The Theory of Ship Motions*

ADVANCES IN APPLWD MECHANICS. VOLUME

18

The Theory of Ship Motions? J . N . NEWMAN Department of Ocean Engineering Massachusetts Institute of Technology Cambridge. Massachusetts

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

222 227

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235 236 237 240 241 242

111. The Boundary-Value Problem A . Exact Formulation . . . . .

B. The Linearized Problem . . . . . . . . . . . . . . . . . . . . . . . . . . C. Linear Decomposition of the Unsteady Potential . . . . . . . . . . . D . Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Slender Ships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Two-Dimensional Green Function . . . . . . . . . . . . . . . . . . B. The Three-Dimensional Green Function . . . . . . . . . . . . . . . . . V. Two-Dimensional Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Radiation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The DiNraction Problem . . . . . . . . . . . . . . . . . . . . . . . . . . C. Applications of Green's Theorem . . . . . . . . . . . . . . . . . . . . . D . LongWavelength Approximations . . . . . . . . . . . . . . . . . . . . VI . Slender-Body Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . The Outer Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Inner Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. The Inner Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII . Slender-Body Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Outer Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Inner Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. The Inner Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. The Long-Wavelength Solution . . . . . . . . . . . . . . . . . . . . . . VIII . The Pressure Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Added Mass and Damping . . . . . . . . . . . . . . . . . . . . . . . . . B. The Exciting Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

244 245 246 249 250 252 253 256 258 258 259 262 265 266 267 268 269 271 272 273 275 277 280

t Preparation of this article was supported by the National Science Foundation and by the Office of Naval Research. 22 1

Copyright 0 1978 by Academic Press. Inc All rights of reproduction in any form reserved. ISBN 0-12-002018-1

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I. Introduction Oceangoing ships are designed to operate in a wave environment that is frequently uncomfortable, and sometimes hostile. Unsteady motions, and structural loading of the ship hull, are two of the principal engineering problems which result. Early research in ship hydrodynamics was devoted primarily to operations in calm water, but a landmark paper “On the Motions of Ships at Sea” by Weinblum and St. Denis (1950) focused attention on this subject, and extensive work has followed. Ships generally move with a mean forward velocity, and their oscillatory motions in waves are superposed upon a steady flow field. The solution of the steady-state problem is itself of interest, particularly with regard to the calculation of wave resistance in calm water; a comprehensive survey has been given by Wehausen (1973). The opposite special case is that of wave interactions with a vessel which has no mean velocity; this topic is reviewed by Wehausen (1971), and more recent numerical solutions are described by Mei (1977). The problem of ship motions in waves can be regarded as a superposition of these two special cases, but interactions between the steady and oscillatory flow fields complicate the more general problem. All three topics are discussed by Ogilvie (1977). In order to predict its motions in waves, a ship may be regarded as an unrestrained rigid body with six degrees of freedom as defined in Fig. 1. The three components of translation are surge parallel to the longitudinal axis, heave in the vertical direction, and sway in the lateral direction orthogonal to surge and heave. Rotational motions about the same axes are roll, yaw, and pitch, respectively. If the unsteady motions of the ship and the waves are of small amplitude, systematic perturbation procedures can be justified, with the leading-order solutions linear in these small amplitudes. Furthermore, the ambient seaway

FIG.1. The coordinate system and six modes of ship motion.

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can be decomposed into individual components which are unidirectional and sinusoidal. In the jargon of this field, spectral analysis makes the study of ship motions in regular waves applicable ultimately to an irregular seaway. This synthesis is well known in other fields of applied mechanics, particularly in random vibrations and acoustics where the analogies are obvious. The first application of spectral analysis to ship motions was made by St. Denis and Pierson (1953). More recent developments are described by Price and Bishop (1974), and in the proceedings of two symposia: Society of Naval Architects and Marine Engineers (1974), and Bishop and Price (1975). The present article is restricted to ship motions in regular waves. The dynamics of ship motions are governed by equations of motion which balance the external forces and moments acting upon the ship, with the internal force and moment due to gravity and inertia. (Hereafter the term “force” will be used in a generalized sense to include the moment.) Assuming the ship to be in stable equilibrium in calm water, its weight is balanced by the force of hydrostatic pressure. Similarly, the steady drag and propulsive force are balanced. These steady forces may be neglected, and our attention is focused on the unsteady perturbations. The principal unsteady force acting upon an unrestrained vessel is due to the hydrostatic and hydrodynamic components of the normal pressure acting on the submerged surface. Additional force components which generally are neglected include the force on the ship’s propeller, viscous forces acting on the submerged surface of the hull, and aerodynamic forces acting upon the ship above the free surface. With the assumption of small unsteady motions, of the ship and surrounding fluid, linear superposition can be applied. Thus we consider separately the radiation problem, where the ship undergoes prescribed oscillatory motions in otherwise calm water, and the diflraction problem, where incident waves act upon the ship in its equilibrium position. Interactions between these two first-order problems are of second order in the oscillatory amplitudes, and may be neglected in the linear theory. The radiation problem may be decomposed further, by considering separately the six degrees of freedom defined above. In each of these modes, outgoing radiated waves will exist on the free surface. The existence of radiated waves implies a complicated time dependence of the fluid motion, and hence the pressure force. Waves generated by the body at time t will persist, in principle for an infinite time thereafter, and the resultant pressure force on the body will act similarly. This situation can be described mathematically by a convolution integral, with the fluid motion and pressure force at a given time dependent upon the previous history of the ship’s motion. In this respect the irrotational flow due

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to unsteady motions of a floating body is analogous to unsteady liftingsurface theory, where in the latter problem the source of “memory” is the shed vorticity in the wake. This aspect of ship motions is emphasized by Cummins (1962) and Ogilvie (1964). Similar effects due to viscous separation are analyzed by Brard (1973). Experimental techniques which account for the convolution effect are discussed by Bishop et a!. (1973) and by Wehausen (1978). For steady-state oscillatory motion a simpler description can be utilized: since the linear pressure field is oscillatory in time, and proportional to the amplitude of forced motion in each mode, the resulting force on the ship hull must be of the form

1

Fi(t)= Re eio‘

6 j= 1

tjtij/,

(i = 1, 2, ..., 6).

(1.1)

Here t j is the complex amplitude of the ship’s oscillatory displacement in each mode of motion, o is the frequency, and t i j is a complex transfer function which depends on the geometry of the ship hull, the frequency w, and the forward velocity U. Re denotes the real part, which is implied hereafter when the time dependence eiof is displayed. The complex transfer function t i j can be expressed in terms of its real and imaginary parts. From the physical standpoint a more useful decomposition is t I.J . = w2a.. - job..- c.. (1.2) IJ IJ IJ‘ The coefficientsaij, bij,cijare real and correspond, respectively, to the force components due to acceleration, velocity, and static displacement of the ship. It is important to emphasize that the coefficients in (1.2) are not constant, but depend on the same parameters as do t i j . In particular, the coefficientsaij, bij, and cij will depend in general on the frequency w, and the representation (1.2) is not meant to imply that t i j is a second-order polynomial in o. The damping codjcients bij in (1.2)are specified uniquely by the imaginary part of t i j . The diagonal elements bii and suitable combinations of offdiagonal elements, can be associated with the work done to oscillate the ship. These can be related to the energy flux in the radiated waves, under the usual assumption of an inviscid fluid. The separation of the added-mass coefficients aijand restoring coefficients cij is somewhat arbitrary. A physically appropriate subdivision can be affected by defining cij = - lim t i j . 0-0

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With this definition, the restoring coefficients cij can be associated with the hydrostatic pressure gradient and, to a lesser extent, with the steady-state dynamic pressure field. In the diffraction problem the ship moves in its steady-state orientation, in the presence of incident plane progressive waves of prescribed amplitude, wavelength, and direction. The resulting oscillatory exciting force on the ship hull is proportional to the wave amplitude A, and can be expressed in the form Fi(t)= AXieimf. (1.4) Here Xi is a complex force coefficient which depends on the wavelength and direction of the incident waves and on the ship geometry and forward speed. In a fixed frame of reference the incident-wave frequency oois related to the wavelength A and wavenumber K O = 2n/A by the dispersion relation. For water of constant depth h, wo = (gKo tanh K0h)”’,

(1.5)

and in the deep-water limit (h -+ 0 0 ) o0= (gK0)”’. In the moving reference frame of the ship, the incident waves arrive with the frequency of encounter o = 1010 - U K o cos

/3I.

(1.7) Here /3 is the angle of incidence, between the phase velocity of the waves and the forward velocity of the ship. In effect, (1.7) introduces a Doppler shift between the wave frequency and the frequency of encounter. The frequency of encounter is reduced infollowing seus (/3 = 0),whereas o is a maximum in head seus (/3 = n). The total oscillatory pressure force acting on the ship hull is the sum of (1.1) and (1.4). The equations for unrestrained motion of the ship in a prescribed incident-wave system follow from Newton’s equations. Since these equations of motion are linear and algebraic in ti,the only nontrivial task is to predict the coefficients in (1.2) and (1.4). The six modes of ship motions can be categorized in terms of the magnitudes of the corresponding restoring coefficients cii. These determine the scale of the natural frequency in each mode and the resulting response characteristics. For surface ships (as opposed to submarines), small vertical motions are opposed by a hydrostatic restoring force proportional to the waterplane area S. The resonant frequency can be estimated by neglecting hydrodynamic forces and equating the restoring force to the product of the ship’s mass

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and acceleration. Since the mass is proportional to the displaced volume V, the heave natural frequency is of order (1.8) where T is the draft. The principal hydrodynamic effect is to increase the effective mass of the ship and to reduce the natural frequency, but the order of magnitude is not changed. The same estimate applies to pitch, since the relevant moments of inertia of the waterplane area and ship’s mass are both proportional to L?.In this respect, heave and pitch are dynamically similar. The existence of a substantial static restoring force ensures that the amplitude of the motions will be relatively small, except near the resonant frequencies. The response at resonance depends on the magnitude of the damping forces and the degree of tuning with respect to the exciting forces. The damping of pitch and heave motions is due principally to the radiation of wave energy. In these modes the damping is generally subcritical but sufficient to prevent highly tuned resonant response. The exciting forces in pitch and heave are significant only if the wavelength is comparable to or greater than the ship length. From the dispersion relation it follows that excitation will occur principally from waves with a characteristic frequency ooIO(g/L)’”. For conventional ships L >> T, and the estimate (1.8) indicates that resonance will occur in combination with significant wave excitation only if the frequency of encounter is substantially greater than the wave frequency. When the Doppler shift (1.7)is considered, it follows that heave and pitch are most severe in head seas. Under these circumstances the acceleration and structural loading on the ship hull are most severe. In most cases, heave and pitch are the modes of greatest practical importance. Moreover, the small amplitudes and large inertial effects of these modes can be described by a linearized theory which assumes that the unsteady motions are of small amplitude and that the fluid is inviscid. Thus, theoretical approaches based on the methodology of applied mechanics have been most successful in predicting these modes. The static restoring moment in roll is small for conventional ships. This is due in part to the narrow beam. In addition, the vertical position of the center of gravity may be positioned to reduce this restoring moment, lengthen the natural period, and hence to reduce the angular acceleration. The radiation damping is weak in roll, especially at the low frequencies near resonance. Thus resonant motions occur with a large amplitude and with significant nonlinear and viscous effects. No satisfactory method exists for predicting the rolling motion of ships with engineering accuracy. The remaining motions are in the horizontal plane, unopposed by (gS/V)”2 = 0(g/T)’/2,

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hydrostatic restoring forces. The response is nonresonant, but motions of large amplitude may occur at low frequencies in following waves. Particularly serious in this context is broaching due to the combined effects of dynamic instability and prolonged unidirectional excitation. As in the case of roll, the modes of surge, sway, and yaw may be influenced significantly by nonlinear and viscous effects. A comprehensive discussion of ship motions in following waves is given by Oakley et al. (1974). Following a brief historical outline in Section 11, the remainder of this article is devoted to a linearized analysis of ship motions in incident waves. Slender-body approximations are applied to the ship’s hull form, without restricting the scale of the wavelength or frequency. All six modes of ship motion are considered. In our analysis the fluid is assumed ideal, with irrotational motion, and unbounded except for the submerged portion of the ship hull and the free surface. Surface tension is neglected. The ship hull is assumed symmetrical, in the “port-and-starboard” sense, about a vertical centerplane which contains the longitudinal axis. The unsteady motions are assumed sinusoidal in time, and they are of small amplitude. These assumptions, and the discussion above, imply that the results may be of the greatest practical value for predicting heave and pitch motions. The exact and linearized boundary-value problems are formulated in Section 111. Fundamental solutions are summarized in Section IV, including the two- and three-dimensional source potentials or Green functions which satisfy the linear free-surface boundary condition. Two-dimensional solutions of the boundary-value problem adjacent to each section of the ship hull are outlined in Section V. The method of matched asymptotic expansions is used with these twoand three-dimensional solutions, to derive a slender-body approximation for the three-dimensional oscillatory flow field. This analysis is carried out for the radiation problems in Section VI, and for the diffraction problem in Section VII. In Section VIII these solutions are used to determine the hydrodynamic pressure force acting on the ship.

11. History Early sailing vessels favored trade-wind routes with following seas and were unable to move with great speed to windward. For this reason, heave and pitch were not of great importance whereas rolling motions in waves were reduced by the stabilizing influence of the sails. The first steamships could not attain high speed in head seas. However,

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the absence of aerodynamic damping increased the importance of roll. This may explain the initial study of rolling motions by William Froude (1861). Subsequently, the importance of pitch and heave motions increased with the power and speed of ships, attracting attention and study by the Russian naval officer Krylov (Kriloff, 1896). Froude and Krylov derived differential equations of motion for the inertial and restoring forces of the ship. No attempt was made to analyze the hydrodynamic disturbance associated with the presence of the ship hull. Only the pressure field of the undisturbed incident waves was considered, and the resultant force on the ship has become known as the Froude-Krylov exciting force. The first significant step to account for the hydrodynamic disturbance due to a ship hull of realistic form was the steady-state wave-resistance theory of Michell (1898). Michell assumed the ship to be thin, with small beams (B) compared to the ship length, draft, and wavelength. In this respect, his approach is related to the solution of the thickness problem in thin-wing theory, which followed subsequently. Michell recognized the possibility of extending his theory to include unsteady motions, but a promised sequel on heave and pitch was never published. Another major advance in accounting for the ship’s hydrodynamic disturbance followed in a study by Lewis (1929), of the added mass associated with hull vibrations in structural modes. In this problem the characteristic frequency is sufficiently large such that inertial effects are dominant, and gravitational forces can be neglected. Thus wave effects are ignored, greatly simplifying the analysis. Lewis assumed the ship hull to be slender, and used a striptheory approach to integrate the hydrodynamic force longitudinally in terms of the two-dimensional characteristics of each transverse section. Not content with this simplified approach, Lewis derived three-dimensional correction factors by reference to the exact solutions for a prolate spheroid. This appears to be the first development of a strip theory in ship hydrodynamics. In an inviscid fluid, the vertical motion of a thin ship is equivalent mathematically to the “wave-maker problem” of a prescribed normal velocity on a vertical plane. The latter problem was solved by Havelock (1929),during a prolific career devoted primarily to the theory of wave resistance. It was not until his last published paper, however, that Havelock (1958) explicitly studied the oscillatory motions of a thin ship. A comprehensive analysis of pitch and heave motions was made in two papers by Haskind (1946a,b). Green’s theorem was used to construct the velocity potential due to the presence of the ship hull, and the necessary Green’s function or source potential was derived. The thin-ship approximation was invoked to solve the resulting integral equation. Haskind treated

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the initial-value problem for arbitrary time dependence, and the special case of sinusoidal motion. A notable feature of Haskind’s work is the decomposition of the velocity potential into a canonical form which includes separately the solution of the diffraction problem and solutions of the radiation problem for each mode of oscillatory ship motion. The thin-ship approximation was reexamined in a critical fashion by Peters and Stoker (1957). A systematic perturbation procedure was adopted with the ship’s beam and the unsteady motions assumed to be of the same small order of magnitude. On this basis the Froude-Krylov exciting force is the only first-order hydrodynamic force. This rather trivial first-order theory essentially confirmed the approach of Froude and Krylov, but cast doubt upon the value of Haskind’s more extensive work. The thin-ship approximation was refined by Newman (1961), with a more accurate statement of the boundary condition on the oscillatory ship hull. A systematic expansion in multiple small parameters was used to avoid the results of Peters and Stoker (1957). Computations, however, of the damping coefficients presented by Gerritsma et al. (1962) did not correlate well with experiments. It is obvious that the (inviscid) hydrodynamic disturbance due to vertical motions of a thin ship is small in proportion to the beam. To avoid this situation, Peters and Stoker (1957) advocated a complementary “flat-ship” approximation with the draft small compared to the beam and length. This leads to an integral equation similar in form to that of lifting-surface theory, but with a more complicated kernel. Moreover, the intersection of the free surface and the ship is a singular region which must be examined with great care. Steady-state solutions have been derived for planing boats, as described by Ogilvie (1977), but unsteady solutions are restricted to the case of zero forward speed. Typical ship hulls are elongated, with the beam and draft of the same small order of magnitude compared to the length. Thus it is logical to develop a three-dimensional approximation analogous to the slender-body theory of aerodynamics. This was done initially for the steady-state waveresistance problem by Cummins (1956); subsequent references are given by Ogilvie (1977). An important restriction resulted from the assumption, not always explicitly stated, that the ship is slender relative to the characteristic wavelength. Thus, the beam and draft were assumed small compared to the wavelength scale U 2 / g as well as the ship length L. Equivalently, the Froude number U(gL)-‘’’*’ was assumed to be of order one. Unsteady solutions based on similar slender-body assumptions and applicable to the prediction of ship motions in waves were derived by Ursell (1962), Joosen (1964), Newman (1964), Newman and Tuck (1964), and Maruo (1967). Here the long-wavelength assumption rZ = O(L)seemed phys-

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ically appropriate and mathematically convenient. In this case, however, as in the thin-ship approach, most nontrivial hydrodynamic effects are of higher order by comparison to the hydrostatic restoring force and the Froude-Krylov exciting force. Moreover, the inertial force due to the body mass is of higher order, with the result that the leading-order equations of motion are nonresonant. Nevertheless, for shiplike vessels without forward speed, this simple longwavelength theory gives reasonable predictions of the pitch and heave motions, as illustrated in Fig. 2. An explanation of this fortunate situation is the disparity between the natural frequencies and the wave frequencies where the exciting force and moment are significant. A different situation results if the ship moves in head seas with forward velocity. Here the domains of resonance and significant excitation overlap, 1.25 X

F = 0.00

0 X

X

F = 0.14

1.00

0.75

l€Sl A

0.50

0

i

0

0.25

I

I

0.00

I

%&I

a

I

5.0

I

2.0

I

1.0

x

I

I

0.5

0.3

c-

L

FIG.2. Amplitude of heave motion, per unit wave height, predicted from the leading-order low-frequency slender-body theory, and compared with experimental measurements for zero forward velocity and a Froude number F = 0.14. This figure is reproduced from Newman (1977), where a similar plot of the pitch motion is included.

23 1

Theory of Ship Motions

due to the Doppler shift toward higher frequencies of encounter. The effect of this shift is indicated by the experimental data with forward velocity in Fig. 2. Naval architects have not awaited a three-dimensional theory of ship motions which is both rigorous and practical. Instead the numerical solutions of a simpler class of two-dimensional problems have been utilized, where the body floats on the free surface and performs small oscillatory motions without forward speed. The wavelength and body dimensions are not restricted, but the solution is most useful, and nontrivial, when both length scales are of the same order of magnitude. This type of problem was first solved rigorously by Ursell (1949) for the heaving motions of a halfimmersed circular cylinder. Various extensions and generalizations have been carried out subsequently, as described in the survey of Wehausen ,(1971). A compendium of two-dimensional results is given by Vugts (1968), with more recent numerical techniques described by Chapman (1977) and Mei (1977). Typical results for the heave added mass and damping of a rectangular cylinder are shown in Figs. 3 and 4. The first utilization of two-dimensional results in a three-dimensional striptheory approximation for ship motions was made by KorvinKroukovsky (1955). This work was self-contained in the sense that all of the

I

0.0 0.0

I

I

0.5

u4-

1.0

I

1.5

FIG. 3. Added-mass coefficients for a family of two-dimensional rectangular cylinders, based on the computations of Vugts (1968). With the normalization shown, the added-mass coefficient is logarithmically infinite for w 0, and also for B/T -,0.

-.

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J . N . Newman 1

I.o

t b33

pe'w

0.5

0.a

FIG.4. Damping coefficients for a family of two-dimensional rectangular cylinders, based on the computationsof Vugts (1968).The scale on the right gives the exciting-forcecoefficient, in accordance with the Haskind relations (5.30).

relevant forces were considered in a linearized analysis of pitch and heave motions in head seas. Use was made of concepts from the slender-body theory in aerodynamics, supplemented by shrewd physical insight, to account for the effects of forward speed. Some refinements and extensive experimental comparisons were provided in a sequel by KorvinKroukovsky and Jacobs (1957). Theoretical workers were slow to accept the striptheory approach of Korvin-Kroukovsky, due to the lack of a systematic and rational derivation. This defect was of less concern to practical naval architects, who recognized the computational simplicity of the strip theory and the generally satisfactory agreement with experiments. In both respects, this nonrigorous approach compared favorably with the thin-ship theory. Typical calculations for heave and pitch are compared with experimental data in Fig. 5. More comprehensive evaluations of the strip theory are given by Salvesen et al. (1970) and Gerritsma (1976). Grim (1960) provided some of the first systematic calculations of the two-dimensional added-mass and damping coefficients for shiplike forms, suitable for use in the strip theory. Not content with this approximation, Grim also proposed a heuristic correction based on the distribution of threedimensional sources on the surface of the ship, with the source strength

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FIG. 5. Heave and pitch motions predicted from strip theory and compared with experiments for a Froude number F = 0.3. (From Newman, 1977.)

determined from the two-dimensional solution at each section. Further corrections were introduced to account for the induced normal velocity of these sources at other longitudinal positions along the ship hull. By comparison to the slender-body results illustrated in Fig. 2, the superiority of the strip theory for predicting pitch and heave motions in head waves can be explained by the practical importance of the resonant frequency regime. From the estimate (1.8) this is precisely the shortwavelength regime where o z T / g = 0(1) and thus, for a slender ship, o24g 9 1. A rational foundation for the strip theory was suggested by Vossers (1962) and Joosen (1964), based on the slender-body assumptions B/L = E << 1, BIT = O(1). These authors showed that in the short-wavelength regime the solution of the three-dimensional radiation problem in the near field adjacent to the ship hull is identical to the strip-theory solution. In the absence of forward speed, this conclusion is fairly obvious: if a slender ship is radiating short waves with slowly varying phase along the length, these waves will be locally two-dimensional, and three-dimensional interference effects are absent from the inner region. The principal effect of forward speed is to introduce the convective derivative Ud/ax, but this is dominated by the time derivative in the high-frequency regime. A systematic analysis of the short-wavelength slender-body problem for heave and pitch was carried out by Ogilvie and Tuck (1969). In addition to

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the leading-order zero-speed striptheory results, Ogilvie and Tuck conThese sistently retained the higher-order terms of relative magnitude E higher-order corrections are linearly proportional to U and thus provide a rational approximation for the effects of the ship’s forward speed. In spite of the work of Ogilvie and Tuck (1969), certain aspects of the strip theory remain unsatisfactory from the rational viewpoint. The principal questions concern the validity of the solution at lower frequencies, the emergence of forward-speed effects only as higher-order corrections, and the intractable nature of the diffraction problem in short incident waves. From the underlying assumptions, it is clear that the strip theory is invalid at low frequencies of encounter. An example is that the (two-dimensional) added-mass coefficient for vertical motions is logarithmically infinite as the frequency tends to zero, as shown in Fig. 3. This is of little importance for predictions of ship motions in head seas since it is the product w’aij which occurs in the equations of motion. Since the strip theory correctly predicts the hydrostatic effects which are dominant as w -+ 0, the resulting equations of motion are also correct for w -+ 0. (However,-this is a consequence of the fact that the Froude-Krylov force is left in its original form; if it is expanded consistently for large frequencies, the result is divergent for w -,0.) In the strip theory, forward speed affects the hydrodynamic force simply by introducing terms proportional to (U/w) and (Ulw)’. By assumption, U / w = O(E’’’),and thus the effects of forward speed are higher-order corrections to the zero-speed leading-order theory. Ogilvie and Tuck (1969) retain terms of relative order E’/’ while neglecting higher-order terms O(E).As such, their theory is mathematically consistent, but at variance with most of the intuitive versions of the theory which retain some additional corrections proportional to (U/w)’. The diffraction problem is complicated in the strip-theory synthesis by the assumption of high-frequency, short-wavelength incident waves. These are rapidly oscillatory along the ship length, and the scattering potential must be sought as the product of a longitudinal oscillatory function times a slowly varying solution of the Helmholtz equation. For head seas the latter problem is singular and a special analysis is required. Various schemes have been devised to circumvent this difficulty, with Green’s theorem used to replace the diffraction problem by a simpler radiation problem. An alternative approach outlined by Newman (1977) restricts the incident wavelength to be intermediate between the ship’s length and beam; the resulting formula for the exciting force is used in Fig. 5. The conventional strip theory is deficient not only for low frequencies, but also for high speeds. A complementary approach initiated by Chapman (1975) uses a high-Froude-number approximation suggested by Ogilvie (1967). The flow at each section along the ship is analyzed in a quasi-two-

’”.

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dimensional manner, with interactions propagated downstream by the freesurface condition. This technique is discussed further by Ogilvie (1977). Chapman’s computations for the sway and yaw response of a thin ship are supported by impressive agreement with experiments. Ship motions are of primary interest offshore in deep water. However, the operation in coastal waters of supertankers, and other ships of deep draft, has increased the importance of the shaliow-water regime. In the slenderbody theory, the effects of finite depth are confined to the outer field for h/L = 0(1),but for depths comparable to the beam and draft the near field is affected and the order of magnitude of the hydrodynamic forces is increased. Beck and Tuck (1972) have studied the latter regime, using shallow-water approximations to simplify the results. The necessity of analytic approximations may be questioned in the present era of numerical fluid mechanics. Only Chang (1977) has reported success with the direct numerical solution of the linearized threedimensional ship-motion problem. Calculations of the added-mass and damping coefficients are presented for a realistic ship hull in all modes except surge. The results show reasonable agreement with experiments, except in the roll mode, and confirm the limitations of the striptheory approximations.

111. ‘Ihe Boundary-Value Problem It is helpful to define three Cartesian coordinate systems, with xo = (xo, yo, zo) fixed in space, x’ = (x’, y’, z’) fixed with respect to the ship, and x = (x, y, z) moving in steady translation with the mean forward velocity of the ship. The space-fixed system xo is the simplest in which to express the free-surface boundary condition, whereas the shipfixed system x’ is the best in which to derive the boundary condition on the ship’s wetted surface. The steadymoving coordinate system x is an inertial reference frame in which the motions are periodic. We take z, = 0 as the plane of the undisturbed free surface, the xo-axis positive in the direction of the ship’s forward velocity, and the zo-axis positive upward. The steady-moving coordinate system is defined by the transformation

x = (xo - Ut, yo, zo),

(3.1)

with U the mean forward velocity of the ship. The shipfixed coordinate system is defined such that x’ = x in steady-state equilibrium.

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A. EXACTFORMULATION With the assumptions noted in Section I, the fluid velocity vector V(x,, t ) is equal to VCP, with the velocity potential CP(xo, t ) governed by Laplace’s equation V’CP = 0 throughout the fluid domain. The fluid pressure p(x,, t ) is given by Bernoulli’s equation p = -p(CPt

+ +vz + gz,) + pa.

(3.2) Here p is the fluid density, g is the gravitational acceleration, and pa is the atmospheric pressure which is assumed constant. In (3.2) and hereafter, when the independent variables ( x , t ) appear as subscripts partial differentiation is indicated. On the submerged portion of the ship’s surface S, the normal velocity is equal to that of the adjacent fluid. The appropriate boundary condition is

-

(Vs - V ) n = 0 on S, (3.3) where V, is the local velocity of the ship’s wetted surface. The unit normal vector n is defined to point out of the fluid domain. The free surface is defined by its elevation zo = c(xo, y o , t). On this surface, the kinematic boundary condition is expressed by means of the substantial derivative DfDt 5 afat V . V, in the form

+

c.

(D/Dt)(C - zo) = 0 on z, = (3.4) Since the position of the free surface is unknown, an additional dynamic boundary condition is imposed, that the pressure on the free surface is atmospheric. From Bernoulli’s equation (3.2) it follows that

+ &ifz+ gz,

=0

on z , = c.

(3.5) This boundary condition can be used to determine the free-surface elevation from the implicit equation

c = - (1/9)(@t+ w2)zo=[.

(3-6) Since (3.5) holds on the free surface for all time, its substantial derivative can be set equal to zero. This gives an alternative boundary condition for the velocity potential, CPtt

+ 2VCP

V a t + $VCP * V(V@ . V@)

+ gCPz, = 0

on zo = [.

(3.7) Equations (3.3) and (3.7) are the principal boundary conditions of the problem, valid on the ship hull and on the free surface, respectively. If the fluid domain contains no other boundary surfaces, the additional requirements are imposed such that V -+ 0 as z, + - 00, and such that the energy *

23 7

Theory of Ship Motions

flux of waves associated with the disturbance of the ship is directed away from the ship at infinity. The latter is the radiation condition. The problem stated above is exact within the limitations of an ideal incompressible fluid. However, the nonlinear free-surface condition precludes solutions without further simplification. Moreover, there are unresolved questions regarding the explicit form of the radiation condition, and the singularities at the intersection of the ship hull and free surface. Further progress requires the fluid motion to be small in some sense. B. THELINEARIZED PROBLEM In the theory of water waves it is customary to assume the amplitude of the oscillatory wave motion to be small by comparison to the wavelength. Equivalently, the slope of the free surface is assumed small compared to unity. With the neglect of second-order terms quadratic in derivatives of @, Eq. (3.7) can be replaced by the linearized free-surface boundary condition

+

on zo = 0. (3.8) Note that this condition is imposed on the mean position of the free surface, since the difference between the value of @ or its derivatives on zo = [ and zo = 0 is a second-order quantity. There are extensive solutions of Laplace's equation and the linearized free-surface condition (3.8), as described especially by Wehausen and Laitone (1960). Of particular importance is the plane-progressivewave of constant amplitude A and sinusoidal profile, for which the velocity potential in deep water is given by Ort gOZ,= 0

@ = ( i g A / o , ) exp[K,(z,

- ix, cos fl

- iy,

sin fl) + io,t].

(3.9)

Here K O is the wavenumber, with I = 27r/K0 the wavelength, oo is the radian frequency in the space-fixed reference frame, and fl denotes the angle of wave propagation relative to the x,-axis. Substituting (3.9) into (3.8) gives the dispersion relation (1.6). In the steady-moving reference frame the velocity potential can be redefined in the form

a

YO7

'07

l) = @(.

+ U t , y, z, l)

+(x3 y7 Thus, in accordance with the Lorentz transformation, '('07

a,= 4(xo - U t , y o , z,, at

t) =

-

(:t

z7

l).

- u - +(x, y, z, t).

(3.10)

(3.1 1)

238

J . N . Newman

Transforming the linear free-surface condition in this manner, it follows that

4,, - 2U4,,

+ U2&, + g& = 0

on z = 0.

(3.12)

In the moving reference frame the plane-wave potential (3.9) takes the form

4 = (igA/oo) exp[Ko(z - ix cos /3 - iy sin /3)

+ iot],

(3.13)

where the frequency of encounter is defined by (1.7). If the ship is stable, and if the amplitude A of the incident wave system is small, the oscillatory motions of the ship and surrounding fluid will be proportional to A. Linearization of the unsteady problem can be justified on this basis. However, suitable geometric restrictions must be placed on the ship hull to ensure that the steady-state disturbance is small. Before restricting the geometry for this purpose, we first consider the general case where only the oscillatory flow is linearized. An overbar will be used to denote the velocity potential due to the steady forward motion of the ship, (3.14)

q x 0 , t) = UI$(x).

The velocity vector of the steady flow relative to the moving reference frame is

w = UV(8 - x).

(3.15)

The boundary condition on the hull surface in its steady-state position S takes the form onS.

W.n=O

(3.16)

In the moving reference frame the nonlinear free-surface condition is

+ w .v(w’) + gI$), = o

on z =

c,

(3.17)

with the steady free-surface elevation given by the implicit formula

e = -(1/2g)(W2

- v”),,,.

(3.18)

The total potential can be written in the form q x , , t ) = 4(xy t ) = UI$(X)

+ cp(x, t),

(3.19)

where the unsteady component cp is assumed small. Neglecting second-order terms in cp, the free-surface boundary condition is

4w

*

+ d), + cp,, + 2 w . vcp, + w V ( W - Vcp) + # V q . v(W) + gqz = 0 on z = [. V(W2)

*

(3.20)

Theory of Ship Motions

239

The corresponding expression for the free-surface elevation is

Here the error is O(cpz), and the last form of (3.21) follows from (3.18) and a Taylor-series expansion. Using this formula to solve for the difference (C - r) gives

c = e - “4% + w . Vcp)/(g + w - W,)l,=i. (3.22) The contribution from the steady terms in (3.20) can be evaluated by expanding from c to c and by using (3.17). Thus, the unsteady velocity potential is governed by the first-order free-surface condition

+ cp,,+

2 w . vcp,

+ w . V(W . Vcp)

+ +Vcp . V ( W ) + gcp, = 0

on z =

c.

(3.23)

If the perturbation of the steady flow due to the ship is neglected, W = - Ui and (3.23) reduces to (3.12). The linearized boundary condition on the hull requires a similar analysis. We begin by decomposing the velocity V, in the form

-

V, = Ui

+ a,

(3.24)

where a = x x’ is the local oscillatory displacement of the ship’s surface and the overdot denotes time differentiation in the reference frame of the ship. Since a is a small oscillatory quantity, this vector displacement can be expressed as a =6 +R x (3.25) XI.

Here 6 and R denote the unsteady translation and rotation of the ship, relative to the origin x’ = 0. Substitution of (3.15), (3.19), and (3.24) in (3.3) gives the unsteady boundary condition cp,=a*n-W*n

onS.

(3.26)

The last term must be retained, in spite of (3.16), since W = O(1) and the difference in the boundary values of W n on S and S is O(a). The two first-order contributions to W n on S are from the rotation of the shipfixed coordinate system, and from the gradient of the steady flow

-

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J. N . Newman

field. After accounting for both effects,

(W * n), 2 ([W

- R x W + (a

*

-

V)W] n)s.

(3.27)

Invoking the steady boundary condition (3.16), and substituting (3.27) in (3.26), it follows that (p,, = [a + R x W - (a V)W] * n (3.28) on S , S. 3

Since each member of (3.28) is O(a),this boundary condition can be applied either on S or S with the difference O(a2). An alternative to (3.28) can be derived using (3.25), in the form q,, = [a

+ (W . V)a - ( a . V)W] - n

on S , S.

(3.29)

The first two terms in brackets give the rate of change of a in a frame of reference moving with the steady flow. Finally, since a and W have zero divergence, a vector identity gives the more compact expression (p,,

= [a

+ V x (a x W)] - n

on S , S.

(3.30)

The boundary condition (3.30) was derived by Timman and Newman (1962) to account in a consistent manner for the interaction between the steady and oscillatory flow fields. In most prior ship-motion analyses an incomplete form of (3.30) was used, which led to an erroneous asymmetry of the coupling coefficients between heave and pitch. If the perturbation of the steady flow field due to the ship is neglected in (3.29), W = -Ui, and thus qn=

[(;

= [a -

-

Ug)a]*n

U(R x i)] * n.

(3.31)

The term proportional to U can be interpreted as the product of the ship's forward velocity and the angle of attack due to pitch and yaw.

c. LINEARDECOMPOSITION OF THE UNSTEADY POTENTIAL Since the unsteady motions are assumed small, the potential (p in (3.19) can be decomposed linearly into separate components due to the incident wave, each of the six rigid-body motions, and the scattered disturbance of the incident wave. With the restriction that the unsteady motions are sinusoidal in time with the frequency of encounter w, the motions of the ship are denoted by (3.32) 5 = (51, 5 2 , 53)eiOt, R = (al,R2, 03)ei"'

= (54, 5 5 , 56)ei0'.

(3.33)

24 1

Theory of Ship Motions

With this notation, the unsteady component of the velocity potential can be expressed as (3.34) Here cpo is the incident-wave potential of unit amplitude, cpo = (ig/o,) exp[Ko(z - ix cos fl - iy sin

p)],

(3.35)

and cp, is the scattered potential such that

(d/dn)(cpo

+ cp7) = 0

on S.

(3.36)

The components cpj ( j = 1, 2, . . . , 6) in (3.34) are the radiation potentials due to motions of the ship with unit amplitude in each of six degrees of freedom. From (3.25) and (3.29), these are governed by the hull boundary conditions cpj, = ionj

+ Umj

on S.

(3.37)

Here the components n j are defined as (n1,

n2, n 3 )

(3.38)

= n,

(n4, n 5 , H 6 ) = (x x n),

(3.39)

and, following Ogilvie and Tuck (1969), (ml, m2, m 3 )= rn = - (n . V)W,

(m4, m,, m6) = -(n

*

V)(x x W).

(3.40) (3.41)

On the free surface, the radiation potentials satisfy the boundary condition (3.23). The same condition applies to the sum (cpo + q7),but since the incident wave satisfies (3.12), the scattered potential cp7 satisfies an inhomogeneous form of (3.23) with derivatives of cpo on the right-hand side. D. SPECIAL CASES

The steady flow field W is a major complexity in the free-surface and hull boundary conditions. Moreover, while it is implicit that W is known in the boundary conditions for cp, the solution of the nonlinear steady-flow problem is beyond the present state of this field. Thus, regardless of whether the steady-flow problem is of direct interest, it must be simplified in order to solve for the unsteady flow. The simplest case of a moving ship is obtained when the hull shape is restricted to be a small perturbation from a plane which contains the x-axis.

242

J . N . Newman

As in the analogous case of thin-wing theory, perturbations of the steadystreaming flow - Ui are small, in proportion to the thickness of the hull, and to leading order the boundary conditions (3.12) and (3.31) apply. Examples are the thin ship and the flat ship, mentioned in Section 11. While it is more appropriate to the study of submarines than it is to surface ships, the assumption that the ship hull is “deeply submerged” should be noted as an alternative to restrictions on the shape of the hull surface. The leading-order steady flow near the body is that for motion in an infinite fluid, and on the free surface the steady disturbance from the body can be linearized. Thus the unsteady problem can be treated with relative ease. This assumption was utilized extensively in the collected works of Havelock (1963), although studies there of the unsteady problem generally were carried out with the incomplete hull boundary condition (3.31) instead of (3.28-3.30). The steady-flow problem also can be simplified if one adopts the “slowship” assumption that the forward velocity is small in some sense. This is a topic of contemporary importance in wave-resistance theory (Baba and Hara, 1977; Keller, 1978). The implications for unsteady motions in waves have not been explored. From the boundary conditions it is clear that as W + 0, the zero-forward-speed case will result, except possibly for a regime where the frequency of encounter is small. Singularities in the steady-state solution may be expected to occur at the bow and stern stagnation points. E. SLENDER SHIPS Finally we consider the consequence of restricting the geometry of the ship hull such that its beam (B) and draft (T) are small compared to the length (L) by factors of order E G 1. It is convenient to choose the unit of length such that L = O(l), whereas (B, T) = O(E).To leading order the components of the factors in (3.38-3.39) reduce to

n = (n1, n 2 , 4 ) , (x x n) = (yn, - zn,, - x n 3 , xn2).

(3.42) (3.43)

The first components of these two vectors are O(E),whereas the remaining contributions are O(1). We shall retain the first components in order to derive nontrivial results for ship motions in surge and roll. In the inner region where (y, z) = O(E),a coordinate stretching argument . can be used to show that gradients in the y-z plane are of order 1 / ~ Except for the diffraction potential rpo + rp,, gradients in the longitudinal direction are 0(1),and the velocity potential is governed to leading order by the

Theory of Ship Motions

243

two-dimensional Laplace equation 4yy

+

422

= 0.

(3.44)

Equation (3.44) applies in particular to the steady potential 0, and from the hull boundary condition (3.16) 6 = O(E2).tThus, to leading order in E, the components in (3.40-3.41) reduce to

m = - ( n 2 a/ay + n3 a/az)v$, -

m4 = - n 2 4 *

m, = -xm3

+ n36y + ym3 - zm2,

+ n3,

(3.45) (3.46a) (3.46b) (3.464

m6 = xm2 - n 2 .

The factors rn, and m4 are O(E),whereas the remaining components of (3.45-3.46) are 0(1), as in (3.42-3.43). The free-surface boundary condition must be dealt with separately in the inner and outer regions. In the outer region (y, z) = 0(1), gradients of the potential are 0(1) in all three directions and, since the steady potential is O(E’), the leading-order free-surface condition is (3.12). In the inner region, the steady free-surfacecondition (3.17) is dominated by the second term, and the elevation (3.18) is O(E’).The leading-order free-surface condition for the steady problem is the “rigid-wall” boundary condition

$,=O

on Z = O .

(3.47)

The resulting inner solution for 6 and the factors mj in (3.45-3.46) are independent of the forward velocity U. The free-surface condition for the unsteady potential rp can be derived in the inner region from (3.23). Using the above results for I$, it follows that

+ u2(Px* + 26,rp,t + 6yyrpt - U 6 y y r p x - 2u6x,(P, - 2U6YrpXY + 4 y 2 ( P y y + 3 6 y 6 y y ( P , + 9rpz

(Pa - 2urpxt

=0

+ O(~rp,~ V r p )

on z = 0.

(3.48)

Since d/az = O ( E - ’ ) ,the last term on the left side of (3.48) is O(V/E).The remaining terms are O(rp), if the time derivatives and gravity are assumed to be O(1). Thus, with the assumption that the frequency of encounter is 0(1), the rigid-wall boundary condition (3.47) applies also to the unsteady potential. I n this case there are no wave effects in the inner region, to leading order in the slenderness parameter E. t Strictly, 6 = O(E’ log essential.

E).

Logarithmic error factors will be deleted unless their display is

244

J . N . Newman

Wave effects can be introduced in the inner problem to leading order if or if o = O(&-(l/’)). In this circumstance the leading-order terms of (3.48)are qtt g q z = 0 on z = 0, (3.49)

a2/atz = O ( E -l),

+

corresponding to the zero-forward-speed condition (3.8). Justification for assuming that o = O(E-(”’)),or that o’B/g = 0(1),can be based on the fact that this is the resonant frequency domain (1.8) for heave and pitch. As noted in the Introduction, this high-frequency regime is of the greatest practical importance for ship motions in head seas. The fact that it leads to a relatively simple free-surface condition and, ultimately, to a correspondingly simple strip theory, is an additional reason for its study, but not the only one. The error in (3.49) is a factor 1 + O ( E ~ /If~ the ) . terms of that order are retained a more accurate free-surface condition results, in the form

+

qff g q Z - 2Uq,,

+ 2$,,qyf+ $Yyqt= 0

on z = 0.

(3.50)

The free-surface boundary conditions (3.47) and (3.49-3.50) lead to two separate theories for ship motions, applicable respectively for long and short wavelengths or low and high frequencies. These are discussed in the Introduction, and in greater detail by Ogilvie (1977). In Sections VI-VII we shall develop a more general approach, which seeks to unify the two separate theories. The velocity potentials q j in (3.34)will be derived from an asymptotic analysis which is valid to leading order in the body slenderness, for all values of o I O(E”(’/’)).The free-surface conditions to be satisfied in the outer and inner regions are (3.12) and (3.49),respectively. With the sinusoidal time dependence indicated in (3.34), it follows that - 0 2 q j - 2ioUqjx in the outer region, and -w2qj

+ U’q,, + g q j z = 0

+ gqjz = 0

on z = o

on z = 0

(3.51) (3.52)

in the inner region. The first term of (3.52) is of higher order for o = 0(1), but not for the extended frequency regime o < O(&-(’/’)). IV. Fundamental Solutions

The interaction of a body with an exterior flow field can be represented by suitable distributions of sources, dipoles and higher order multipoles. In the simplest example, a point dipole may be used to represent the uniform flow past a circle or sphere. More complete multipole expansions are applicable for other body shapes, with the singularities situated at a point or on a line

Theory of Ship Motions

245

within the body. In the most general case, sources and normal dipoles may be distributed in a continuous manner on the body surface, with Green’s theorem used to derive integral equations for the unknown source strength or dipole moment. The source potential is known also as Green’sfunction and will be denoted here by the symbol G. This is the fundamental singularity, since dipoles and higher-order multipoles can be derived from the source by differentiation. The elementary three-dimensional source potential is G = - (47cr)-’ with r = 16 - x I the distance between the source and field points. An analogous result holds in two dimensions, with G = (27c-l log r. In both cases the normalizing factor is such that the source generates a unit rate of flux. It is possible to use the elementary source potential for free-surface problems, as shown by Yeung (see Bai and Yeung, 1974).In linearized problems, however, it is common to use a modified source potential satisfying the free-surface condition, radiation condition, and (for infinite depth) the condition of vanishing at z = - co.When expressed in terms of this singularity, the velocity potential will satisfy all of the boundary conditions except that on the body. For a particular body geometry and normal velocity one then seeks appropriate distributions of surface sources and/or dipoles, or interior multipole expansions, so as to satisfy the body boundary condition. Source and multipole solutions which satisfy the linearized free-surface condition are described systematically by Wehausen and Laitone (1960, Section 13). In slender-body theory the source and dipole potentials are particularly useful for the outer solution, where the body boundary condition is absent. Typically, the outer solution consists of sources and transverse dipoles, distributed on the longitudinal axis of the body. For a slender ship these singularities are on the free surface, and the net flux associated with vertical motions implies the need for sources, whereas lateral motions of the ship hull in relation to the surrounding flow are represented by transverse dipoles. Before considering the three-dimensional source potential we first discuss the simpler result for two dimensions, which will be used subsequently for the inner solution. A. THETWO-DIMENSIONAL GREENFUNCTION

For two-dimensional flow in the y-z plane, the free-surface condition (3.52) is applicable. The corresponding source potential is given by Wehausen and Laitone (1960,Eq. 13.31). In the present notation, with the source point at the origin in the free surface,

1 GZD(y,z) = -- lim 2R p + o +

2‘”cos(k’y)

IOm dk’ k’

- (w - ip)2/g *

(44

J. N. Newman

246

The parameter p can be interpreted as a Rayleigh viscosity coefficient, representing a fictitious dissipation which suppresses incoming waves at infinity. Alternatively, this parameter can be associated with a complex frequency such that the factor ei(a-ip)fgrows slowly from a state of rest at t = - GO.In either case, the limit p -,O + determines the contour of integration in the complex k'-plane, such that the radiation condition is satisfied. Since the imaginary part of the pole k' = (a- ip)2/g is negative, p may be set to zero in (4.1)if the contour of integration is deformed to pass above the pole at k' = w2/g = K . An alternative form for (4.1)is in terms of the exponential integral

defined such that the complex parameter u is exterior to a branch cut along the negative real axis. After a reduction it follows that G,,(y,

1 1 Re{eK('+'Y)El(Kz+ iKy)} + - ieK(z-ilyl). (4.31 2n 2

Z) = --

The residue term in (4.3)results from deforming the contour in (4.1)to avoid the branch cut. The asymptotic properties of the two-dimensional wave source can be obtained from the corresponding approximations of the exponential integral (Abramowitz and Stegun, 1964,Eqs. 5.1.11 and 5.1.51). For small values of Kr the sourcelike logarithmic singularity is displayed in the approximation

Here y = 0.577. . . is Euler's constant, and (r, 8) are polar coordinates such that y = r sin 8, z = - r cos 8.The error in (4.4)is a factor 1 + O(K2r2).For large values of K Iy I the asymptotic approximation of the exponential integral confirms the outgoing two-dimensional plane waves in the form G z D &eK('-'IYI) for K l y l 9 1. (4.51 B. THETHREE-DIMENSIONAL GREENFUNCTION

The three-dimensional source potential which satisfies the outer freesurface condition (3.51) corresponds physically to a source of oscillatory strength, moving with constant velocity U.This source potential was derived initially by Haskind (1946a),and subsequently by Brard (1948).The solution is described by Wehausen and Laitone (1960,Eq. 13.52)and by Lighthill

247

Theory of Ship Motions

(1967). With the source point at the origin on the free surface, this source potential is given by 1

G(x, y, z ) = -2lim 8ff p + o +

j

m

0

2n

k‘dk‘

0

d8

exp[k‘z + ik’(x cos 8 + y sin 8)] k‘ - (o- ip + Uk’ cos 8)’/g ’

(4.6) Ultimately we shall distribute these sources along the longitudinal axis, and an inner approximation of this distribution will be required. For this purpose Fourier transforms are particularly helpful. Thus we shall analyze the Fourier transform of (4.6), in the form G*(y, z ; k) =

[

m

dxeikXG(x,y,

‘-m

(4.7)

2).

After substituting the solution (4.6),and using generalized harmonic analysis to perform the integral in (4.7), it follows that G*(y, z; k,

1 . lim 4n p + o +

K) = - -

j-,

du

exp[z(k2 + u’)”’

(k’

+ iyu]

+ uz)1/2- (o- ip - Uk)’/g’

where u = k‘ sin 8. Here, for future convenience, we define = (a - Uk)’/g.

(4.9) The parameter p may be set to zero in (4.8),if the contour of integration is deformed to avoid the poles at f(K’ - k’)’/’. For k < K the poles are symmetrically situated on the real axis, and the sign of the imaginary part as p + O + is such that the contour should pass above or below the pole according as u(w - U k ) is positive or negative, respectively. For k > K the poles are imaginary, and no deformation of the contour is necessary in (4.8). The value of G* for k = 0 is the longitudinal integral of the threedimensional source potential, which reduces t o the two-dimensional source potential (4.1). To confirm this we note that K

1 G*(y, z ; 0,K ) = -411

exp[zlul + iyu] du j-, (uI - ( o - i p ) ’ / g

= GzLdY,

4.

(4.10)

Approximations similar to (4.4) and (4.5)can be derived for the transform G*(y, z ; k, K). For this purpose we shall assume that k = O(1). An asymptotic expansion of (4.8) for K r -g 1 is derived by Ursell (1962, Eq. 2-19), in the special case U = 0. In the present notation, Ursell’s result

J . N . Newman

248

can be expressed in the form G*(y, z ; k, K ) E (1/2n)(1

+ Kz)[log(* Ik I I ) + y + ( I 1 - k Z / K ZI ) - ( ' I 2 )

1

( K / l k l )- II - K Z + KyO , )cosh-' ( K / l k l )+ ni COS-'

(4.11)

where the upper or lower expression in braces is applicable according as K/ I k I 5 1, respectively. The error in (4.11) is a factor 1 + O ( K 2 r Zk2r2). , This approximation is analogous to (4.4),except for an additional homogeneous solution of the form (1 + K z ) F ( K / l k ( ) . Generalization of (4.11) for U # 0 simply requires that the parameter K is replaced by K and also requires that the conjugate contour of integration is used if w - Uk < 0. After a straightforward reduction it follows that G*(y, z ; k ,

K)=

G*(y, z; 0, K ) - (1/2n)(1 + K z ) f * ( k , K , K )

+ O ( ( K - K ) r , K2r2,k z r z ) ,

(4.12)

where G*(y, z ; 0, K) is defined by (4.10),and f * ( k , K , K) = log(2K/ I k I ) + ai - ( I 1 - k Z / K Z I ) - ( I i z )

1

.\

cos-'(K/lkl) - a \cosh-'(K/ I k I ) + ni sgn(w - U k ) '

(4.13)

The asymptotic approximations (4.11) and (4.12) are valid for K r < 1, irrespective of the magnitude of K. This can be confirmed by writing (4.8)in terms of the nondimensional variables Ky and Kz before deriving (4.11).By the same argument, the requirement that k = O(1) can be replaced by the less restrictive assumption that k/K = O(1). The functionf* defined by (4.13)tends to zero for K % 1, with the limiting behavior

f* = log(K/K) + O ( K - ' ) = O(K-('/')).

(4.14)

A complementary approach is required for the short-wavelength regime K B 1. First we deform the contour of integration in the expression (4.8)for G*, into the upper or lower half of the complex plane u + iu, according as y 2 0. The contour can be deformed ultimately to a large semicircle I u + iu I = 00, except for a branch cut I u I > I k I along the imaginary axis. There is no contribution to the integral from the large semicircle, but one must include the residue from the pole situated in the appropriate half-plane

Theory of Ship Motions

249

for p > 0. For short wavelengths such that o - Uk > 0 and final result of this procedure is the expression G*(y, Z ; k,

K)

= )i(l - k’/K’)“’’’

K

> 1 k 1, the

exp[Kz - i Iy I (K’ - k’)”’] exp[-u lyl

+ iz(u’ + iK

k2)”21.

(u’ - k2)”’

(4.15)

For large K z K, the last term can be approximated by

where K l is the modified Bessel function, and the error is O(Ky)-’. After expanding the modified Bessel function for small kr and substituting the result in (4.16), we obtain the approximation 1 2

G* = - i(1 - k’/K’)-(”’’ exp[icz

-i

1 y I (K’

- k’)”’]

cos 9 ++ O(KY)-’. 211Kr (4.17)

Comparison with the limiting value of (4.17) for k = 0 gives the result G* z GZD,

with the error a factor 1 + O(k’y/K, K1’’y, ( K y ) - ’ ) .

(4.18)

V. TweDirnensional Bodies In the inner region close to a slender ship hull, its boundary surface is approximated by a long horizontal cylinder having a two-dimensional profile defined by the local cross section of the ship. We shall refer to such a cylinder as a “two-dimensional body.” In the radiation problem of forced oscillatory motions, in the y-z plane, the inner flow is governed by the two-dimensional Laplace equation (3.44). The resulting velocity potential is denoted by 4 ( y , z ) to distinguish this from the outer three-dimensional potential q ( x , y , z). In both cases the complex timedependent factor eio‘ is implied. For oblique waves incident upon a two-dimensional body, the threedimensional diffraction potential can be expressed as the product of a sinusoidal function of x, and a two-dimensional function @(y, z). The latter is governed by the Helmholtz equation, reducing to Laplace’s equation in the special case of beam seas. There is an extensive literature on wave radiation and diffraction by two-

250

J . N . Newman

dimensional bodies. Numerical results have been obtained for a variety of body profiles, using several different methods of solution. These are described in the surveys of Wehausen (1971) and Mei (1977).Our discussion in this section is limited to the derivation of analytic properties which are needed subsequently in the three-dimensional slender-body analysis. A. RADIATIONPROBLEMS

Forced motions of the three-dimensional ship hull in sway (j= 2), heave (j= 3), and roll (j= 4) can be related directly to the same motions of the two-dimensional body in the y-z plane. Pitch and yaw motions will be

related ultimately to appropriate translations of the two-dimensional body in heave and sway, respectively. The remaining surge mode (j= 1) corresponds to a dilation of the two-dimensional body, with normal velocity proportional to the longitudinal component of the unit normal vector on the ship hull. Thus we must consider the four radiation problems of surge, sway, heave, and roll. In each case the two-dimensional Laplace equation (3.44) and free-surfacecondition (3.52) apply. With the normalization (3.34), the corresponding potentials satisfy the boundary condition

4Jn. = iconj

(j= 1, 2, 3, 4)

(5.1) on the body profile. Restricting the body to be symmetrical about y = 0, the surge and heave potentials are even functions of y, whereas the sway and roll potentials are odd. Each boundary-value problem is completed by imposing a radiation condition of outgoing plane waves at y = k 00 and by requiring that the motion vanish for z + - 00. Following an approach introduced by Ursell(1949), we shall express the solutions for surge and heave in the form

c m

+j=ajGzD+

m=l

cos 2mO a j m ( 7

K

+ (2m-

1)

cos(2m - l)e rZm-l

(j= 1, 3).

(54 In this expansion G,, is the two-dimensional source potential (4.1), and the higher order multipoles have been combined to form the “wave-free potentials” in braces. The source strength aj and coefficients aimare unknowns which must be determined from the boundary condition (5.1) on the body surface. In practice this leads to an infinite system of simultaneous equations, which can be truncated and solved by numerical methods. Ursell (1949) proves that this process is convergent for a circular body profile. For more general body profiles the expansion (5.2) is valid for symmetric mo-

25 1

Theory of Ship Motions

tions exterior to a circle of constant radius which encloses the body, as shown by Ursell (1968a). Since the wave-free potentials in (5.2) vanish at infinity, the radiated waves are associated only with the source term. Using (4.5) with (5.2) it follows that

4J. N- 2 l j a . eJ K ' z - ' l Y l )

( i g / o ) A eK(Z-'IyI)

as lyl

+GO.

(5.3)

Here Aj = t(o/g)aj (j= 193) (5.4) is the complex amplitude of the radiated wave. Since the potential & j is normalized for a unit amplitude of motion, (5.4) is nondimensional. Similar results hold for sway and roll. The appropriate singularities can be obtained by differentiation of (5.2). After redefining the unknown coefficients it follows that

(5.5)

Here the unknown coefficients p j , ah are determined from the boundary condition (5.1). Since the radiated waves are associated only with the wave dipole, c$j

z

(ig/o)A eK('

'

iY)

as y - ,

+GO,

(5.6)

where A = -+ i ( o K / g ) p j

( j= 2,4).

(5.7)

Hereafter we shall assume that these two-dimensional radiation potentials are known. The hydrodynamic pressure then may be determined from the linearized form of Bernoulli's equation (3.2), p = - iop4e'"'. (5.8) Ignoring the restoring force cij in (1.2), due to the hydrostatic pressure, the remaining components of the pressure force are associated with the addedmass and damping coefficients. These can be computed by integration of (5.8) over the submerged portion of the body profile, o Z A i j- iwBij = - i o p

jp n i 4 j dl.

(5.9)

Here A i j and Bij denote the two-dimensional values of the added-mass and damping coefficients, and P denotes the submerged portion of the body profile.

252

J . N . Newman

Other properties of the added-mass and damping coefficients are derived by Wehausen (1971). Included in that survey are the computed results for a family of rectangular cylinders, including the coefficients for heave results reproduced here in Figs. 3 and 4, as well as similar results for sway and roll. These calculations and additional results for other body profiles are due to Vugts (1968).

B. THEDIFFRACTION PROBLEM If a progressive wave system of the form (3.35) is incident upon a twodimensional body, the fluid motion will be periodic along the body axis. The effect of forward speed along this axis can be ignored, since the body is two-dimensional. Therefore the analysis can be performed in a frame of reference fixed with respect to the fluid, and there is no need to distinguish the wave frequency oofrom the frequency of encounter o. With the incident-wave potential defined by cpo = ( i g / o ) exp[K(z - ix cos

/I - iy sin /I)],

(5.10)

the scattered potential cp7 is subject to the condition that the total potential cpo + ip7 has zero normal velocity on the body. Since the flow is periodic in the x-direction, we can write cpj(x, Y , z ) = Oj(y,z)e-"",

where

j = 0, 7,

1 = K cos /I

(5.11) (5.12)

is the longitudinal component of the wavenumber. With these definitions, the boundary condition (3.36) takes the form 07,, = io(n, - in, sin j?)exp[K(z - iy sin /I)],

(5.13)

on the body profile. After substituting (5.11) in the three-dimensional Laplace equation, the two-dimensional functions Oj satisfy the Helmholtz equation

+

Ojyy Ojzz- PO,= 0.

(5.14)

Both Oo and O7 vanish for z -,- 00 and satisfy the free-surface condition KOj - Ojz= 0

on z = 0.

(5.15)

The scattering function O7 satisfies a radiation condition of outgoing waves for y + +a. Expansions similar to (5.2) and (5.5) can be applied to the symmetric and antisymmetric portions of 07,except for the singular case sin /I = 0, where

253

Theory of Ship Motions

I

the two-dimensional solution is unbounded for I y --* 00. The proof is given by Ursell (1968a). The appropriate wave-free singularities which satisfy the Helmholtz equation involve the modified Bessel functions K,,,(lr);these are exponentially small at infinity. The corresponding source function which satisfies (5.14) can be derived from the Fourier-transformed three-dimensional Green's function G*, by setting U = 0 and k = 1 in (4.8). With these substitutions, and ignoring the wave-free functions, it follows that (5.16)

where Z, denotes the source strength and M , denotes the dipole moment. The radiated waves in the far field can be derived by substituting (4.17) in (5.16), with the result

I

CD, 2 *(iC, csc

p I f K M , ) exp[K(z - i I y sin /3 I )I,

(5.17)

for y -, f00. Since the incident wave amplitude is unity, the radiated wave amplitude at y sin /3= - 00 is equal to the rejection coejjcient R = ( 0 / 2 g ) ( C ,csc /3+ i K M , ) sgn(/3).

Similarly, the total wave amplitude at y sin mission coeficient T=1

/3 = + 00

(5.18)

is equal to the trans-

+ ( 0 / 2 g ) ( C , csc /3 - X M , ) sgn(/3).

(5.19)

In (5.18) and (5.19), --R < /3 < A. If the body profile is symmetric about y = 0, the dipole moment M , is an odd function of b, whereas the source strength C,, and hence R and T, are even functions of /3.

C. APPLICATIONSOF GREEN'S THEOREM Green's theorem may be applied to the pair of functions (Yl, Y 2 )in the plane x = constant, in the form

f (YlY2n- Y2Yln)dl = C

(Y1V2Y2 - Y2V2Yl)dS.

(5.20)

S

The closed contour C is the boundary of the simply connected domain S . The surface integral vanishes if, in this domain, both functions satisfy the same governing equation (3.44) or (5.14). We shall apply (5.20) to the fluid domain in the y-z plane, between the body profile P and a pair of vertical contours at y = k 00. These are connected by horizontal lines at z = 0 and z = - 00 to form a closed contour.

254

J . N . Newman

There is no contribution from the horizontal lines to the contour integral in (5.20),provided that the functions Y i satisfy the linearized free-surfacecondition (5.15) and vanish for large depths. Thus (5.20) gives the result (Y1Yzn - Y2Yln)dl =

-I

0

dz[YIYz, - Y z Y l y ~ Z ? m . (5.21)

-m

In this form, Green’s theorem can be used to derive various relations for the forces acting on the body and for the characteristics of the radiated waves at infinity. In the simplest example, we apply (5.21) to two solutions of the radiation problem +i and dj. Since these are subject to the radiation condition at infinity, the right side of (5.21)vanishes. After using the boundary conditions (5.1) and comparing the left side with the pressure force (5.9), it follows that the added-mass and damping coefficients are symmetric, i.e., A, = Aji and BII. . = BI..t ’ If (5.21) is applied to the radiation potential +i and its conjugate $i,the left side of (5.21) is proportional to the damping coefficient Bii. The contribution from the right side is nonzero. After a reduction using (5.3)or (5.6),it follows that Bii

I

= (pg2/co(’3) Ai 12

(i = 1, 2, 3, 4).

(5.22)

Alternatively, this relation can be derived from energy conservation. If (5.21) is applied to the diffraction solution Oo + (P, and its conjugate, there is no contribution from the left side due to the body boundary condition. The contribution from the integration at infinity gives the familiar result IRIZ+ ITl2=1. (5.23) This relation also can be derived from energy conservation in an obvious manner. Alternatively, if (5.21) is applied to the diffraction potential with an angle of incidence 8, and the conjugate of the diffraction potential with angle of incidence R + 8, it follows that

RT

+ RT = 0.

(5.24)

Here the symmetries noted after (5.19) are used, and (5.24) holds only for a body profile symmetrical about y = 0. Further relations can be obtained by applying Green’s theorem to suitable combinations of the radiation and scattering problems. TO preserve the Helmholtz equation for both functions, we define a class of radiation problems where the normal velocity on the two-dimensional body surface is periodic in the same manner as (5.11). This corresponds physically to a

Theory of Ship Motions

255

forced sinuous motion which propagates along the cylinder with phase velocity w/l.To be more specific, “generalized radiation functions” are defined to satisfy the same conditions as 0,, except on the body profile where

ajn= i o n j

( j = 1, 2, 3, 4).

(5.25)

These functions can be expanded in the same manner as the scattering potential, with source strength C j for the symmetric modes ( j= 1, 3) and dipole moment M j for the antisymmetric modes (j= 2, 4). The radiated waves in the far field may be defined in a similar form to (5.17). For y - , +a, Q j z (ig/w)Ajexp[K(z - iy I sin B I )I, (5.26) where Aj

= 9(0/g)cjlcsc

BI

Aj = -*i(oK/g)Mj

( j = 1, 3),

(5.27)

( j = 2, 4).

(5.28)

The same results hold for y + - 00, provided the sign of (5.26)is reversed for = 0, the wave amplitudes (5.27-5.28) reduce to the corresponding values defined by (5.4) and (5.7). If Green’s theorem (5.23) is applied to the diffraction function a0 and the generalized radiation function Oj, one obtains the result

j = 2,4. When cos

iw

’

sin /3( ) jp nj(Oo+ 0,)dl = -i(g/o)’Jj ( -sin.

c

+

for j =-l , 3 - 2, 4

).

(5.29)

The left side of (5.29) is proportional to the linearized pressure force on the body profile, due to the interaction of the fixed body with the incident waves. With the notation (1.4) it follows that the two-dimensional exciting force X j can be related to the wave amplitude of the generalized radiation function, in the jth mode, by means of the formula

(5.30) This is the two-dimensional form of a more general result known in ship hydrodynamics as the Haskind relations. Equation (5.22)can be combined with (5.30) to relate the damping coefficient and the magnitude of the exciting force, as shown in Fig. 4. A different result follows if the generalized radiation function is replaced by its conjugate Gj. In this case - io

jp nj(mo + (D7)dl = i ( g / o ) * J j ( R_+ T )( -Isinsin. BIj?)

for

0 1. j=l 3 ’ = 2, 4

(5.3 1)

256

J . N . Newman

Adding (5.29) to (5.31) gives the relation (5.32) These linear equations can be solved for the reJection and transmission coefficients R and T in terms of the ratios Aj/Aj.A corollary is that the phase of the radiated waves (5.27-5.28), and hence the phase of the exciting force (5.30), is equal to the argument of -h(R f T) for symmetric or antisymmetric modes, respectively. Since R and T are independent of the particular modes, the two phase angles are likewise invariant with respect to the distribution of normal velocity on the body profile. Finally, we combine (5.32) with (5.18-5.19),and obtain relations for the source strength X, and dipole moment M , ,

C, = -(g/u)~sin~ l ( + 1 Aj/jj) M , = -i(g2/03) s g n ( ~ ) ( l -Zj/Zj)

( j = 1,3),

(5.33)

( j = 2,4).

(5.34)

The relations derived in this section from Green’s theorem are special cases of more general results which are summarized by Newman (1976).

D. LONGWAVELENGTH APPROXIMATIONS When the wavelength is long compared to the dimensions of the body profile, the free-surface condition can be replaced to leading order by the “rigid-wall” boundary condition (3.47). In this limit the source strength for surge and heave can be determined from a continuity argument,

I,ah

dl = -4Zj

( j = 1, 3).

(5.35)

Substituting the boundary condition (5.l), it follows from geometrical considerations that

XI = -2ios’(x),

(5.36)

-2ioB(x).

(5.37)

x 3

=

Here S(x) is the submerged area of the body profile; S’(x) = dS/dx; and B(x) is the beam, or width of the body profile at the waterplane. Derivations of these results are given by Newman and Tuck (1964, Appendix 2). The dipole moment for sway can be determined by considering the body profile plus its image above the free surface, in an infinite fluid. For this

257

Theory of Ship Motions “double-body” the dipole moment is given by the formula M , = -2io(S

+ m,,/p),

(5.38)

where 2m,, is the added mass of the double body. Equation (5.38) results from a theorem due to G. I. Taylor, which is rederived and extended by Landweber and Yih (1956).The dipole moment due to rolling motion can be derived in a similar manner, noting that on the double body the normal velocity for roll is an even function of z. The corresponding dipole moment is given by M4 = 2iw(szB

+ B3/i2 - m,4/p).

(5.39)

Here zB < 0 is the vertical coordinate of the center of buoyancy, or the centroid of the submerged profile, and 2mZ4 is the added-mass coupling coefficient for the double body, between roll and sway. The same estimates apply for the source strength and dipole moment of the generalized functions Oj.The boundary condition (5.13) for the scattering potential implies the long-wavelength approximation O, z -03

+ i sin POz

(5.40)

and, hence, from (5.36-5.37),

x, z 2ioB(x), M , z 2 4 s + m z z / p )sin /3.

(5.41) (5.42)

The amplitudes of the radiated waves can be found using (5.27-5.28). These results are summarized as follows:

A, z -iKS’Icsc 81, A, z - K 2 ( S + m,,/p), A, z -iKBIcsc 81,

(5.43) (5.44)

(5.45)

,T4 z K’(SZ, + ~ ~ / -1 m,4/p). 2

(5.46)

More accurate approximations for the phase angles of the radiated waves follow from (5.33-5.34) and (5.41-5.42) in the form arg(Aj) z

IL --

2

+ K B I csc fl I

arg(d,) z IL - K 2 ( S + m z , / p ) I sin /3 I

( j = 1, 3),

(5.47)

( j = 2,4).

(5.48)

These can be used with the Haskind relations (5.30) to approximate the exciting forces, and with (5.32) to obtain approximations for the reflection and transmission coefficients.

J . N. Newman

258

VI. Slender-Body Radiation Hereafter the slender-body assumption is invoked for the ship hull with the slenderness parameter E defined as the ratio of the beam (B) or draft (T), divided by the length (L). It is convenient to presume a length scale such that L = O(l), and thus (B, T) = O(E).Following the method of matched asymptotic expansions, approximate solutions for E 4 1 are derived separately in the outer region (y, z) = 0 ( 1 ) and in the inner region (y, z) = O(E).These two separate solutions then are required to match in a suitable overlap domain E 6 (y, z) 4 1. In this section we analyze the radiation problem for each mode of rigidbody motion. The corresponding velocity potential 'piis defined? by (3.34). The three-dimensional Laplace equation and linearized free-surface condition (3.51) apply in the outer region, together with a radiation condition and the requirement that the solution vanishes for z + - 00. The inner solution is governed by the two-dimensional Laplace equation (3.44), the linearized free-surface condition (3.52), and the boundary condition (3.37) on the ship hull. The method of matched asymptotic expansions has been applied to the radiation problem for a slender ship by Newman and Tuck (1964), for the case where the characteristic wavelength I = O(1), and by Ogilvie and Tuck (1969) for I = O(E).Here we seek a more general "unified" approach, which is valid for all wavelengths 1 IO(E). The objective of a unified theory requires a careful analysis of the matching error. For this reason the errors in the inner and outer solutions will be estimated, and the overlap region will be chosen to minimize the largest of these. The accuracy of each solution will be indicated by an error factor 8, with logarithmic factors neglected in these estimates. A. THEOUTER PROBLEM

Since the ship hull is collapsed onto the longitudinal x-axis as E + 0, the outer solution can be constructed from a suitable distribution of singularities on this axis. For the symmetric modes (j = 1,3,5) the three-dimensional source potential (4.6) is the appropriate singularity, whereas the antisymmetric modes ( j = 2, 4, 6) require axial distributions of transverse dipoles. (These statements are somewhat intuitive, but they will be confirmed ultimately by matching the results with the inner solution.) t Throughout this section the integer j takes all values from one to six, unless otherwise noted.

Theory of Ship Motions

259

The strength of the source and dipole distributions will be denoted by

4 j ( x ) and d j ( x ) , respectively. The potential for a source at the point x = l is obtained by shifting the longitudinal coordinate in (4.6), and the transverse

dipole potential can be derived by differentiating with respect to y . Thus the outer solution is expressed in the form

with the convention that 4 j = 0 if j is even, and d j = 0 if j is odd. The Fourier transform of (6.1) is derived from the convolution theorem in the form

(

)

+d.* a G*(Y, 2 ; k, K), (6.2) ’aY where G* is defined by (4.74.8). Assuming the source and dipole distributions vary slowly along the hull, the Fourier transforms 4j* and dj* will tend to zero if k 9 1. Thus it is reasonable to assume that k = O(1), in seeking asymptotic approximations of (6.2). Inner approximations of (6.2) for small values of the coordinates ( y , z) can be constructed from the results of Section IV. For small values of Kr, the approximation (4.12) is used to give ‘pi*

= qj*

Here G z D andf* are defined by (4.1) and (4.13), and the error in (6.3) is the factor d = 1 O ( ( K - K ) r , K 2 r Z ,k2r2). (6.4) Alternatively, for K 1 y ) 9 1, (4.18) gives the approximation

+

with the error factor

d = 1 + O ( k Z y / K ,K’”y, ( K y ) - ’ ) .

(6.6)

B. THEINNER PROBLEM In constructing the inner solution we shall ignore temporarily the matching requirement, replacing this by the two-dimensional radiation condition. The result can be identified with the strip-theory synthesis, and a superscript (s) is used to distinguish this solution.

260

J . N . Newman

In view of the boundary condition (3.37) on the body profile, the striptheory solution can be expressed in the form

cpp = + u$j $Ij

(6.7)

= iconj,

(6.8)

where, on the body profile, +j,,

$.Jn = m i .

(6.9) The factors n j and mj are defined by (3.42-3.43) and (3.45-3.46), respectively. The two-dimensional potentials 4j and $ j are governed by the Laplace equation (3.44),satisfy the free-surface condition (3.52), and vanish as 2 - -aoO. These boundary-value problems and their solutions do not involve the forward velocity U. The two-dimensional potentials $ j ( j = 1,2,3,4) are analyzed in Section V,A. Analogous results can be derived for d j if the factors mj are known. In practice a numerical solution is required not only for the two-dimensional potentials, but also for the factors mj.Hereafter, we shall assume that these are known. The remaining potentials for pitch ( j = 5 ) and yaw ( j = 6) follow from the definitions of n j and mjr

cpp = - xcpy + (U/iw)&, cpt)= xcp$’ - (U/iw)&.

(6.10) (6.11)

In general the matching requirement with the outer solution will differ from the condition of outgoing radiated waves satisfied by the strip-theory potentials. Moreover, the potentials (6.7)are unique (assuming unique solutions of the two-dimensional radiation problems), and depend only on the local cross-sectional geometry of the hull. Since the outer solution includes a longitudinal function of x which depends on the three-dimensional shape of the ship hull, a more general solution is required in the inner region. In the classical slender-body theory of aerodynamics, and also in the long-wavelength case where the rigid free-surface condition (3.47) holds in the inner region, the inner solution is generalized simply by adding an arbitrary “constant” C, which may depend on x. In the present case, the free-surface condition (3.52) requires a nontrivial homogeneous solution. Since the homogeneous solution satisfies a boundary condition of zero normal velocity on the hull, it can be identified physically with the scattering of incident waves by the fixed body. A solution which is symmetric about y = 0 can be derived by combining two waves of equal phase incident from opposite sides of the body. An antisymmetric solution can be derived similarly, with incident waves of opposite phase. In either case standing waves will exist in the far field of the two-dimensional problem. An alternative derivation of the homogeneous solutions follows by

261

Theory of Ship Motions

observing that the boundary conditions (6.8) are purely imaginary, hence is a homogeneous solution. Similarly, from (6.9), Im($j) is a homogeneous solution which will differ from Re(4j) by a multiplicative constant. With an arbitrary multiplicative factor, only one homogeneous solution is required in each mode, and we shall take this to be (4j+ Bj). With this choice, the general form of the inner solution is given by 'pj

= Cpy

+ Cj(X)(4j+

(6.12)

Bj).

Here C j ( x ) is a function to be determined by matching with the outer solution. The outer approximation of the inner solution (6.12) can be derived from the expansions (5.2) and (5.5). Since the radial derivatives of these expansions are O(1) in the inner region, as r + O(E), (6.13)

ajm= O(E'"'O~, &'"pj).

Thus, the wave-free potentials can be neglected for r B E , and the only contribution to the outer approximation of the inner solution is from the source or dipole. For the potential 4 j ( j = 1, 2, 3, 4) the source strength 01,3or dipole moment pz,4is defined by the two-dimensional solution. These definitions are readily extended to the potentials for yaw and pitch where, in accordance with (6.10-6.11), os = - x o 3 and p6 = x p z . With the same convention as in (6.1), the outer approximation of the potential 4j is

(6.14)

A similar representation can be applied to the potentials

G

$j,

-

DjGzD.

(6.15)

For convenience in subsequent expressions we also define the differential operators

+ UDj, R j = Dj + Dj. S j = Dj

(6.16) (6.17)

With these definitions, the outer approximation of (6.12) is q j

z SjGzD + Cj(DjGzD + D j G z D )

= (Sj

+ C j R j ) G z D- iCjDjeKzcos K y .

(6.18)

262

J . N . Newman

The last result follows from (4.3). The Fourier transform of (6.18)is given by 'pj* E [S?

+ (C,Rj)*]G2D- i(CjD,l)*eKzcos K y .

(6.19)

The errors in the inner solution may be summarized in the following manner for r & E. The two-dimensional Laplace equation and body boundary condition involve second-order errors in the ratio of longitudinal to transverse gradients, or a factor of 1 + O(kr)'. The two-dimensional freesurface condition (3.52) contains an error factor 1 + O(K'I2kr). Neglect of the wave-free potentials involves the error factor 1 + O(EZ/r2). Thus the cumulative error in (6.19) is the factor

d = 1 + O(k2r2, K1I2kr,E2/r2).

(6.20)

C. MATCHING

The inner and outer solutions are matched in a suitable overlap domain 6 r 4 1 to determine the unknown source strength and dipole moment of the outer solution and the coefficients C , in the inner solution. Initially it will be assumed that the overlap region is close to the ship, in terms of the wavelength, and thus Kr 4 1. This assumption will become invalid for short wavelengths, at which point a separate approach will be adopted. Matching of the inner and outer solutions is carried out in the Fourier domain. Thus we equate (6.3) to (6.19), and obtain the relation E

= - (S,+ CjRj)*G2D- i(CjBj)*eKzcos K y .

(6.21)

The dominant terms in this matching relation are the antisymmetric contributions associated with the dipole potential, of order l/r, and symmetric contributions from the source potential, of order log r. First we consider the antisymmetric terms in (6.21), corresponding to the modes j = 2, 4, 6. Equating the factors of the dipole terms gives dj*

=

[pj

+ Ufij + C j ( p j+ pi)]*

( j = 2,4, 6).

(6.22)

The long-wavelength approximation (5.48) can be used with (5.7) to show that the dipole moment p, is imaginary with the error a factor 1 + O(K2c2). Thus the interaction term proportional to C , may be neglected in (6.22). After inverting the Fourier transforms, d, = p j

+ Ufi,

( j = 2,4, 6),

(6.23)

Theory of Ship Motions

263

and the dipole moments are identical in the inner and outer solutions. The remaining antisymmetric part of (6.21) is a higher order contribution from the last term, proportional to sin Ky. To leading order the antisymmetric solution is strictly two-dimensional in the inner region and given correctly by the striptheory approach. This is a familiar situation in slender-body theory, where lateral body motions without a net source strength contain no longitudinal interactions. Ogilvie (1977) refers to this as a “primitive” strip theory. Next we consider the symmetric modes, which are dominated in (6.21)by the source potential. Equating the factors of G z Dgives a relation for the source strength qj* = ( S j

+ CjRj)*

( j = 1, 3, 5).

(6.24)

Equating the remaining terms in (6.21) of order one gives qj*f* = 2ni(CjDj)*

( j = 1, 3, 5).

(6.25)

A comparison of (6.4) and (6.20) indicates the cumulative error in (6.24) and (6.25) to be the factor

d=1

+ O(k2r2,K112r,K 2 r 2 ,c2/r2).

(6.26)

The optimum location of the overlap region may be selected to minimize this error. For sufficiently small values of K, the dominant contributions to (6.26)are O(K”*r, E2/r2).The optimum value of r is such that these two errors are equal and thus

,.=

0(&2/3~-(1/6)

).

(6.27)

With this choice, the maximum error in (6.26) is O ( E ” ~ K ” ~provided ), K &-(li2). For K > E - ( ’ / ~ ) , the dominant contributions to (6.26) are O ( K Z r 2E2/r2). , Equating these defines the optimum overlap region r =O(E/K)”~.

(6.28)

The corresponding error in (6.26) is O(EK). As K + O(E-l ) a separate matching must be constructed from the shortwavelength inner approximation of the outer solution. Proceeding on this basis with (4.18), (6.21) is replaced by

G z D2 ( S j + C j R j ) * G z D- i(CjDj)*eKzcos Ky.

(6.29)

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J. N . Newman

The solution of (6.29) is the strip-theory result (6.30)

cj*= 0,

(6.31)

in accordance with the short-wavelength analysis of Ogilvie and Tuck (1969). The error factor in (6.29-6.31) is 8 = 1 + O(kzrz,K”*r, c2/r2, ( K y ) - ’ ) .

(6.32)

The maximum error in (6.32)is O(K-(1’3’)if the overlap region is defined by r = O(K-(5/6)). Since the alternative error in the long-wavelength matching is O(EK),the short-wavelength matching should be adopted if K 2 E - ( ~ / ~ ) . At this stage we observe that the long-wavelength results (6.23-6.25) are consistent with the striptheory results (6.29-6.31) for K 9 1, sincef* vanishes in accordance with (4.14). For this reason (6.23-6.25) are valid in general, for all wavelengths such that K IO(E-’), and will be used exclusively hereafter. The inverse transforms of (6.24-6.25) can be expressed in the form qj = Sj

2nicjDj =

+ CjRj,

J”L qj(<)f(x - <) d<,

(6.33) (6.34)

f o r j = 1,3,5. The kernelf(x) is the inverse transform of (4.13).Elimination of C, gives an integral equation for the outer source strength:

= g,(x)

+ Ukj(X)

( j = 1, 3, 5).

(6.35)

Here the two-dimensional source strengths have been substituted for the differential operators using (6.14-6.17). The relations (5.4) and (5.32) can be used to replace the factor ( 0 , / 8 ~ + 1) by (1 - R - T),where R and Tare the two-dimensional beam-sea reflection and transmission coefficients. The kernel in (6.35) is identical to that which accounts for longitudinal interactions in the long-wavelength slender-body theory (Newman and Tuck, 1964). On the other hand, the source strengths 0, and 6, are derived from the striptheory solution. Thus the integral equation (6.35) provides a blend between the two limiting theories, such that the result is valid for all intermediate wavelengths.

Theory of Ship Motions

265

The term containing the integral in (6.35)tends to zero for the two limiting regimes K = 0(1) and K = O ( E - ’ ) . For long wavelengths the factor ( a j / S j 1 ) = O ( K E )from (5.4) and (5.43).For short wavelengths the kernel vanishes, in accordance with (4.14).To leading order it follows that

+

qj

E bj

+ u&j.

(6.36)

This approximation can be refined by iteration. D. THEINNERSOLUTION

At this stage the inner solution (6.12) is determined. The two-dimensional potentials 4j and t$j ( j = 1, 2, 3, 4) must be obtained numerically at each section along the hull. The coefficients C j in (6.12) vanish for ( j = 2, 4, 6), and are otherwise determined from (6.34) in terms of the outer source strength qj. The latter is determined from the integral equation (6.35) or by approximation from (6.36). The resulting inner solution can be expressed in the form

The “unified” inner solution (6.37) is the principal result of our analysis. This velocity potential is a linear superposition of the strip-theory solution (6.7),and the homogeneous solution (4j + $ j ) . The homogeneous solution is multiplied by the same longitudinal integral of the outer source strength which appears in the long-wavelength slender-body theory. In the long-wavelength regime K = 0(1),the inner free-surface condition is the “rigid-wall” boundary condition (3.47). This governs the twodimensional potentials 4j and $, and thus qy).The homogeneous inner solution is a constant and the last term in (6.37) is a function only of the longitudinal coordinate x. The outer source strength qj is given explicitly in terms of the net flux at each section, in accordance with (5.36) and (5.37). Longitudinal interference is accounted for by the kernel f (x - t),but transverse interactions are neglected. This is the “ordinary slender-body theory” described by Newman and Tuck (1964) and by Ogilvie (1977). In the short-wavelength regime K = O ( E - ~ the ) , kernel is of higher order and (6.37) is dominated by the potential cpy). Transverse interference is accounted for in this two-dimensional potential, but longitudinal interactions are negligible. This is the striptheory solution. The unified potential (6.37) is valid for all wavenumbers K I O(E-’ ) . In the special case of zero forward velocity this solution reduces to that derived

266

J . N . Newman

by Mays (1978) and outlined by Newman (1978). The latter reference also treats an analogous problem in acoustic radiation. The unified result may be compared with an “interpolation solution” derived by Maruo (1970) for the case U = 0. Maruo’s approach is rather different, but the only change in the final result is that the homogeneous solution in (6.37) is replaced by (1 + K z ) , and the amplitude of the twodimensional striptheory potential is modified accordingly to satisfy the boundary condition on the body. Maruo’s source strength is governed by an integral equation similar to (6.35), with the kernel simplified by the restriction to zero forward velocity. VII. Slender-Body Diffraction

In this section we consider the diffraction problem for a slender ship, in the presence of incident waves. With an incident wave of unit amplitude defined by ( 3 . 3 9 the scattered potential q7 must be determined such that the total potential ‘po + q 7 satisfies the boundary condition of zero normal velocity on the ship hull. Our approach here is similar to that used for the radiation problems in Section VI. The conditions in the opening paragraph of that section apply. The principal difference is with respect to the boundary condition on the ship hull, which for the scattered potential is given by ‘pTn =

cos p - in, sin

p) exp[K,(z - ix cos p - iy sin p)]

-iw,(n,

- in,

on S.

(74 Here oo is the incident wave frequency, in a fixed reference frame, and K O = o o 2 / g is the corresponding wavenumber. The diffraction problem is simplified by the absence of forward-speed effects in (7.1), by comparison to the factors mjand the corresponding solutions $ j in the radiation problem. On the other hand, the normal velocity (7.1) is oscillatory along the ship’s hull, at a rate which is proportional to the longitudinal wavenumber component K O cos p. In the beam-sea case (cos b = 0),the right side of (7.1) is slowly varying for all values of K O I O(E-’). Thus the beam-sea diffraction problem can be treated in an identical manner to the radiation problems, except that the striptheory solution in the inner region corresponds to the two-dimensional scattering problem. The boundary condition (7.1) is simplified also for head or following seas (sin = 0),but this simplification is deceptive. Head seas propagating along

Theory of Ship Motions

267

the two-dimensional body in the inner problem are diffracted over a transverse width that increases without limit. Thus it is not possible to derive a conventional striptheory solution in the inner region. The singular nature of the head-sea diffraction problem was established by Ursell (1968a,b). Ursell’s proof states that head seas cannot propagate along an infinitely long cylinder in a periodic manner, unless the diffraction potential is unbounded at large distances from the cylinder axis. A detailed analysis of the head-sea diffraction problem has been carried out by Faltinsen (1971) for the case where the incident wavelength is O(E). To leading order the incident wave is canceled in the near field by an equal and opposite longitudinal wave. Ursell’s unbounded solution is utilized in a higher order inner solution, and matched with the outer solution in a consistent manner. A singularity is encountered at the ship’s bow, of the sort which generally occurs in short-wavelength scattering problems. Maruo and Sasaki (1974) present a modified approach intended to remove this singularity. Both solutions are discussed further by Ogilvie (1977, 1978). For a ship moving in head seas with U = O(l), the frequency of encounter o is increased by the Doppler shift (1.7). The regime of resonant pitch and heave motions (1.8) coincides with incident wavelengths of order cl/’, intermediate in scale between the ship’s length and transverse dimensions. In this regime the head-sea problem can be analyzed in a relatively simple manner from the long-wavelength slender-body theory. A solution of the diffraction problem will be derived from the unified slender-body approach which was developed for the radiation problems in Section VI. This theory is intended to apply for incident wavenumbers K O IO ( E -’), but we shall concentrate on the regime where K O I O(&-(”’)). Beam seas will emerge as a relatively simple limit, but for head seas the unified solution is singular for all wavenumbers. The alternative longwavelength assumption will be used to provide a simple remedy for this defect. A more fundamental extension of the unified theory is warranted to include head seas, but this task is left for future research.

A. THEOUTERPROBLEM Since the scattering potential (p7 differs from the radiation potentials of Section VI only with respect to the body boundary condition, the boundaryvalue problems in the outer region are identical. The outer solution (6.1)is applicable directly to the potential (p7, with unknown source strength q , and dipole moment d 7 . The Fourier transform of the outer scattering solution is given by (6.2), with j = 7.

J . N . Newman

268

B. THEINNER PROBLEM

The oscillatory longitudinal factor of the boundary condition (7.1) suggests expressing the inner solution for the diffraction potential in the form (p7 = Q, exp( - iKox

cos fl) = Q exp( - dox).

(74

After substituting in (7.1), and neglecting the longitudinal component n,, the function Q, satisfies the boundary condition On= -iwo(n, - in, sin

fl) exp[Ko(z - iy sin fl)]

on S.

(7.3)

Since (7.3) is slowly varying in the x-direction, the same behavior is expected of Q,. Substituting (7.2) in the three-dimensional Laplace equation and neglecting longitudinal derivatives of Q then gives

+

a,,

-

lo% = 0.

(7.4)

Thus Q, is governed by the two-dimensional Helmholtz equation. The leading-order free-surface equation can be derived from (3.49) or (3.52), onz=O.

KoQ,-Q,=O

(7.5)

Here (1.7) has been used to replace the frequency of encounter by the and K O = wo2/g. incident-wave frequency oo, The problem defined by (7.3-7.5) is identical to the two-dimensional diffraction potential Q7 derived in Section V,B, except for the appearance of the frequency oo in place of o.Thus the striptheory solution of the inner problem is the solution of the two-dimensional diffraction problem, which will be denoted here by Q7. The general solution of the inner problem is given in a form analogous to (6.12): @ = @7

+

(74

Since the real and imaginary parts of the boundary condition (7.3) are antisymmetric and symmetric, respectively, the general homogeneous solution is given by @(h)

= C,(@,

+ as)+ C,(@, - a,).

Here

and (Cs, C,) are arbitrary constants.

(7.7)

269

Theory of Ship Motions

The outer approximation of this inner solution can be obtained from (5.16) in the form

G*(y, Z, lo, KO)

+

CsE7 - C , R 7

(C*- G*).

(7.9)

After using (4.15) to evaluate the last factor, and taking the Fourier transform, it follows that

C s z 7- C a M ,

-)aYa * eKoz cos(Koy sin B).

(7.10)

This outer approximation of the inner scattering potential is analogous to (6.19) for the radiation problem. The error in (7.10) is the factor (6.20), with K O substituted for K , and with the Fourier parameter k replaced by lo if lo > k. Since (D is slowly varying with respect to x, the significant domain for (7.10) is k = O(1).

C. MATCHING The Fourier transform (7.10) of the inner solution is to be matched with the transform (6.2) of the outer solution. Recalling the oscillatory factor in (7.2), it is necessary to shift the transform parameter from k to k - lo in (7.10), or alternatively to shift from k to k + lo in the outer solution. Adopting the latter approach, (6.2) is rewritten in the form (P7*(Y,z ; Q = [q7*(jt)+ d,*(Q(a/ay)lG*(y,

2;

IT, 3,

(7.11)

where K=k+lO

(7.12)

and K‘ = (0 - UE)’/q = (00 - Uk)’/g.

(7.13)

Matching of the inner and outer solutions is performed by equating (7.10) and (7.11) in a suitable overlap region. Since the transforms G* in these two

270

J . N . Newman

solutions contain different arguments, it is not possible to match their factors directly. Instead, the long-wavelength inner approximation (4.12) of G* is used in the solutions, with (4.4), to derive the relation 1 - q,*(E)(l + Kz)[log(Kr) + y + ni -f*(& II-, K) - Kz + KyO] 2n 1 +d,*(E)(l + Kz)[(sin O)/r + KO] 271 1 z - [C, + Cs(C7+ X,)]*(l + K,Z) 2n [log(Kor) + y + ni - f * ( l o , K O ,K O )- Koz + KoyO]

1 +[ M , + C,(M7 - &f7)]*(1 + Koz)[(sin O)/r + KoO] - ilcsc flleKoZ 2n

. [(Cs&)* cos(Koy sin fi) + (C,l\si,)*(Ko sin fi) sin(Koy sin /I)].

(7.14)

Proceeding as in Section VI,C, the antisymmetric terms in (7.14) are matched by neglecting the higher-order interactions, proportional to C,, with the result d7*(E) = M7*(k). (7.15) After inverting this Foilrier transform the outer dipole moment is given by d 7 ( x ) = M 7 ( x )exp( - ilox).

(7.16)

Once again there is no interaction in the antisymmetric solution, and the only difference between the inner and outer dipole moments is the oscillatory factor in the definition (7.2) of the inner solution. The dominant symmetric terms in (7.14) are proportional to log r. Equating these, (7.17) q7*(Q = [C7 + CS(C7 + %)I* Using this result, the remaining symmetric terms in (7.14) can be equated to give the relation 274CsC7)* = 47*(l;)f7*(k9 lo, 4, (7.18) where from (4.13),

[

f,*(k, lo, k) = 1 sin fl I log( I l o / E l ) - ( 1 1 - R / i 2I ) - ( ' I 2 )

cos-'(i?/~E~) - II + xi sgn(oo- U k ) (cosh-l(i/ I

+ [cosh-'(sec

/3)

+ nil.

(7.19)

Theory of Ship Motions

27 1

Equations (7.17-7.18) are analogous to (6.24-6.25) in the radiation solutions. The error factor in (7.14-7.18) can be deduced in a similar manner to (6.26). With k = O(l), the error is a factor

I = 1 + O(KO2r2,&'/r2).

(7.20)

~ ' ~this ; choice The optimum overlap region is defined by r = O ( E / K ~ )with the error factor is 1 + O(&KO).A separate analysis must be performed for K O-+ O ( E - ' ) ,using (4.17) to approximate G* in (7.10) and (7.11). We shall not repeat that analysis here, since the results are similar to the radiation solutions, and serve only to confirm the validity of (7.15-7.18) for the extended regime KO < O ( E - ' ) . The Fourier transforms (7.17) and (7.18) can be inverted, after noting the shift in the parameter (7.12). The result is a pair of simultaneous equations, q7 = [C,

+ C,(C7 + C,)] exp(-il,.x),

(7.21) (7.22)

wheref7(x) is defined by its Fourier transform (7.19). From (7.21), the outer source strength can be expressed in a similar form to (7.16), q7(x) = Q7(x) ex~(-ilox),

(7.23)

with Q7 a slowly varying unknown function of x. Elimination of C , from (7.21) and (7.22) gives an integral equation for Q 7 ,

Q ~ ( x-) ( 2 4 - ' ( & / x 7

+ 1) '[L Q 7 ( 5 ) f 7 ( ~ - 4 ) d4 = C7(x).

(7.24)

This integral equation is similar in form to (6.35). Once again the integral term in (7.24) vanishes in the limits K OE 4 1 and K O B 1 so that to leading order (7.24) is approximated by Q7b)

= C7(x).

(7.25)

D. THEINNERSOLUTION The solution in the inner region follows from (7.6-7.8), with the coefficient

C, determined from (7.22) and the (antisymmetric) coefficient C , = 0. The

result is of the form

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J . N . Newman

Here @, is the solution of the two-dimensional diffraction problem outlined in Section V,B, X7 is the corresponding source strength, and QS is the symmetric part of @., In all cases the incident-wave frequency wo and wavenumber K O apply to the two-dimensional solution. In view of the definitions (7.12-7.13) and (7.19), this inner solution is not dependent on the frequency of encounter o,or the corresponding wavenumber K , but depends instead on the frequency wo and wavenumber KO.A physical interpretation for this distinction is given by Newman (1977). For short wavelengths the kernel in (7.26) vanishes, and the striptheory result follows. This two-dimensional solution has been studied by Bolton and Ursell (1973), Choo (1975), and Troesch (1976). E. THELONGWAVELENGTH SOLUTION Since the solution @, of the two-dimensional diffraction problem, is singular in head or following seas (fl= 180" or 0", respectively) the unified solution (7.26) is not well behaved in these limits. In the long-wavelength regime this difficulty does not arise. The essential change is to simplify the inner boundary-value problem by assuming that lo and K O are much less than E - ' in (7.4) and (7.5). The two-dimensional Laplace equation (3.44) and rigid free-surface condition (3.47) follow. The source potential for tbe inner solution is (271)- log r, and the homogeneous solution is a constant. With these simplifications the outer approximation of the inner solution (7.9) is replaced by 1 (7.27) @ 2 - (Z, M, log r C,(x), 271

$)

+

+

with the source strength C7 and dipole moment M , given by(5.41)and (5.42). Matching follows by equating the Fourier transform of (7.27) to the left side of (7.14). The relation for the outer dipole moment (7.15-7.16) is unchanged, but (7.17-7.18) are replaced by (7.28)

47*(lt) = Z 7 * ( k ) ,

2nC,*(k) = q,*(Q[log(K)

+ y + xi - f * ( &

2, K)].

(7.29)

After inverting these transforms and substituting the source strength from (5.41), it follows that q7(x)= - 2iw0 B ( x ) exp( - ilox),

2nC7(x)exp( - ilox) = q7(x)[log K

+ y + nil -

L

(7.30)

q7(5)f(x - t) d5. (7.31)

Theory of Ship Motions

273

Here the kernelfis the inverse transform of (4.13)' as in (6.34-6.35). As intended, the source strength q , and constant C, are well behaved in (7.30-7.31) as sin fi + 0, and these results can be utilized for head and following seas. On the other hand, the factor log K in (7.31) suggests difficulties for large wavenumber. This is a defect of the long-wavelength slender-body theory which is absent from the unified results.

VIII. The Pressure Force The six components of the pressure force (Fl, F,, F 3 )and moment (F4, F s , F 6 ) can be expressed in the form

F, =

ljpn, dS

(i = 1, 2,

. . . , 6).

(8.1)

S

Here S is the submerged portion of the ship hull, and the factors n, are defined by (3.38-3.39). The pressure p is determined from Bernoulli's equation (3.2), and can be separated into hydrodynamic and hydrostatic components. The hydrostatic pressure force is analyzed by substituting p = -pgz in (8.1). The nonvanishing components (i = 3 , 4 , 5) are analyzed by Wehausen (1971) and Newman (1977). The results are simplified for linearized motions of a ship hull which is symmetrical about y = 0. Further simplification follows by adding the gravitational force and moment due to the ship's mass, and assuming equilibrium when the unsteady motions vanish. The latter assumption implies that the ship's weight is balanced by the buoyancy force pgV and that the horizontal coordinates of the center of gravity and center of buoyancy are coincident. With these restrictions, the force components due to the sum of the hydrostatic pressure and the ship's weight are given by F3 = -pg(t3SOO - tSSlO)e'"',

(8.2)

+ ~ ( Z B- Z G ) ] C ~ ~ ' ~ ' , = pg{s10t3 + v(zB -

F4 = -Pg[So, FS

[sZO

zG)]<5}ei0'.

(8.3) (8.4)

Here zB and zG denote the vertical coordinates of the centers of buoyancy and gravity, and

sij = ljX'yJ dx dy, where the integral is over the plane z = 0, interior to the ship hull.

274

J . N . Newman

The steady component of the hydrodynamic pressure follows by substituting the velocity field W(x) in Bernoulli’s equation. From symmetry, the only contributions to (8.1) are for (i = 1, 3, 5). The steady portion of Fl is the (negative) wave resistance, which is balanced by the propulsive force. The steady vertical force and pitch moment are balanced by static “sinkage and trim,” which modify the equilibrium position of the ship hull but d o not contribute directly to the unsteady force or motions. Hereafter we consider the unsteady component of the hydrodynamic pressure force, with the usual assumption that the oscillatory motions of the ship and the fluid are small. Neglecting second-order terms in Bernoulli’s equation (3.2), the pressure is given by

An additional contribution to (8.1) results from the oscillatory position of the ship’s surface S with respect to the mean surface S. From a Taylor-series expansion the total oscillatory pressure in (8.1) is p = -p(q,

+ w . vq + +x . vwz)s.

(8.7)

The last term in (8.7) gives a force proportional to the unsteady displacement of the ship, and hence an additional contribution to the restoring coefficients cij in (1.2-1.3). This force is due to the unsteady motion of the ship within the steady pressure field. A similar contribution from the oscillatory change in the upper boundary of S is noted by Timman and Newman (1962). For a slender ship, the last term of (8.7) is approximated by

The resultant force from this change in pressure is O ( E )relative to the restoring-force components (8.2-8.4). For horizontal translation the contribution from (8.8) is zero and the only first-order effect from (8.8) is a sway force, and moment, due to a static yaw angle. The latter are analogous to the aerodynamic lift force and moment on a slender body, with important effects on the low-frequency steering maneuvers of ships. For oscillatory motions in waves these generally are neglected. The remaining contributions to the linearized unsteady pressure force are due to the added-mass and damping coefficients defined in (1.2) and the exciting force defined by (1.4). These are discussed separately below, using the solutions for the radiation and diffraction potentials derived in Sections VI-VII.

Theory of Ship Motions

A. ADDED MASSAND

275

DAMPiNG

With the notation of (l.l), our task is to evaluate the transfer function t . .= - p

jj(ioqj+ W . Vqj)nidS.

(8.9) Here Bernoulli’s equation has been used in the form (8.7), together with (3.34) and (8.1). The real and imaginary parts of (8.9) give the added-mass and damping coefficients defined by (1.2). The term in (8.9) proportional to the steady velocity field W can be transformed by means of a theorem due to Tuck (Ogilvie and Tuck, 1969, Appendix A), IJ

1’1

(W . V q j ) n ,dS = - U

1s q j m i d S

-

U

S

S

fqj&ni C

dl.

(8.10)

-

This result follows from Stokes’ theorem, and the fact that V W = 0. The last integral in (8.10) is over the boundary of S, i.e., the intersection of the ship hull with the plane z = 0. Since the rigid free-surface condition (3.47)l applies to &, the line integral is of higher order and can be ignored. Substituting (8.10) in (8.9) and using (6.7) and (6.12) for the unsteady potential q j , it follows that t I.J. = - p

jj(ion, - Umi)[c$j+ U J j + C j ( 4 j+ Bj)] dS.

(8.11)

S

The two-dimensional potentials 4j and $ j are defined in Section VI,B, and the interaction coefficient Cj( x)is determined from the source strength of the outer solution by (6.34). Equation (8.11) may be rewritten in the form

where

(8.13)

- mi4j) dl,

T!;)= - p U P

TI;’ = pU2 T $ ) = -pCj

P

mi$j dl,

1 (ioni P

-

(8.14) (8.15)

Umi)(+j+ $ j ) dl.

(8.16)

276

J . N . Newman

From Green's theorem (5.21), and the boundary conditions (6.8-6.9), TIIO.') = T$)g2),

(8.17) (8.18)

(8.19) TI;) = (- 2ipgAjCj/K)(Ai + UA,). Equation (5.32) can be used with (6.34) and (6.37) to show that the contribution from (8.19) to the integrated force (8.12) is symmetric with respect to the indices i and j. Thus the total three-dimensional force (8.12) satisfies the reverse-flow theorem of Timman and Newman (1962), i.e., ti;) = t);), where the superscript denotes the direction of the forward velocity. The local force (8.13) can be related to the two-dimensional added-mass and damping coefficients (5.9), and the contribution to (8.12) is the zerospeed striptheory result. The contribution from (8.15) is similar, but requires a separate analysis of the potentials 4j.There is no contribution from (8.16) to the fluxless modes (j = 2, 4, 6) where C j = 0. From (8.18) the integrals defining TI;) vanish for i = j , and give nonzero contributions only for the cross-coupling coefficients. For coupling between heave ( j = 3) and pitch ( j= 5), we note from(3.43) and (3.46b) that 4, = -x&, and m, = -xm3 + n3. From (8.18) it follows that 7''') - - " ( 15 )3 (8.20) 35 - (W4TS"J. By a similar argument, for sway and yaw, T':Q =

- Thy = - (U/io)T$"J.

(8.21)

Thus these contributions to the cross-coupling coefficients can be expressed in terms of the "zero-speed" potentials 4j. In the short-wavelengthstriptheory regime, the interaction coefficientsC j vanish, in accordance with (6.31). This leaves only the integrals (8.13-8.15) to be considered. For heave and pitch there are two variations of the resulting formulas. In the intuitive approach, gradients of the steady-state velocity field are neglected with the result that the only nonzero elements of mi are m, = n3 and m6 = -nz. Thus the two-dimensional potentials 4, vanish except for iw4, = and = -&. With these simplifications in (8.12-8.15),

(8.22) (8.23)

277

Theory of Ship Motions

It’

I- J;. -

-

T&”Jxdx & ( U / i o ) t 3 , ,

(8.24)

t55

=

S, T&’Jx’ dx + p(U/o)’t33,

(8.25)

t66

=

jL T$’iX2 dx -t

(8.26)

b53

p(U/W)2t22.

These equations are essentially? identical to the striptheory results for heave and pitch derived by Salvesen et al. (1970) and by Newman (1977). Somewhat different results are obtained in the systematic approach of Ogilvie and Tuck (1969),as noted in the Introduction. With o = O(&-(”’)), the moment proportional to (U/w)’ is discarded as a higher-order effect. Additional cross-coupling terms are added to (8.24) from the free-surface boundary condition (3.50). The heave force t 3 , is unchanged from (8.22). The comparative merits of these different approaches to strip theory have not been resolved, although Faltinsen (1974) has shown that the comparison with experiments is improved by using the Ogilvie-Tuck cross-coupling coefficients. The results (8.12-8.16) are not limited by the strip-theory assumptions and apply more generally for all frequencies and wavelengths. The principal complication is that the forward-speed potentials and the factors mjmust be determined, as well as the interaction coefficients Cj. Such calculations have not been performed in the general case U # 0. For zero forward velocity (V = 0), calculations of the heave and pitch coefficients by Mays (1978) show good agreement with “exact” threedimensional computations for slender spheroids with E < $. Computations of the added mass and damping at zero velocity also have been performed by Maruo and Tokura (1978), based on the “interpolation solution” of Maruo (1970) which is described in the closing paragraph of Section VI. Maruo and Tokura show good agreement of their results with experimental data. The difference between Maruo’s theory and the unified solution is of little practical significance for U = 0, and both offer substantial improvement relative to the ordinary slender-body theory and the strip theory.

4,

B. THEEXCITING FORCE

The exciting force (1.4) is the result of the pressure associated with the diffraction potential. The coefficient X i in (1.4) can be interpreted as the t In Salvesen et al. (1970), a “transom-stem correction” is introduced for ships where the after end of the hull is not pointed; the validity of this correction is questionable. In Newman (1977) a different coordinate system is used, and the expression for the pitch moment contains an error in the sign of the term proportional to (V/o)*.

J . N. Newman

278

complex amplitude of the exciting force due to an incident-wave system of unit amplitude. With the diffraction potential substituted in (8.9), this coefficient is given by

xi= - P = -P

\jni(io + W V)(cpo + jjs (ioni - Umi)(cpo + *

(p7) dS

‘s

( ~ 7d )s,

(8.27)

where the last form follows from (8.10). A direct evaluation of the exciting-force coefficient follows by substituting the inner solution (7.26) for the diffraction potential: Xi = -p

j exp(il,x) dx L

P

(iconi - Urni)[@,

+ O7+ C,(x)(Os + a,)]dl. (8.28)

The contributions proportional to the factors ni can be evaluated directly from the two-dimensional zero-forward-speed exciting force (5.30). The contributions from the factors mi depend on the steady-state solution, and the interaction coefficient C , is dependent on the forward speed. For the fluxless modes (i = 2, 4, 6), and also in the short-wavelength regime, C , does not contribute to (8.28) and the exciting force coefficients are linear functions of the forward velocity. Since Stokes’ theorem has been used in the last form of (8.27), the integrand in (8.28) cannot be interpreted as the local force. To emphasize this distinction we note the special case of a long “parallel middle body” where the ship hull is cylindrical and, in the inner region, W = - Ui. Neglecting the interaction coefficient, the only effect of the gradient operator in (8.27) is on the oscillatory factor exp(ilox ) , with the result that (iw + W - V) = io,,and the local exciting force is independent of the forward speed. This is confirmed physically by the fact that a long cylinder may be moved axially (in an inviscid fluid) without affecting the local pressure field except near the ends. In this connection we recall the discussion following (7.26). Green’s theorem can be applied to the three-dimensional potentials in (8.27), or alternatively to the two-dimensional functions in (8.28). In both cases the result is a form of Haskind’s relations, with the solution of the diffraction problem replaced by an appropriate radiation solution. To derive Haskind’s relations in the three-dimensional form, following Newman (1965), the boundary conditions (3.37) are combined with (8.27) to give

xi = - P

jf s

CPG)((P~

+ ( ~ 7 d) s .

(8.29)

279

Theory of Ship Motions

Here the superscript (- ) denotes the reverse-flow solution of the radiation problem, with negative forward velocity. This radiation potential satisfies the free-surface condition (3.51) with U replaced by - U . After an application of Green’s theorem to (8.29) with the boundary condition (3.36) imposed,

Jj (cp{,’cpo

- cp,cp~-’) dS. (8.30) s Here a line integral similar to the last term in (8.10) is neglected. In this form, the total exciting force on the ship is expressed in terms of the solution of the radiation problem. The inner solution for cpj-’ may be substituted, with the integral performed over the mean surface of the ship hull. Alternatively, this integral can be performed over a closed surface at large distance from the ship, where the far-field asymptotic form of the radiated waves may be used. McCreight (1973) has extended the Haskind relation (8.30) to the striptheory regime of Ogilvie and Tuck (1969), including the higher-order terms in the free-surface condition (3.50). Equation (8.30) is unchanged, but the integral cannot be evaluated at infinity due to the inhomogeneous freesurface condition in the higher-order solution. McCreight notes that the leading-order contribution to (8.30) is the Froude-Krylov force. The next term in a systematic perturbation expansion can be expressed in terms of the zero-speed two-dimensional potential 4. If Green’s theorem (5.21) is applied to the two-dimensional functions in (8.28), we find after a reduction using (5.33) that

Xi = - p

Xi = i ( p g / K , ) I sin and

. [l

X i = -i(pg/K,) sin

B

fl I

L

exp(il,x)(& - U j i )

+ C,(AIi/zi - Ji/Ji)]dx I

exp(il,x)(& - U A , ) dx

(i = 1, 3, 5), ( i = 2, 4, 6).

(8.31) (8.32)

JL

Here the local exciting force at each section is expressed in terms of the wave amplitude of the generalized radiation function &i and the corresponding forward-speed function &. $or the antisymmetric modes the exciting force follows directly from (8.32) but for the symmetric modes (i = 1, 3, 5) one must determine the interaction coefficient C , of the diffraction problem, by solution of (7.21-7.22). If the inner solution of the three-dimensional radiation problem is substituted in (8.30), the result is an integral along the length involving twodimensional solutions of Laplace’s equation. By comparison, the integrands

280

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of (8.31) and (8.32) contain solutions of the two-dimensional Helmholtz equation. It is not obvious that these alternative integrals for the total exciting force will be equivalent, even in the asymptotic sense for small values of the slenderness parameter. Computations of the heave and sway exciting forces have been made by Troesch (1976) for the case U = 0, using these two alternative expressions; the results are in satisfactory agreement with each other and with experiments. In the intuitive strip theory initiated by Korvin-Kroukovsky (1955), thZ local exciting force at each section is derived from a “relative-motion” assumption. This states that the exciting force can be expressed in the same form as the pressure force of the radiation problem, but with the ship’s velocity and acceleration replaced by the relative motions of the incident wave, at a suitable mean depth. (In accordance with G. I. Taylor’s theorem, there is an additional component of the exciting force, equal to the product of the incident-wave acceleration and the mass of fluid displaced by the hull.) In this relative-motion approach the pressure force of the radiation problem is expressed in terms of the added-mass and damping coefficients, evaluated at the frequency of encounter w. Similar expressions for the exciting forces are derived by Salvesen et al. (1970), using the two-dimensional Haskind relations. In modified results derived by Newman (1977) the added-mass and damping coefficients are evaluated at the incident-wave frequency wo. An argument in favor of this modified approach is that, along a parallel middle body, the exciting force is independent of the forward velocity of the ship. REFERENCES ABRAMOWITZ, M., and STEGUN,I., eds. (1964) “Handbook of Mathematical Functions.” U.S. Gov. Print. ON., Washington, D.C. BABA,E., and HARA,M. (1977). Numerical evaluation of a wave-resistance theory for slow ships. Proc. Int. Con$ Numer. Ship Hydrodyn., 2nd. pp. 17-29. Univ. California, Berkeley. BAI,K. J., and YEUNG,R. W.(1974). Numerical solutions to free-surface flow problems. Proc. Symp. Nav. Hydrodyn., 10th ACR-204. pp. 609-647. ON. Nav. Res., Washington, D.C. BECK,R. F., and TUCK,E. 0. (1972). Computation of shallow water ship motions. Proc. Symp. Nav. Hydrodyn., 9th ACR-203, pp. 1543-1587. ON. Nav. Res., Washington, D.C. BISHOP,R. E. D., and -Ice, W. G., ads. (1975). Proc. lnt. Symp. Dyn. Mar. Vehicles Struct. Waoes. Inst. Mech. Eng., London. BISHOP,R. E. D., BURCHER, R. K., and PRICE,W. G. (1973). The uses of functional analysis in ship dynamics. Proc. R.SOC.London, Ser. A 332,23-35. BOLTON, W. E., and URSELL, F.(1973). The wave force on an infinitely long circular cylinder in an oblique sea. J . Fluid Mech. 57, 241-256. BRARD,R. (1948). Introduction a l’etude theorique du tangage en marche. Bull. Assoc. Tech. Marit. Aeronaut. 47, 455479. BRARD,R. (1973). “A Mathematical Introduction to Ship Maneuverability,” Rep. No. 4331. Nav. Ship Res. Dev. Cent., Bethesda, Maryland.

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