the wave elevation. It is well known that the first-order wave height (3.23) far downstream takes the simple form (l)(x) = a * cos vx, (3.27) where a is the first-order wave amplitude, and x is the x-coordinate so selected that the wave has no phase shift. The second-order part (3,24) can be shown to have the following form far downstream (2) 1 2 (1) 2 (X2 ) = 5 cos v(X —r) - - va2 + v 1))2 (5.28) which by (3.27) becomes 2 (X) = & ~ cos v(_-T) + 2 va2 * cos 2vH, (3.29) where 5 is of order c2 and T is a phase shift. From (3.27) and (3.29) it can be realized that far downstream we have two Stokes waves. One of the form 1 2 nl()= cos v+ + v cos 2vx + c... (55o)

-22and the other, which has a phase shift T and an amplitude 6 of second order in a, has the form 02(X) = 6 cos V(7-T) +..., ( 1) where both (3.30) and (3.31) are correct to the second order in a. From the theory of Stokes waves (Appendix A) it follows that the velocity potential for these two Stokes waves can be written as v(y-b) 01(X y) =-aUe sin vx, (3.32) v(y-b) 02(x,y) =-6Uev ) sin v( - T). (5.33) Assuming now that the potential and the wave elevation far downstream can be written as the sum of the two Stokes waves above, we have that = 1 + 02 =-Ue(b){c sin v- + 8 * sin V(-T)}, (3534) = Tl + e2 = a cos v~ + 2 va cos 2vx + 6 cos v(X-T). (.355) Note that the potential (3.34) and the wave elevation (3.35) are valid far downstream, and that both are correct to the second order in a. The "exact" formula for the wave resistance, as derived in Appendix B, is 1 b+r(xo) 2 2 1 2 R = (Xy) + y(Xoy)]dy + gpq (xo) (56) Y~~~~~~~~(.6

-23where 0(x,y) is the "exact" velocity potential, r(x) is the "exact" wave elevation, and xo denotes any vertical plane behind the body. Evaluating this expression far downstream using the velocity potential (3.34) and the wave profile (3.35) it is shown in Appendix F that the wave resistance correct to the third order is R = - pg[ 2 + 2a5 cos VT]. (3.37) 4 2 Since by (3.35), the wave height correct to order a is H = n(o) - (jr) = 2a + 25 cos VT, (.388) the resistance R can also be expressed in terms of the wave height: R = g(H) (3~39) 4 2 where H is the wave height obtained from the second-order wave theory. This is exactly the same result as obtained in Appendix A using a secondorder Stokes wave theory. A plot of the first- and second-order wave resistance is seen in Figure 6, where the first-order resistance is obtained from the well known relationship R = pga2 (3.40) with Gz denoting the first-order wave amplitude far downstream. Again we see the large difference between the first- and second-order theory for the lower speed range.

-240.03 / / LOW SPEED / 07 / 0,~ o0 2-s —/-_ - o. oo L j / gj-~ *0.6 / 0.01.i/ // I /.2 / 3,/0.5 / / II co -< T/ _ 2.0 2.5 3.0 w -) f t/sec / 0.4 0.4 f 12-oSuegn/ 0.3 / 0.3 0.2 -1 — 0.2 f O.1 —- - — f/ ------- FIRST ORDER 0.1 /. —---- SECOND ORDER 2 3 4 5 6? SPEED U, ft/sec ---- Figure 6. First- and Second-Order Wave Resistance Curves for 1.25-Foot Submergence.

-25C. Accuracy of Mathematical Representation of Body We assumed in Chapter II that the body could be represented by its singularities in a uniform flow without accounting for the free surface. The boundary condition on the cylinder wall is by this assumption only correct to the first order. In order to be correct to the second order, we should really have modified these singularities such that the same body would be generated in a fluid with a linearized free surface. This is one of the critical assumptions in this work and it was therefore felt that an investigation of its accuracy should be included. A computer program was written which could trace the streamlines around the singularity distribution including the linearized free surface. This is a rather time consuming computation, especially because we a priori do not know the value of the stream function at the stagnation points. The streamlines were therefore only traced for two speeds, V = 4 ft/sec. and V = 6 ft/sec., both with the same submergence b = 1.25 ft. Each case took about ten minutes on the IBM 7090 computero The results are shown in Figure 7. It is clearly seen that when the linearized free-surface effect is included, the body is no longer closed, and that the forward and aft stagnation points are on different streamlines. This effect was first pointed out by T.O. Tuck (1965). However, the author is of the opinion that for lower speeds the given singularity distribution does represent the body fairly well and can be used with reasonable accuracy for a second-order theory. For larger speeds it is obvious that the

-26SPEED =4 ft/sec AND SUBMERGENCE = 1.25 ft ABOUT INCH | —----- ~I FT SCALE - SPEED =6 ft/sec AND SUBMERGENCE =1.25 ft j- ---- I FT SCALE — i —Figure 7. Effect of Linearized Free Surface on Flow Around Assumed Singularities.

-27body representation cannot be used for a second-order theory. For a submergence of 1.25 feet the use of this body representation for speeds up to about 4.5 feet per second, and higher speeds for deeper submergences, seems justified, as can be seen from Figure 7. It is interesting to note that the body can be made closed simply by applying a small angle of attack and hence without introducing any new singularities. In other words, a closed body can be obtained by an appropriate rotation of the line on which the weleven sources are located. The amount of rotation necessary to close the body will obviously depend on the velocity and the submergence of the cylinder.

IV. EXPERIMENTAL RESULTS Experiments were conducted in order to see how well this second-order theory can predict wave elevations and wave resistance for submerged two-dimensional slender bodies. The experiment was performed at The University of Michigan model tank (360'x20'x9') on an eleven-foot two-dimensional strut seen in Figure 8. The model's cross-sectional offsets are given in Figure 1 on page 11. The instrumentation for measuring horizontal drag force and wave elevation is schematically shown in Figure 9. The horizontal drag force was measured on the middle, two-foot section of the model by water proofed strain gauges mounted on two small cantilever beams (see Figure 10). The wave elevation was measured by a.0016 inch capacitance wire attached to a small carriage which could move on a ten-foot rail as shown on the picture in Figure 11. The change in capacitance of this wire as caused by changes in water elevation was registered by a highly sensitive capacitance bridge (Wayne Kerr B541) from which results were traced on an x-y recorder. The x-coordinate was used for position of carriage and y-coordinate for wave elevation. With this capacitance wire, we were able to obtain the wave profile to a good accuracy, about +.03 inch for speeds up to 4.5 ft/sec. No records could be taken at speeds higher than 6 ft/sec, however, due to an air cavity formed behind the wire. At speeds between 4.5 and 6 ft/sec the wire did not give as good accuracy as desired so the wave heights were also checked mechanically. Wave profiles were recorded for a total of five submergences (b = 0.50', 0.75', 1.00', 1,25', vad -28

-29Figure 8. Picture of Foil with Supports. 00l0 I0 I!l "-*ooooo 0-0 - 00 CAPACITANCE X- Y RECORDER STRAIN BRIDGE X-Y RECORDER GAUGE BRIDGE 8~~ I Figure 9. Instrumentation for Measuring Horizontal Drag Force and Free-Surface Elevation.

-30Figure 10. Test Section of Foil with Strain Gauges. Figure 11. Cart and Ten-Foot Rail with Capacitance Wire.

-311.50') and nine speeds for each submergence (V = 2.0, 2.5, 3.0... 6.0 ft/sec). Every profile included in this report was obtained from at least two differenct runs in order to check the results, and a total of about two hundred runs were performed. The wave resistance was obtained from the model test data in two different ways: 1. By subtracting the horizontal drag at 4.5-foot submergence from the total horizontal drag at the other submergences (assuming no wave resistance at the 4.5-foot submergence and that there is no interaction between wave and viscous resistance, and that the viscous drag is the same at a deep submergence as for a submergence where waves are created). 2. From the derived equation R =- Pg()2 4 where H is the actual measured wave height, and recalling that this relation is correct to the third order in wave height. The wave elevation curves for 1.25-foot submergence are shown in Figure 12. Good correlation in wave height can be seen between measured waves and second-order theory for speeds up to 4.5 ft/sec. The figure shows, on the other hand, some discrepancy between measured and theoretical wave lengths. We note especially the excellent agreement

-52U = 2.5 ft/sec U 5',,f:o' 3-.5',ft'.o' s.5'e 0.2" 1." U =4.5 ft/sec -1.3" 1.3" ^ 4U 5.5 ft/sec ~~~~~~~1.5"~~~~"' FIRST ORDER THEORY I ARROW INDICATES Figure 12, \. Wave Elevation Curves for 125-Foot Submergence. MEASURED1 WAVeEevto Cuvsfr12-otSbegne

-33at the lowest speed where the difference in elevation from first- and second-order theory is extremely large. The wave resistance curves plotted in Figure 13 show exactly the same trend. We do observe from these figures a poor agreement at speeds above 4.5 ft/sec, however. This is believed to be caused by the inaccurate mathematical representation of the body at higher speeds and also probably due to some viscous effects. The wave resistance curve obtained from horizontal drag measurements seems to be rather high at the lower speeds, which seems to indicate that the viscous drag is not the same at 4.5- and 1.25-foot submergence; but that it increases as the body gets closer to the free surface. This is most likely due to an increase in the velocity of the fluid particles next to the body as a result of the free surface. Furthermore, this proves how extremely difficult it is to determine the exact wave resistance from drag force measurements. The author, therefore, strongly believes that we shouldnot compare wave resistance curves, but rather wave profiles when checking the validity of theoretical work. Figures 14 and 15 show the wave profiles and the wave resistance curves for 1.50-foot submergence and as expected the agreement is even better at this larger submergence. Another interesting result is that the waves start to break at submergences smaller than 1.25-foot. This can clearly be seen from the wave profiles plotted in Figure 16 and from the pictures shown in Figure 17, both for one-foot submergence (about three times the thickness of the body). Many theoretical investigators have assumed that submergences

-34-.03: 5 I LOW SPEED / _.7 // / -FRS -O;E- T E R.02 ——.' /. M, I LO I I 7.01 / / // / A_______.5,./ /.// / S 0_ I-....... J 2.0 2.5 3.0 ft/sec /' W..4':y r___ C.4 U).1.3.3 w I /.:// I.- //.2 I _ _.2 / -.-FIRST ORDER THEORY ~/ /I ----— SECOND ORDER THEORY FROM MEASURED VMVE /7........,... DIFFERENCE BETWEEN HORIZONTAL DRAG AT 1. 25- FOOT AND 4.5-FOOT SUBMERGENCE 2 3 4 5 6 7 SPEED U, ft/sec Figure 13. Wave Resistance Curves for 1.25-Foot Submergence.

-35of this magnitude were large enough to apply small disturbance theory. E. O. Tuck (1965) used twice the thickness as his largest submergence. This wave breaking, however, is such a highly nonlinear phenomenon that no perturbation theory can predict its occurrence. We note that for the one-foot submergence the waves break for speeds between about 2.5 and 5.5 ft/sec, and that for lower speeds when the waves do not break, the second-order theory agrees fairly well with measurements. Figure 18 is included to show that at even smaller submergences (b = 0.75 ft) the waves break down completely leaving no regular waves behind the body, but only some kind of a "hydraulic jump." This has previously been observed by E. V. Laitone (1954) and by Parkin, Perry, Wu (1956).

-36-.4 0" 2' ~ 2'4 5 U = 4.5 ft/sec 7 _______ X X f5 75,\ U = 5.5 ft /sec 0 1.2' 1//2' \ 3' I%' 5 U =.5 ft/sec "/ F__igri______ Wave OEln CORv fr O15F TRAILINS bDegE Figure 14. Wave Elevation Curves for 1.50-Foot Submergence. Figure 14. Wave Elevation Curves for 1.50-Foot Submergence.

-370.04 -0.5 0I0 > / /0 LOW SPEED / 0.03 / I I f 0.02 / 0.4 // 0.01 -ft 0 o' 0.3 2.5 3.0 3.5 Z tf/sec Irl__r / _ cr! I I J w 0.2 __ 0.2 / 1' 0.1 0.1 /I/ ------ FIRST ORDER THEORY i// / ----- SECOND ORDER THEORY /. —---- FROM MEASURED WVES 0 2 3 4 5 6 7 SPEED U, ft/sec Figure 15. Wave Resistance Curves for 1.50-Foot Submergence.

-385"6. U= 2.5 ft/sec.5' / 1.5' 2 2.5'.8" U 3.0 ft/sec / /.8" \/ / /\\ I / 2"0- U=4.0 ft/sec > /- N 5. 3.0' / 60'- 5' U= 5.0 ft/sec / 2" -21(-' -|-_-_ FIRST ORDER THEORY | ARROW INDICATES.......SECOND ORDER THEORY T LOCATION OF TRAILING EDGE - MEASURED WAVE Figure 16. Wave Elevation Curves for One-Foot Submergence.

-39U = 4.00 ft/sec U = 4.50 ft/sec U = 4.92 ft/sec U = 5.00 ft/sec Figure 17. Pictures of Breaking Waves (Submergence = 1.0 ft). _'_~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.~:i He _D~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~iiii ii-iiiiiii ii

-40/ - I'* 2'/',3 46 475 \ FIRST ORDER SECOND ORDER MEASURED WAVE Top View of Foil and Breaking Wave Figure 18. Breaking Wave (Submergence = 0.75 ft and Speed = 3.50 ft/sec).

V. CONCLUSIONS This work clearly proves the importance of the effect of nonlinearity at the free surface, and it shows how much better results can be achieved by the second-order theory than by the linear theory. The speed range and submergences are also indicated for which linear and for which second-order theory may be applied. The following conclusions can be drawn for free-surface flow past submerged two-dimensional bodies: 1. The second-order theory gives very good results for slender bodies at moderate speeds and submergences larger than three and a half times the thickness of the body. 2. At very low speeds the second-order theory predicts wave heights several times the size given by the linear theory. The agreement between experiment and second-order theory is surprisingly good at these very low speeds. 3. For the purpose of checking the validity of the theory a comparison between theoretical and measured wave profiles should be used rather than a comparison of wave resistance data. 4. Due to the wave breaking phenomenon at smaller submergences, the perturbation theory should not be used in general if the submergence is smaller than three times the thickness of the body. 5. Both the linear and the proposed second-order theory give less satisfactory results at high speeds. This may be due to the inaccurate -41

-42mathematical representation of the body at these speeds and also probably due to viscous effects. We should be careful in extending any of these conclusions to the three-dimensional ship problem. It seems reasonable to conclude however, that nonlinear free-surface effects are most likely more important for surface ships than previously believed. The author should like to suggest that furtherresearch be carried out in this area, and he feels especially that the three following problems deserve attention: 1. A consistent second-order theory, satisfying the cylinderwall condition to the same order of accuracy as the freesurface condition, should be applied such that better results can be obtained at higher speeds. 2. A thorough investigation should be made of the viscous effects on free-surface flow past a cylinder by means of wake study and possibly also by applying boundary-layer suction control. 5. Work should be extended to include three-dimensional submerged bodies. From this extended research, we may be able to draw conclusions of great importance to the surface ship case.

APPENDIX A STOKES WAVES AND THEIR APPLICATION TO WAVE RESISTANCE PROBLEMS It was first shown by G. G. Stokes (1847) that the velocity potential and the stream function given by = Ux - UeVY *~ sin vx E = Uy - Ue Y cos vx - C2 (A-1) is not only the linearized wave solution, but that it also satisfies the free-surface conditions correctly to the second order for the case of infinite depth. Lord Rayleigh (1876) pointed out that the solution (A-l) can also be shown to be correct to the third order. In this work, however, we will only be concerned with first- and second-order terms. The equation of the wave-profile (E = 0) follows from (A-l) by successive approximation: C2 C - +e cos vx =- + (l + v +...) cos vx U U = + v P2 + P cos vx + 2 *2 cos 2vx +.... (A-2) U 2 2 For convenience, let us set C2 = - 2 U2, (A-3) such that the wave-profile becomes 1 2 = P cos vx + v ~ cos 2vx.... (A-4) -43

-44We note that (A-4) coincides with the equation for a trochoidal wave to the given order of accuracy. The solution (A-i) with the wave-profile (A-4) must, in order to be valid, also satisfy the condition of constant pressure (p = 0) at the free surface. Applying the Bernoulli equation 2 ra + gradgy = C (A-5) we have that 1 U2l - 2vpeVy. cos vx + v2 2e2v + gy = C. (A-6) At points on the line = - - vP2 + Pe V * cos vx we therefore have (g - vU2) + 2 C12 (A-7) 2 where only terms to the order P have been included. Hence, the pressure is constant (p =0) on the free surface if v =. (A-8) Let us now investigate if it is possible to have a Stokes wave behind a submerged two-dimensional body in a uniform stream. The two following conditions must be satisfied. 1. The pressure at the horizontal plane y = -h as h - co, must be the same far upstream and far downstream. 2. Conservation of mass: The inflow through a section upstream must be equal to the outflow through any section downstream. The pressure at a depth y = -h far downstream is by the Bernoulli equation (A-5)

-45+ 1 U 2{-2vPe-v cos vx + v22e2h} - gh = 0,(A-9) P 2 and therefore when h is very large P = pgh. (A-10) Hence, in order to satisfy the pressure condition 1., we have that the undisturbed free surface far upstream is given by y = 0 The conservation of mass condition requires that o f U dy J fxdy fd -h X l im - lim (A-11) h-> o h h ->oo h where the left hand side is o U dy lim Ud = U (A-12) h -> o h and the right hand side f (dy 1 -h U1V v lim h lim H(U - UVev cos vx + Uh) h -4 oo h-> < = lim (U 1 ) = U. (A-13) h -- 0o 2 h Hence, condition 2 is also satisfied, and a Stokes wave is therefore a possible wave behind a two-dimensional body. Having concluded this we may now obtain the wave resistance for any two-dimensional body in terms of the Stokes wave far behind the body. The "exact" formula for the wave resistance, as derived in Appendix B, is 1 = (xy) + ( + gp (x) (A-14) R [-x(xo,y) + y (xo,y)Ody + gp (x 2 - oo y2

-46where xo denotes any vertical plane behind the body. Evaluating this expression far downstream, using the velocity potential (A-l) and the wave elevation (A-4), it is seen that the wave resistance correct to the third order in P is R pgp2. (A-15) Let us now compare the wave resistance (A-15) obtained by the second-order Stokes wave to the resistance obtained by the linear theory. The linearized velocity potential and wave elevation are ^ = Ux - UaeVY sin vx (A-16) a = a cos vx where a is the linearized wave amplitude. The well known wave resistance for the linear theory is therefore 1 2 R= pgc (A-17) It is very important here to distinguish between the linearized wave amplitude a and the second-order Stokes wave amplitude P. If these amplitudes are mistakenly set equal we have by (A-15) and (A-17) the incorrect conclusion that the linear theory and the second-order Stokes wave both yield the same wave resistance. In fact we have that P =a + ~2) (A

-47and in general the complete second-order problem must be solved in order to obtain P correct to the second order. In other words, the Stokes wave theory only gives us the form of the potential (A-l) and the shape of the wave (A-4) correct to the second order, however, the magnitude of the wave elevation cannot be obtained without solving the complete second-order problem. A very useful result can be obtained from the theory of Stokes waves. By the wave profile equation (A-4) it follows that H pf ='7 where H is the wave height. Therefore we have by the wave resistance formula (A-15) that R = 1 g(H2 (A-19) Applying this result to experimental work, we have that Equation (A-19) gives the wave resistance from measured waves correct to the third order in wave height.

APPENDIX B DERIVATION OF "EXACT" WAVE RESISTANCE FORMULA AND NOTE ON VARIABLE RESISTANCE PARADOX Consider a two-dimensional body in a uniform stream of velocity U, and let us apply the momentum theorem to the fluid region bounded by the plane x = a far ahead, another plane x = c behind the body, the bottom y = -h, the surface of the body, and the free surface. The horizontal force on the body is then given by 2 2 R = f [p+pT ]dy- f [p+PT dy, (B-l) x=a x x= where 0 = Ux + 0. Introducing the Bernoulli Equation (2.5) and the continuity condition f/ x dy = f x dy, (B-2) x=a x=c we have that R = pg f -y dy + pg f y dy + / f (0 2 )dy - f ( dy (B-) x=a x=c 2 x=a Y x 2 x=c y x And as a -a- it follows that the "exact" wave resistance formula under the assumption of ideal fluid, irrotational flow and neglecting surface tension is P (~) 22 R = P c [-02(c,y)+2(c,y)]dy + gn (c). (B-4) Where we note that the plane x = c may be taken at any distance behind the body and that ) is the wave elevation measured relative to the undisturbed free surface far ahead of the body. -48

-49This expression is "exact" and therefore when applied to a linear problem will yield the wave resistance correct to the second order; and when applied to a potential correct to the second order, it will give the wave resistance correct to the third order. S. D. Sharma (1963, p. 255) shows that if the continuity condition (B-2) is not applied, however, the wave resistance becomes 22+ pU U R = P / [-0 +2 ]dy + 2+ pU Udy - pU (U-0x)dy (B-5) X=C x x= a x=c c He then introduces the linearized velocity potential and wave profile and gets the "variable resistance paradox" R = Pg c + pga2 sin vx. (B-6) This paradox shows the kind of erroneous conclusions which follow from a first-order theory if the equations are applied directly without a specific knowledge of the order of magnitude of each term involved. (B-6) was obtained using a linear theory, hence continuity is satisfied only to the first order, and therefore any second-order terms resulting from the continuity condition have no meaning and should have been disregarded by Shama. In order to make this work more complete, we also will derive the "exact" wave resistance formula from energy considerations following closely Wehausen and Laitone (1960) and F. John (1949). Let -(t) be a region occupied by an ideal fluid with boundary S(t). The energy of the fluid contained in V(t) is given by E = P f [2 | grad I2 + gy] di (B-7) e(t)

-50Using the Bernoulli equation for unsteady flow t +: Igrad |1 + ~ + gy = A(t), (B-8) we may express E as E = f (-p-pt )dw, (B-9) f(t) t where 0 has been so defined that A(t) may be set equal to zero. If follows by differentiation with respect to time t, that dE dt = P( grad * grad Otd~ + f (p+Pt)VndS, (B-10) dt V(t) u S(t) where Vn is the velocity of the boundary surface S(t) in the direction of the exterior normal. The integrand in the volume integral can be written in terms of the divergence as grad $ ~ grad t = grad(<t grad 0) - $tV 2. (B-ll) Therefore by Green's theorem we have dE p f [% 6n + Vn(P+Pt )]dS (B-12 dE'p - ^tll-'-V ~ ^ *.(B-12) dt S(t) n n t For the two-dimensional problem of interest here we will let the surface S(t) be represented by a plane M: x = -a-Ut far ahead, another plane N: x = c-Ut behind the body, the bottom, the surface of the body B, and the free surface. The energy within this region is constant, and one obtains, with 4(x,y;t) = 0(x+Ut,y): 0 = - p dS + f pu + U[ |grad 012 + pgy]dS B ] d + f pU - U[| |grad I|2 + pgy] dS, (B-l ) NJ axan 2

-51where the first integral is equal to R U, R being the force on the body and where the second integral vanishes as a --. It follows therefore that the "exact" wave resistance formula is [ + 2(cy) + (c), (B-14) R= -h [-c o (,y) + o (i,y)]dy + tn (c) m (B-t) 2 -h x y 2 which is identical to (B-4) obtained by the momentum theorem.

APPENDIX C EXPANSION OF WAVE INTEGRAL In this appendix an expansion for the following integral I() = - P.V du - ire (C-1) 1-u will be obtained and discussed. Clearly this expression is analytic in Re(() > 0. Suppose first that ~ is real and positive and let w = - (l-u)~, then 00 -u~ r _ e-' i _o-w dw 1-u w 1 1-e-W 0 -w /~ ew_ = e fl dw- dw d + + w dw} o w 1 1 w 0 = y + in, +.n (C-2) n=l n.n Therefore the expression (C-l) can be written as 00 n 1() = -e (7 + in 5 + is + Z ). (C-3) n=l non' This relation has been proven on the assumption that ~ is real and positive, but both sides are analytic functions when Re(0) > 0, and so the relation holds at least in the half-plane Re(Q) > 0. Let = ivz = v R e where R = +y2 and = tan - ( ), then the imaginary and real part of the expansion (C-3) takes the following forms +0 (vR)n Im I(ivz) = eVY{y + in vR + n= n n cos nO) sin vx - (a+ + (VR)n sin n~) cos vx} (c-4) n=l n * n -52

-55Re I(ivz) = - eV{(+r+n+ ( R) cos nO) * sin vx n=1 n. *n oo (vR)n + (y + ln vR + Z n n! sin nG) * cos vx}. (C-5) n=l For z = x + iO it can be shown that (C-5) becomes Re I(iv(x+iO)) = - cos vx {y+ln vx + ( Vxn( l) } n=l 2n-2n' 0 v 2)2n-l(_l)n + sin vx { (l+sign x) + 2 +n=1 (2n ) (2n-l): -6) Using the usual notation, 00 00 u2n( n Ci u = -cos du = y + n u +n u2n u n=l 2n-2n! usin u 00 2n-l(_)n Si u = sin u du = - iu= Uito n=l (2n-)f.(2n-l) it follows that Re I(iv(x+iO)) = - Ci vx * cos vx + (r(l+sign x)+ x - Si vx) * sin vx (c-7) Note that as vx increases from zero, Ci vx and Si vx tend very rapidly to their asymptotic values 0 and, respectively. Hence, lim Re I(iv(x+iO)) = 2jr sin vx, x —0oo (c-8) lim Re I(iv(x+iO)) = 0. x-> -00

APPENDIX D DERIVATION OF PRESSURE FUNCTION In this appendix we wish to write the function f(x), defined in (2.16) by f 1 = (1)12 - Im(w(l)) Re i(w ) + ivw(l)] f(x) = c Re i(w- + zz ) 2 z zz Z in terms of the first order complex potential (3.5) w(l)() = m ln(z-9) + m ln(z-S-2ib) + m I{iv(z-j-2ib)} By definition (3.6) m -ik(z- -2ib) I = J k-v dk The derivative of this integral is,m -ik(z-_-2ib) d i /e-ik(z- -2ib) I = -i f e dk-V-f idk J -- -- d o o k-v or I = 1 - ivI. (D-1) z z-~-2ib Hence the derivative of the potential is w(l)(z) m 1 m 1 m..-n^ Wz (z) z-= 2i (z —2ib) v I{iv(- )} (D-2) and by (D-l) the second derivative becomes (l) -m 1 m 1. vm 1 w1 (z ) =_ 2J -t) + 2~ (z-S-2ib)2 + i + (z-S-2ib) - v2 I I(iv(z —2ib)} -54

-55so (iw) V(1)) = im 1 + im 1 zz z 2i (z )2 21i (z- -2ib)2 mv 1 mv 1 23r z- -2ib 2it z-_ Hence, Re (iw(1) - vw(1)) 2 mb(x-) mvx-) zz z y=-b (t((x-_)2+b2)2 [ (x-_ ) +b] and from (D-2) w(1)(x+ib) = -m ib _ - iv Iliv(x- -ib)}, (D-3) z it (x- )+b2 = so (l)l2 = (Re w(1))2 + (Im w())2 z z z Im I)2+, m b m 2 v Im I ) ( I m ) 2b + -v Re I) Therefore, the function f(x) can be written in the form f(x) U2{ 2v (2U Im I) + 2[2U (( 2 v Re I)]2 mU m U 2b(x-0 +b + 4[m Im I][2U ( 2x-0 + v )]} (D-4) 2~TU 2mrU ((x-) +b ) (x-)2 +b2 where I = I{iv(x-|-ib)}

-56APPENDIX E COMPUTER PROGRAM FOR FIRST- AND SECOND-ORDER WAVE PROFILES DIMENSION T(15),FACT(30),FAKT(60),-INTt(100),SUM(99),F(100), -________ 1 FUN K120)GQLGN1 30.-O.GFN2991,tG.E..N3(1hL)GIFN410 ), XVV ( 100 ) INTEGER J,I,K,ME,M,P,N. —..XEClTE ___R-_ EXECUTEE.R AAP. _..................___ THROUGH AAA, FOR J=O0,lJ.G.10 _._____ WEWHNEVNER J.G..-7... —... ---- -. _. -__ _ ----- -__- _........._____ _ ____ T( J).04......... __ _-QErLtsE...,..............~....^_._..,......... _.. _______________ T(J)=-.04*(5.+J*8./7.)/24...-. AAA.............ED O___NI I.ONAL............._ __ ___ _...... FACT(1)=1. -_._-_T__EIHRLLHB..R___ _=__F. t __l __tJ FG ~ 25___ BBB FACT(J+1)=FACT( J ) / (J+l)...K..._________________________, THROUGH CCC, FOR J=1,1,J.G.50 CC __ _. _ K_+1)=FAKJ KT(.J )*10.L / J __+ __ __ THROUGH LLL, FOR I=O,1,I.G.89 ___,___ Wh..S.WHF.NEN iVER.L..._ _ L 82..._.___.___.___.._ __..._._____ __ XS=(4.1-I*.2)*1.288..,._.._......JE.....DI.N.,...G,. _8_.5......... XS=+( (I-86)*.03+.C1)*1.288 ___._ __._._ MH.ERiLIS.E..................................................................._ XS=-((I-82)*.03+.C1)*1.288 _ _____ E...ND.F COND I T.IONAL........___.._... -___..___........ —WHENEVER XS.G.O.O... __,-.......4.___,_.E3i416/Z. -....._.._......... __. __. __...__ _ _ OTHERWISE __._ _ _ P_ _ _i=? 1.59*3.1416... __.~ _. _ _. _.... END OF CONDITIONAL ___ _______ WHENEVER (I.G.18.AND. I.L.23).OR. I.G.81 ME=4 OR WHENEVER I.G.22.AND. I.82 __ ME=25 (oTHER W I SE_ ________________________ OTHMERW ISE ME=13 END OF CONDITIONAL SUMA=O. SUMB=O. THROUGH MPM, FOR N=1,1,N.G.ME M=2*N P=2*N-1 ______WHENEVER ME.E.25 TRANSFER TO KKK SUMA=SUMA+(-1).P.N*(XS ).P.M*FACT(t)/M SUMB=SUMB+(-1).P.N*(XS ).P. P*FACT( P) /P TRANSFER TO MMM KKK SUMA=SUMA+(-1 ).P.N (XS /10. ).P. M*FAKT(M)/M SUMBUMBSMB+(-1).P.N*(XS /10.).P.P*FAKT(P)/P MMPM - CONTINUE XA=.ABS.XS LL INT(I)=-COS.( XS)*(.5772+ELOG.( XA)+SUMA) 1 SIN. ( XS)*(PI+SUMB) THROUGH EN2, FOR I=01TI.GG.20 EN2 GFN( I)=INT( I)

-57THROUGH EN31 FOR I=O, 1I.G.60 _____ EN3 GFN2(I)=INT(21+I) THROUGH TO3, FOR =61 1, I.G.80 T03 GFN2(I)=2.*3.1416*SIN.(1.288*(-.1-1*.2)) THROUGH EN4, FOR I=O,_ I.lG.3 ___ _ ___ ____ GFN3( I )=INT(82+I) __ 4_ GFN4( I) = INT ( 86+ I ) _ __ ___ _____ _ _ _________ START READ FORMAT SAME,B,U,A VECTOR VALUES SAME = $3F10.5 *$ ___ PRINT FORMAT JOHN,BU,AA VECTOR VALUES JOHN = $1H1, S20 2HB= F5,.2, S _5 2HU _lF5.2_S5 2H A= 1 F6.5 *$ _ L=32.2/(U*U) ______ Y=B YR=Y-2. *B YBS=YB*YB ___ PXX.(EPX=EXP, (LYB) _ _ _ TOEX=2. EPX _~___ _THROUGH EEE, FOR I =0 1,1..G.80_ ___ __ X=(4.-.2*I )1.288/L WHENEVER I.G.54 _______.. ME=51 OTHERWISE._..___. __........_............. ME=26 _ END OF CONDITIONL AL.___................... SUM(I)=C. THROUGH EEE, FOR K=Ol,1,vK.G.lO______ XK=X-K/10. R= SQRT (XK*XK+YBS 1 __ _ _ _ - RATO=XK/(-YB) _____ __J__ ____ ETATAN. R ATO) _ _= —-— _ _-. I- -) SUMA=O. SUMB=O. _ _ __.... ____-. THROUGH FFF, FOR M=1,1,M.G.ME ___ ____WHENEVER, ME E ~ 26 _ PART=(L*R).P.M*FACT (M)/M __T E__R ____OTHERW ISE __ _ _ __________ ---------- PART=(L*R/1O.).P.M*FAKT(M)/M END OF CONDITIONAL SUMA=SUMA+PART*COS. ( M*TET) FFF SUMR=SUMB+PART*SIN.(M*TET) __ AFUN=(.5772+ELOG.(L*R)+SUMA)*SIN. (LXK) BFUN=(3.1416-TET-SUMB)COS. (L*XK)___ EEE SUM(I)=SUM(I)+T(K)*(ATN1.(Y,XK)+ATNI.(YBXK) 1 +TOEX (AFUN+BFUN ))_ PRINT FCRMAT ENES VECTOR VALUES ENES = $1HOS18,1HXS9,7HFUNK(X) __ THROUGH GGG, FOR I=O,1,I.G.81 X= (4.1- I*.2)*1.288/L WHENEVER I.G.52 ME=51 OTHERWISE ME=26 END OF CONDITIONAL SUMC==O SUMD=O. THROUGH III, FOR K=O,1,K.G.10 XK=X-K/1O0. R=SQRT. (XK*XK+YBS)

-58__RATO=XK/(-YB)__ _.____ TET=ATAN. (RATO) N F V = XK* X K+YBS__ NEVS=NEV.P.2. _ - ___ SUMA=O. _____._.___ SUMB=O. THROUGH HHHL FOR M=1,lMG. ME WHENEVER MEE.E26 P AR T=( L* R),P P.M * FAC.I.LM_ _ _ _ OTHERWISE __ __ __________PPART=(L* R/10). PLM.*FAKTI M)M_ M__ END OF CONDITIONAL _-_____ - s __SUMA= SUMMMA+P R (TET) -_ _________S___.._.M.E.___ HHH SUMB=SUMB+PART*SIN.(M*TET).....__s = S UM C +T ( K EPX.(_i5.7L+f. G. L*R ) + SUMA ) *S I N. ( L *XK ) 1 +(3.1416-TET-SUMB)*COS. (L*XK)) ___ _SUMD=SUMDO+TK ) *B/NEV -L*EPX*_{ " 3.1416+TET+SUMB) S IN ( L*XK) 1 +(.5772+ELOG.(L*R)+SUMA)*COS.(L*XK)))..__. _ _ _ SUME=SUME+TK ) * (2. B*XK/NEVS+L*XK/NEV) F(I)=+U*U*(2,*(L*SUMC).P.2.+2.*SUMD.P.2.+4.*SUMC*SUME) ___GfG____ _ PRINT FCRMAT _N..I Xt F..(F )__ VECTOR VALUES NS = $1HO,S15,F6.2,F14.5 *$ __ ___ ___- THROUGH ENI_ FOR I=0_-O 1 I.G.8_________ 1__1 EN1 FUNK(I)=F(I) ____ _ HER =__32. Z*_ L2 _-.__ __ 5, _ ________ THROUGH TO1, FOR I=82,1,I.G.10O T IC 1 Et_11~........, ______.________.-.,....... THROUGH RRR, FOR J=O,1,J.G.91 __RRR B.JJ__,_X 1... 2l._JJ2= 841-L L-128_L ____ _ ____ TERM=.02*(-.5772-ELOG.(.01288)+1. )1.288/L ______ _PAC-. L2* ( 1.288/L)/3. _____. --— _8 —---- ___ _ SPASE=.C3*( 1.288/L)*(3./8.).RIL.I.....ECR.MAT. JQ B.ti....U At._-__t. PRINT FORMAT HANS.__.__ ___ VECTOR VALUES HANS = $1HOiS18.,iHXtS9_t6HYFIRSTI S8R8HFIRSTSQR __ i S5,7HYSECOND,S8,5HYWAVE *$ -__ _ TlHROUGH I QJ FOR I=-Oll.G.9_ _ _.9__ X=(4.-.2I )*1.288/L CCDO PRINT FORMAT KIM X1_ SUM( I ) VECTOR VALUES KIM = $1HO,S15,F6.2,F14.5 *$ THROUGH TTT, FOR N=10tltN.G.80 ____ X=(2.-(N-10.)*.2)*1.288/L NV=2*(N/2) INT1=O. THROUGH NNNI FOR M=O,1,M.G.20 WHENEVER M.E.O.OR. M.E.20 SM=1.O OR WHENEVER 2*(M/2).E.M SM=2.0___ ___________-____ OTHERWISE SM=4.O END OF CONDITIONAL NNN INT1=INTI+GFN1 (M)*FUNK (N+21-M) SM INT2=O. THROUGH-00, FOR P=Ol.,P.G.NV WHENEVER P.E.O.OR. P.E.NV SM=1.0 OR WHENEVER 2*(P/2).E.P SM=2.0 OTHERWISE

-59SM=4_.0 _ END OF CONDITIONAL nr0 c inITNT2= INT2+GFN2(P)*FUNK(N-P)*SM INT3=0. INT4=O. THROUGH QCQ, FOR K=O,1,K.G.3 WHENEVER K.E.O.OR. K.E.3 SM=1.0 OTHFRW IS SM=3.0 END OF CONDITIONAL__ XR=+(K*.03+.01)*1. 288/L+X XL=-(K*. 03+. 01Q)*1.288/L+X YR=TAB.(XRXV(O),FUNK(O)t,1,t 3t92,MIC) YL=TAB.(XL,XV(C) FUNK(O) t1. 13,92,MIN) WHENEVER MIC.E.1.0.AND. MIN.E.1.0 TRANSFER TO PPP _____ OTHERWISE ~~~_ ____TRANSFER TO END END OF CONDITIONAL PPP _INT3=INT3+GFN3 ( K)*YR*SM CCQ INT4=INT4+GFN4(K)*YL*SM YF=TAB. (XXV( ),FUNK(0), 11,392,MIK) WHENEVER MIK.E. 1.0 TRANSFER_ TO SSS _ OTHERW I SE __TRANSFER TO END END OF CONDITIONAL SSS INlTC5=YIERM......................_ ___ INTC1=INT1*SPAC __ ____ S.___.._...._ I NTC..2_I1N_.S _AC...._.______ INTC3=INT3*SPASE ___INTC4=I NT_4~*SPA_,.__.ASE __............ INTC=INTC +INTC2+INTC3+INTC4+INTC5 __ __ YVPAK=LC / 3. 1416*U*U) ______ ______ _ YSQ=SUM(N )*SUM(N )*L __ YWAVE=SUM(N )+YSQ+YPART TTT PRINT FORMAT NDS,X,SUM(N ),YSQ,YPART,YWAVE V YLe.O._ALtES NDS = $1H0,S15,F6.2s4F14.5 *$ TRANSFER TO LAST END. PR!NTI FTRMAT PER.___ _ _ VECTOR VALUES PER = $1HO,S10,7HNO GOOD *$ LAST TRANSFER TO START ___ _ END OF PROGRAM

APPENDIX F WAVE RESISTANCE COMPUTATIONS The "exact" formula for the wave resistance is b+R=(x) 2 2 1 R = 2P0 [-Px((xoy) + 0Y(xoy)]dy + gpr (xo). (F-1) 2 -0 0 2 The wave elevation and the potential far downstream are 1 2 ](x) = cos vx + 2 va cos 2vx + 5 cos V(X-T) 0(x,y) =-Ue (Y-b){c sin vx + 6 sin v(x-T)}. Using these expressions, both correct to the second order in wave amplitude, it is clearly seen by (F-l) that the wave resistance will be correct to the third order. So including terms up to third order, we have 2 (x) = a o c vos vx + v cos vx * cos 2vx + 2a6 cos vx * cos v(x-r) or 2 1212 (x) = 2 U + + a cos 2vx + va3 cos vx' cos 2vx + 6b cos VT + as cos 2vx * cos vT + a6 sin 2vx * sin VT, (F-2) and 2 + i2 2u2 2v(y-b)r 2 2 ( + 02) = v2u2e2v( -)a2 cos2 vx - 2o6 cos vx * cos v(x-T) + a2 sin2 vx + 2a6 sin vx * sin V(x-T)} or 2 + 02) = v22 e2v(y-b) {_a2 * cos 2vx - 2a6 cos 2vx * cos VT - 2a5 sin 2vx * sin vT}. (F-3) -60

-61Applying Equations (F-2) and (F-3) in the resistance formula (F-l), we llave 1 = 1, U {-2co[ s 2vx - 2a6 cos 2vx * cos V - 2a6 sin 2vx * sin VT} * 1 (1 + 2va cos vx) + 1 gp{1 CL+ I a2 cos 2vx + vaC cos vx * cos 2vx 2v 2 2 7 + a6 cos VT + a6 cos 2vx' cos VT + a5 sin 2vx' sin T}. Rewriting this as R -= 2 gp{- 1 y cos 2vx - c6b cos 2vx * cos VT - c5 sin 2vx * sin VT 2 2 - v cos 2vx ~ cos vx + 1 c2 + 2 a cos 2vx + voa cos vx * cos 2vx 2 2 + a6 cos VT + a6 COS 2vx * COS VT + a5 sin 2vx ~ sin VT} it is seen that the wave resistance takes the form R = L gp { + 2a6 cos VT}. (F-4) g4(F~

REFERENCES Bessho, M., "On the Wave Resistance Theory of a Submerged Body," The Society of Naval Architects of Japan, 60th Anniversary Series, Volume 2, (1957), 155-172. John, Fritz, "On the Motion of Floating Bodies I, Communications on Pure and Applied Mathematics, Volume 2, (1949, 15-57.Laitone, E. V., "Limiting Pressure on Hydrofoils at Small Submergence Depth," Journal of Applied Physics, Volume 25, (1954), 625-626. Lamb, Sir Horace, Hydrodynamics, Sixth Edition, New York: Dover Publications, 1945. Parkin, B. R., B. Perry and T. V. Wu, "Pressure Distribution on a Hydrofoil Near the Water Surface, Journal of Applied Physics, Volume 27, (1956), 242-240. Rayleigh, Lord, "On Waves,n Philosophical Magazine, Series 5, Volume 1, (1876), 257-279. Sharma, S. D., "A Comparison of the Calculated and Measured Free-Wave Spectrum of an Inuid in Steady Motion," International Seminar on Theoretical Wave-Resistance, Volume 1, University of Michigan, (1965), 201-270. Stokes, G. G., "On the Theory of Oscillatory Waves," Transactions of the Cambridge Philosophical Society, Volume 8, (1847), 441-455. Tuck, E. 0., "The Effect of Non-Linearity at the Free Surface of Flow Past a Submerged Cylinder," Journal of Fluid Mechanics, Volume 22, (1965), 4o01-414. Wehausen, J.V. and E. V. Laitone, Handbuch der Physik, Volume 9, Surface Waves, Berlin: Springer Verlag, 1960. -62