Energy relation for the wave resistance of ships in a seaway

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by Hajime Maruo

(To be presented to the Seminar


Theoretical Wave Resistance



The mechanism by which the resistance is increased under the influence of ocean waves has been discussed by various researchers for more than twenty years, but no theory evolved could give a

con-sistent description to the source of the resistance increase until recent years. This was a result of the complicated nature of the fluid motion around a ship in a seaway which had been defying the mathematical analysis. The first attempt at a theoretical approach seems to be attributable to Haveloek.1 He discussed the horizontal force acting on a thin ship moving among waves with uniform velocity. His conclusion was that the extra force due to the .existence of the wave was periodic, so there was no resistance augmentation in the

order of approximation An attempt was made by Kreltner2 who pro-posed a semi-empirical formula which gave the increase of resistance

in terms of the wave reflection at the surface of the ship. An

analytical and rigorous calculation was made by Havelock3 with respect to the average force which was brought by the reflection of waves. It showed a resistance too small to give the extra resistance of a ship among waves. Some years later, Havelock4 made an attempt to give

the resistance augmentation in a different way. The close correlation of increase of resistance with the shipts oscillations, especially pitching, has been recognized as a matter of fact, since Kent's5 resistance experiments in regular waves, Havelock was the first one who established a theory which gave the relationship between the




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crease of resistance and the oscillation of ships, though the basic idea by which Havelockts theory was developed was due to Watanabe.6 The latter proposed a theory of the drifting force experienced by a ship rolling in beam seas. Therefore, the theory may be called the drifting force theory. It assumes the force acting on the ship hull to be calculated from the fluid pressure in the ocean wave which is not disturbed by the ship. This assumption is similar to the hypo-thesis of Froude7 and Krilov8 in the theory of shipTs oscillations. According to the drifting force theory, the ship experiences a

con-tinuous force independent of time in the direction to which the waves propagate, if a phase lag exists between the oscillation of the ship

and the excitation by the wave, even though the pressure itself is purely periodical. Havelock applied it to the heaving and pitching of a ship in head seas and found a simple relation between the re-sistance augmentation and the amplitude and phase of the

oscilla-tions. At one time, Havelockts theory was regarded as the most reliable one of existing theories about the resistance increase due to the seaway and some numerical examples were reported by St. Denis.9 However, physical significance was still obscure because of the

formalism in the deduction of the formula, and there was opinion that the underlying principle of the theory was quite doubtful, such as the criticism by Russian scientists'0 who blamed it as violation of the principle of energy conservation. There is another theory de-rived in a different way. Hanaoka1' studied the resistance when a

ship was moving in a calm sea with a uniform velocity but making heaving and pitching motions under the excitation of some external mechanism such as an oscillator. He called the increment of

re-sistance in this case the nonuniform wave rere-sistance, He showed


that the principle of energy conservation was attained in this case and some years later, this was reproduced by Newinan12 One may feel sorne confusion by the coexistence of two theories, which stand on quite different theoretical bases

ThIs short essay aims to unravel the entanglement between two theories and give a lucid answer to the origin of the resistance augmentation due to the seaway.

The Energy Consideration for a Ship in Regular Head Seas

Though the fluid motion near the ship is very complicated, the fluid motIon far from the ship shows a comparatively simple

nature, and an energy analysis is possible which gives correct in-formation about the resistance experienced by the ship. This sort of analysis was employed first by Havelock13 In discussing the wave resistance of ships in a calm sea, Havelock assumed two vertical planes, one of which in front of a ship moving on a calm sea with uniform velocity and the other far behind it, and derived the wave resistance from the energy variation in the fluìd contained between these planes. Recently Newman extended HavelockTs method of analysis to an oscillating ship. A similar analysis can be applied to the ship in a seaway, but as the waves generated by the ship also propa-gates sideways in this case, vertical planes are not suitable as an

energy control surface. Instead of vertical planes, a cylindrical surface with a vertical axis and radius large enough to surround the ship is taken as the control surface.

In the first place, the cylindrical surface is assumed to be fixed and the ship is moving with a uniform velocity U. After a unit time has elapsed, the ship comes to the position of distance U from the initial position. Now we compare the energy of the water


Inside the cylinder at the beginning and that at the latter instant. If we assume fcr the moment that the fluid motion around the ship is stationary in order to simplify the exposition, the change of the energy inside the cylinder is attributed solely to the change of the position of the ship

If, on the other hand, the cylinder is assumed to move with the ship, the cylinder will corne to a new position at a distance U from the original one after a unit time, We will designate the initial


of the cylinder by C and the new position by 02. As the fluid motion around the ship is assumed stationary, the energy inside the surface 02 at the latter instant must be the same as that inside C at the beginning0

Now, returning to the former case that the cylinder is fixed at the position C1, the energy insIde it changes as the ship moves to the new position. Here let us designate the space inside C excluding the portion inside 02 by I, the space inside 02 excluding the portion inside C by II, and the common space inside both C and 02 by III. The energy contained in each portion after a unit time has elapsed is designated by E1, E11 and E111 respectively. As the fluid motion relative


the coordinates fixed to the ship is assumed unchanged, the energy inside C at the beginning and that inside

02 after a unit time must be the same. The latter is E11 + E111. Since the energy inside Cl after a unit time is E1 + E111, the net varia-tion of the energy in a fixed cylinder at C during a unit time is given by

= E1 + E111 - (E11 + E111) = E1 - E11 (1)

If the cylinder is assumed to move with the ship, the above quantity is identical with the energy efflax which is carried away by



the fluid going across the surface. As the energy inside the fixed cylinder changes, he net variation of the energy must be surplied from outside according to the principle of energy conservation.


energy supply is partly brought in by the fluid flow at the boundary, i.e., the cylindrical surface, We shall write this


It is positive if the energy flows inwards across the surface, but it may become negative in some cases. Then the difference


-must be supplied by the ship. 1hen a ship


towed by a constant force which overcomes the resistance R, the effective work done in a unit time is RU and is the rate of the energy supplied to the fluid Then the princIple of the energy conservation gives the following


RU (2)

3t t

This relation holds also in the case where viscous forces exist, if the energy dissipation by the viscosity is taken into account. In the present analysis, however, the effect of viscosity is assumed un-important so that the energy is carried away only by the potential motion of the fluid. We can write




where E/t and

W/t are both determined by the fluid motion at the

cylindrical surface if the motion is stationary. If we take the radius of the cylinder very large, the resistance is determined by

the fluid motion at a great distance from the ship.

Now turning to the ship among waves, the motion of the ship and that of the fluid changes from time to time. Even in the

non-stationary motion, the relationship between the resistance and the energy variation given before still holds in principle, but the term



ßurface alone because the fluid motion within the cylinder moving with the ship changes periolically. As the resistance of a ship

among waves is understood in the sense of time-average of the hori-zontal force, the time-average of each term of the equation is suf-ficient to th analysis of the resistance. If the motion is periodic the time-average has a definite value which is independent of time,

so that the energy relation is discussed similarly to the case of

calm sea.

Let us consider a ship moving under a constant towing force in regular head seas. Though the forward speed changes periodically, the speed of advance is defined as the average velocity of the ships The periodical change of the velocity is a component of the shipTs

oscillations, i.e., surging. In generai, the ship makes oscillations with six degrees of freedom by the action of the wave, but in head seas,

the oscillations have three modes, heaving, pitching and surging.

Instead of a ship moving with forward velocity U, we first consider the case that a ship is floating on a stream of velocity U in the

opposite direction to the advance velocity of the ship, the average position of which is sustained at a fixed position by a constant towing force, and a regular train of waves is superimposed on the uniform stream. As the towing force keeps the ship at a stationary position, it does no work on the average. Therefore there is no ex-ternal force that supplies energy, except the hydrodynamic action.

Let us consider two modes of oscillations, heaving and pitching, the amplitudes (double) of which are Z and respectively. The heaving is excited by the periodic force of amplitude F and the pitching by the periodic moment of amplitude N. There are phase lags between the motion and the excitation. They are for heave and


for pitch. The phase lag in the oscillation is the consequence of the dar'iping term in the equatirn of motion The damping force and moment consume the energy. In the absence of the vIscous forces, the energy


transmitted by the wave which is possibly generated by the ship. The energy consumed or transmItted by the wave In a

complete cycle can be calculated at once as follows:


FZ sin e ±


IVW sin e2

So the average rate of work which must be done to maintain the motion is





e e

where Te is the period of encounter. Since the motion of the ship is excited by the wave pressure and no other source of energy exists,

the energy must be supplied by the incident waves. if the incident wave is not disturbed by the ship, or there is no scattered wave, the

energy brought into the cylinder by the wave is carried away by the same wave out of the cylinder. Therefore there is no supply of

energy. Consequently the scattered wave again plays the role of

supplying energy If the wave is produced by an oscillating ship In calm water, the wave can only disperse the energy. Accordingly, the scattered wave cannot supply energy by itself. The only possibility of energy supply Is the interference between the incident wave and the scattered wave. The principle of the energy conservation in-dicates that P1 is identical with the energy taken Into the cylinder per unit time as a result of the interference between the incident wave and the scattered wavë

Next, let us consider another case in which there is a uniform stream of the same velocity as the wave celerity c but the

direction of the flow is opposite to the propagation of the regular


wave. In this case the wave superimposed on the uniform stream does not propagate and steady motion of the fluid is obtained. The motion of the ship, which corresponds to uniform motion with velocity U in the opposite direction of the propagation of wave of celerity c, is the motion with uniform velocity U + c in the system now being

con-sidered. Here the regular train of incident wave does not propagate but form just a wavy surface of the stream upon which the ship is

moving. Accordingly, there is no energy transmitted by the incident

wave. The energy is carried away across the vertical cylinder which encircles the ship by the wave generated by the ship and there is nothing to compensate for it but the work done by the towing force T

which overcomes the resistance to propel the ship with velocity U + c. The wave generated by the ship is constituted of a wave system which accompanies the ship maintaining a constant pattern, and another wave system which is the consequence of the oscillatory motion and is

itself periodic. As the energy transmitted by the wave is a quadratic functional of the wave motion, we can regard the resistance as a sum of two components corresponding to the energy transmission by each wave system.

Accordingly, we can write



The component AR is the extra resistance as a consequence of the seaway. The work done by the part of the towing force which overcomes AR is

consumed by the transmission of energy through the periodic wave and its rate per unit time is identical with P1. If there is no supply of energy elsewhere, the following relation exists.

AR(TJ + c) = P1


There is a relation between the period of encounter and the wave length X as follows in the case of head seas.




U+ e

If the waves generated by the ship are identical with those which accompany the heaving and pitching of the ship in calm sea, we can equate the right-hand side of eq. with the left-hand side of eq. 6

Dividing both sides by U + c and applying the relation (7), we find the formula

AR = (71/kX) FZ sin c + (r/4X) M

Slfl 62


This equation coincides exactly with the formula given by Havelock, who derived it from the drifting force theory Thus there is no contradiction in the principle of energy conservation in spite of the criticism stated before. This relation is not, however, an exact one. The energy efflux P1 corresponds to - W/t of eq. 3 in

the present system of motion, but as the result of the ship's motion with velocity U + e, another term which corresponds to E/t must be

considered. Therefore we must write instead of eq. 6 AR (u + o) = P1 + iE/t

=P +P

1 2

We have assumed that the waves generated by the ship are identical with those produced by the heaving and pitching oscillations of the ship in a calm seau If the oscillation of the ship is excited by some external mechanism such as an oscillator, the latter exerts work to overcome the damping forces and the average rate of work done in a unit time

is P1. Hanaoka studied this case and found a change in resistance

against the forward motion as a result of the oscillation. Let us designate lt by ARG. The energy transmitted by the wave, which is generated by the ship is P1

+ P2

in a unit time, while the energy supplied to the ship by the external forces, that consist of the towing force and the oscillator, is




± AR2


+ c)

Therefore, the following relation exists. P1 + AR9


+ c) = P1

+ P2

That is

P2 = AR2


+ c)

Combining eqs. 9 and 11, we obtain

AR = AR1 + AR2

Thus the increase in resistance is the sum of the force obtained by the drifting force theory and the resistance due to the oscillation in a

calm sea, the nonuniform resistance by Hanaoka.

It has been assumed in the above discussion that the waves generated by the ship are identical with those produced by the same

ship constrained to oscillate in a calm seas However, the waves ac-companying a ship in a seaway are somewhat different from the above because of the undulation of the sea surface. We can take account of the effect of the seaway by assuming a fictitious deformation or snake-like motion of the ship hull. In fact, Hanaoka applied this technique when he attempted to extend his nonuniform wave resistance theory


the resistance among waves. Though this idea can be applied to the component AR2, there is a difficulty in its application to AR1, the drifting force.

Giving up the theory of drifting force and that of nonuniform wave resistance, we can obtain the total increment of the resistance AR directly from an energy analysis of the wave generated by the ship together with the incident wave. As a detailed analysis is given in ref. 1k, there is no intention to reproduce the procedure of analysis


In order to obtain an analytical expression, the problem is analyzed by means of the linearizinr' procedure, assuming that the


wave height is small compared with other quantities such as the wave length or the ship length. As the energy is a quadratic functional of the fluid velocity, the added resistance is of second order with respect to the wave height Let us designate the ship's length by

L, the beam by B, the wave amplitude (= 1/2 wave height) by r , the density of water by p and the acceleration of gravity by g . Then

the following expression is proposed for the increment of the


L\R = pgr2


K (13)

where K is a dimensionless coefficient which is a function of the wave length, the ship's speed and the ship form.

Resistance in an Irregular Seaway

The discussion in the preceding section has dealt with regular head seas, but a similar analysis applies to regular oblique seas and also to following seas In order to apply the theory to the actual seaway which is irregular and complicated, we assume the possibility of linear superposition. The seaway is assumed to be constituted of a superposition of plane regular waves of infini-tesimal amplitudes with different frequencies arranged in various directions. The amplitudes of the component waves are distributed over any value of the frequency and direction, and the manner of the arrangement is defined by the energy spectrum of the sea. The

assumption of linearity results in the ship oscillation being linear with respect to the motion of the sea surface. This means that the amplitude of any component of oscillation Is obtained by a linear

operation. In contrast the increase of resistance due to the seaway is a quadratic quantity with respect to the wave. If the amplitude of the wave element in the direction x having the circular frequency


energy spectrum of the sea is defined in the following

- - r. du dX = (dr)2 (1k)

of the compound wave is represented by the integration of




- ii- called the cumulative energy density. The sea surface is

= i (w,X)dw dX sin cosy + y sinx) + uit + 5(wX)] (16)

is the phase of the wave element in the direction X with

fency w.

Since the shipTs oscillations are linear quantities, the ship is a linear superposition of the oscillations each wave element. The waves generated by the shipTs

::ao:ns are also linear and their superposition is possible; The

the shipts oscillations is not

w hut

we , the circular

encounter. Accordingly, the frequency of waves generated must be defined

by the

frequency of encounter.

Though w

have a one-to-one correspondence in the plane head seas, there relation in the three-dimensional irregular seaway. Though

spectrum of the seaway is defined in the w-x plane, the reaoion with which the resistance is concerned, must be

dis-ohew-) plane.

In the transformation from We_X plane to w is a multivalued function of We Then cross products ponents of different w but with the same we give

time-e-- terms and do not vanish when the time-average is taken.

prevents an analytical expression for the time-average of 'es3-arìce in terms of the spectrum of the sea. If the problem


is discussed in a statistical sense, the expression for the average increase of resistance can Le obtained by assuming a random phase.'5

We designate the increment of the resistance in a regular plane wave of amplitude dr, circular frequency w and heading angle X by the following form.

dR = pg(dr)2



(17) Then the average increase of resistance in the irregular seaway de-fined by the energy spectrum, eq. 11-,is expressed as follows:

AR = pg


K(w,x) E(w,x)dw

dX (i8)

o o


Havelock, T. H.: Proc. Roy. Soc. 161A, 1937 Kreitner, H.: T.I.N.A. 81, 1939

Havelock, T. H.: Proc. Roy, Soc. 175A, l90


Havelock, T. H.: Phil. Nag.

33, 192

Kent, J. L.: TI.N.A. 6, 1922

Watanabe, Y.: T.I.N.A. 8o, 1938 Froude, W.: T.I.N.A. 2, i86i Krilov, A. N.:T.I.N.A. 37, 1896

St. Denis, M0: T.S.N.A.N.E. 59, 1951 Pershin, V. I and Voznessensky, A. I.:

Proc0 Symposium on Ships in a Seaway, 1957

Hanaoka, T.: Journal of Zesenkiokal (J.z0K.) 9k, 195k Newnian, J. N.: Journal of Ship Research, 3-1,

1959 Havelock, T. H.: Proc. Roy Soc. lkkA, 193k

1k. Maruo, H.: Bulletin

Faculty of Eng Yokohama Nat. Univ 9, 1960

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