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A note on the linearized theory of wave resistance for accelerated motion


Academic year: 2021

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EXTRA/T DES MEMO/RES SUR LA- MECANIQUE DES FL offerts A M. D. Riabouchinsky a rorcasion de son Juba Scien fique


Lab. -v.




Studiecentrum T. N. 0.

voor Scheepsbouw en Navigatie Md. Scheepsbouw.




by J. K. LUNDE




Skipsmodelltankens medd.

nr. 27









The linearized theory of wave resistance for displacement ships in

accelerated motion on the surface of a deep inViced and incompressible

- fluid has been discussed by HAVELOCK [1], [2], [3], [4], [511 and

SRETENSKY [6] whilst the corresponding theory in the case of a shallow fluid and in the case of an infinitely long shallow canal filled With fluid has been discussed by the author [7]. The solfition in the last case may,

perphaps, be taken as the general one when considering motion in a straight line,_ because from it we may deduce solutions of particular

problems such aS accelerated, as well as steady motion in a deep canal, a

shallow fluid or a deep unrestricted fluid; The essential points of this

generalization will therefore be recalled as Well as aniplifial.

The exact problem of wave resistance is by its nature a hard one. On the theoretical side we are faced with finding a potential functiOn

Which satisfies the equation of continuity, the condition at the hull

of the ship, as well as a certain non-linear boundary condition at the free

fluid surface whose shape is not given but which must be determined as part of the solution. Because of this, the problem is linearized. On

the other hand, on the 'practical side, we have the difficulty of assessing the true wave resistance of any body moving on or below the free surface

of a real fluid.


The classical -linearized problem of waves on a deep fluid due to a moving disturbance is an old one and dates back to KELVIN. He

theo-retically examined the waves produced by a-concentrated pressure/point

moving on the surface of a fluid by regarding the displacement of a point in the free surface as due to a series of pressure impulses applied

along the course of the pressure point.

The approximate solution for a slender ship form in steady motion was, however, given by MICHELL [8] in a- classical paper, which was unfortunately overlooked and forgotten for many years. His procedure was to consider a semi-infinite uniform stream of fluid with a free upper surface and centrally bounded by a vertical plane parallel to the stream.'

He solved the problem for the steady motion produced by a given

dis-tribution of normal perturbation velocity over this plane. A ship of

narrow beam placed in the stream was then pictured, as producing a

normal perturbation velocity outwards on the two sides of amount given

approximately by the product of uniform stream velocity and the

hori-zontal gradient of the water lines of the ship. This was then treated as a

given distribution of horizontal perturbation velocity outwards on the two sides of the longitudinal vertical centre-line. section of the ship.

Such a discontinuity of normal perturbation velocity is, however, equi-valent to a corresponding distribution of sources and sinks over the same

vertical plane, and .Michell's result is accordingly a particular case of

the more general solution given by HAVELOCK [9] for a distribution of sources and sinks over the curved surface of a ship in steady motion.

It is with a distribution of sources and sinks representing the influence of the ship that this paper will be concerned.


' In any practical problem the motion will have started at some time

and after a period of acceleration the speed may, for example, become constant whereupon the waves eventually establish themselves fully. We follow Kelvin's suggestion and regard the motion as a limit of a

succession of small steps, at each step an impulse being applied. Each

of these will start a series of ring-waves travelling out in all directions. The total effect at any time is then obtained by summing up the effects

due to all the .preirious elementary steps.

Let the x-, y-axis of a right hand, orthogonal reference frame coincide with the undisturbed free surface of a body of inviced and incompressible


fluid with the z-axis directed vertically upwards. Suppose the fluid at

rest fills the space di,

z 0, that the x-axis coincides with the direc-tion of modirec-tion and that the reference frame moves with the ship. If the

-velocity of the ship, varying with the time t, is c (0, and if .1:1 (x ,.y, z, t) is

the Velocity potential of the absolute irrotational motion the pressure

equation referred to the moving frame may be written:

1 1 1

(p p')±


gz .1)/

-2- c G(I),

where p denotes the pressure, p the density, g the acceleration due to

s gravity, p' the atmospheric pressure at the free surface (supposed constant)

and g the magnitude of the fluid velocity.relative to the moving frame.

, As the components u, v, w of the perturbation are given by Ox;

Ou, --"4311z in the x-,y-,z-direction respectively we have that (1) may be written:



_L_ _1(Dx21

(I) 4. cr,z

, 2 u2 2) gz c (Dx =- G (t)

and neglecting small quantities of second order, that is the resultant

perturbation velocity square, we get the linearized pressure equation:

1 1


1:13x ± Cot gz.


At the free surface z =-- We have p = p' and (2) becomes:

= - (4)/ c

(1)x)zc-Expanding (1:0 in a Taylor series_as follows:

(x, g,


(x, y, 0,1) [(Ipz (x, y, z, O]z.L0

and neglecting the second term on the right hand side in comparison with the first, we may regard (3) as fulfilled at z = 0 rather than z

The general kinematical boundary condition., which must be satisfied

by the fluid, is that the velocity relative to the surface of a particle

lying in it, must be wholly tangential. Thus; if F (x, y, z, I) = '0 is the equation of the boundary surface and Ili V, W are the Components of

the fluid velocity relative to the moving frame this condition iS given by

Ft + UFx VF/j WFz = O. (4)




In our case the free

surface is

given by z

(x, y, 0;1), thus F x, y, z, t) (x, y, 0, 1) leading to:


(c (I)x) 1:Dg - (I)z = 0.

As in the foregoing we neglect small quantities of second order, that is the nen-linear terms, and we have to the adopted order of



+ i==.-

6 (5)

for z 0..

Combining the kinematical and pressure boundary conditions (5), (3) we have:

(1)11 g + c2 (I)xx c (I)xt

= 0,


where c =

Thus (6) expresses the condition to be satisfied by 41),

. the perturbation velocity potential, at the average free surface given

by z = 0.

Making 'use of Havelock's method [2] and emploYed by him for a

deep fluid, we imagine a source of strength m suddenly created at the

point (0, 0,


at the time t 7 0, maintained fOr a short interval of time 8 -r and -then annihilated. Assuming the fluid continued both above the


free surface and below the bed, the velocity potential is given by =. As the condition to be satisfied at the bed is that the normal fluid velocity

is zero here or, (I). =-. 0 for z d, we have to introduce a._ second

source of strength .m. at the image point (0, 0, 2 d, +1) in the bed and the velocity potential of these two becomes:


OSO exp 1K[i us 7 (z + f)HdK


= x2 + y2 + +

1)2, r22 = x2 + g2 (z,+ 2 d fy,

= x cos 6 y sin 0

and where the first integral tetm requires



0 and the second

z + 2 d1

I >

0. It will readily be seen that this potential gives zero

normal velocity at the bed and also satisfies the equation of continuity. exp K [i tu

(z.H- d1

f)] dK, (7)



Whilst the sudden creation of the source and its image in the bed

may be compared with an impulsive motion yet the free surface condition is still that the impulsive pressure is zero ,here, or (I) = 0 for t 0.

As the fluid is supposed continued above the free surface this now becomes

the condition to be satisfied at the plane z = 0 which coincides with

the free surface. We therefore add a third term to (7) in order that the initial free surface condition shall be satisfied. This term must; fur-thermore, satisfy the equation of continuity as well as the condition of

the bed by itself. It is therefore clear that we seek a term of the type:

_rtd 0 F (0, K) cosh [K (z di)] exp (iK us) dK

and from the initial surface condition, i.e. 41) = 0 for z = 0, we find:

(d,-- /)1

F (0, K) m cosh [K7r cosh K d, exp ( K d1).

The initial velocity potential- due to the sudden creation and

annihi-lation of the source now becomes' the real part:

m m m d ccc dosh [K (d,. f)] 7.1

-I- r2

7r cosh K

x cosh [K (z d,)] exp [IC (i us d,)] dK, (8)

which causes a vertical velocity w at the surface z = 0 in the unlimited

fluid. As our system is maintained a short interval of time 8 .7, the

vertical' displacement of the surface from its original position z = 0

will be given approximately by Pf=0 =


(w)z=0 assuming the vertical

coordinate / to be large, thus:

()z=° =8 T

d O [1 + tanh (K di)] cosh [K (di 1)]



x exp [K (i d1)] K dK. (9)

Since an impulse is to be regarded as an infinitely large force acting

during an infinitely short interval of time the effects of all finite forces during this interval are to be neglected: Thus, gravity does not come into play during the impulsive motion.

We now imagine the fluid above the surface Pt=0 removed. The effect of the creation of a source which is maintained a. short interval

of time is that the free surface is left with a known elevation given by (9). The subsequent motion of this elevation will now have to be worked out,


under the action of gravity. In fact (9) may be regarded as an,initiat

displacement of the fiee surface without initial -velocity. The problem is then a well-known one in which we have to determine the subsequent motion.

As the source system is not yet in motion the surface elevation is given by (3) where we put c = 0, or:





where is the elevation at any time. From (9) we have that the motion

at any subsequent time due to the initial elevation (9) is given by the

velocity Potential:



cosh [K (z di)] cosh [K (c11f)]

TC 30 - cosh (K di) tanh1/2(K

x [1 -F tanh (K (JO] sin [1 (gK tanh K (4)1/2] exp [K (iUS di)] 102 dK, (11)

where the real part has to be taken. It will be seen that (11) satisfies (6) when c = c = 0 is substituted in that expression. The velocity potential given by (11) may also be obtained from (9)_simply by observing that a

wave of the usual type cos (Kx) cos ft (gK tanh K d1)]'I has a velocity potential given by:

g'1. cosh [K (z di)] cos (Kx)

I K-tanhK d0 1/ (K tanh K cil)/. cosh (K d,) sin [ (g


We want, however, to find the velocity potential at any tine due

to a source created at (0, 0, f) at the instant t = 0"and made to travel

with any velocity c (1) in the positive direction of the axis of x. If the source was created suddenly at the timeT, we would have to replace t

by t

T in (11). Thus the resultant velocity potential of a moving

source of strength m (1) can be arrived at by summing or integrating the disturbarice of the small steps as given by (11) and adding (8) or:

m m (t) cn d 0 c`c cosh [K (di f)]

30 cosh K

X cosh [E. (z di)] exp [K (i d1)] dK

(T) d cn d 0 c cc cosh [K (z di)] cosh [K (di f)]

J-Tr cosh (K cli) tante/. (K di)




x [1 H- tanh (K di)] sin [(I T) (gK tanh K di)1/.]


where r, r2,zu have the usual meanings and:



c(T)dTi cos 0 + g sin 0

and where x is measured from a moving origin vertically over the source. It will readily be seen that (12) satisfies the condition at the free surface given by (6), the equation of co-ntinuity and giving zero normal velocity

at the bed.

Consider now a continuous distribution of sourtes and sinks.

Replac-ing x, y by x g k and taking a (I) to be the surface density of.

the source strength at the point (Ii, k, f) on the surface S within the liquid, the velocity potential may be obtained from (12) by integration,

Canal Walls

Fig. 1

since the problem is made linear. We shall, however, only consider a

very slender ship for which the distribution may be taken to coincide

with the vertical centre-line plane g 0 of the ship. We assume further

that this plane coincides with the centre-line plane of a canal of depth d,


and width b. The vertical canal walls are then given by y =

The only condition to be satisfied at the canal walls is that the fluid

velocity component normal to the walls is zero. Since the ship is assumed

- to be moving in the centre of the canal the condition at wall can be satisfied by placing an image ship at the image point; that is, its vertical centre-line plane would coincide with the plane bp These ships, however,


will cause some sort of flow along the canal wall y = -2- and in order to satisfy the condition along this wall we assume image ,ships at y b,

and at y = 2 b,.

These, however, will in turn disturb the flow along


the wall y -2- and in order to satisfY the condition here we now- have to assume two more image ships at y =- 2 b, and at y = 3 L., and so on.


of ships whose centre-lines are placed at y

b, ±.2 b,

nb,, as indicated in figure 1 and the velocity potential now becomes:




(t) ds rn, 4:131 8 S, I()


77d 0 Cx_ c°shJo cosh K di


x cosh [K (z exp [K (i yin .d1)] dK

43,t (.0 d ST, d 0 cosh c[Koszo±c ddi1))1tacohnlii.[-K(K(dd10-1)]

x [1 tanh (K di)] sin [(t T) (gK tanh K di)11.]

x y exp [K (i 7344 d1)] Ki/. dk,


- where:,

(x h)2 + (y + f2,

rri2 = (x h)2 + (y

nb,)2 + (z + 2 d, 29

Uhl h) cos 0 (y n bi) sin 0,



+torc (T) d COS 0 (y rib].) sin 6.

The velocity potential given by (13) for the linearized problem is

quite general' for motion along the centre-line of a canal and from it. We

may deduce the velocity potential for the particular cases mentioned

in the introduction. These particular cases will however, be considered later in connection with the expression for the wave resistance.

It may, be shown that (12) or (13) leads to the so-called practical

solution with the main wave-pattern only, to the rear of the body

advanc-ing into still fluid.


We consider a very, slender ship, symmetric about the longitudinal,

vertical centre-line Plane y = 0 and moving with a speed c (1). If the

form of the hull is given byy f (x, z)we substitute F -= f (x, = 0

in (4) giving (c u) fx + v wz = 0, where u, v, w are obtained from (13).

The usual first approximation And on which Michell based his investi-gation is to neglect \ ulx and w/z in comparison with the other terms


when considering a very slender form and to write v cix Which

is assumed to apply to the centre-line plane instead of on the hull

- y = (x, y).

The form is now replaced by a continuous distribution of sources and sinks over this plane and it will readily be seen that such a distribution gives a normal velocity. 2 77 a on each side .of the plane.

From the foregoing the source density

is approximately given by

v c


It shourd be noted that by- using

27: TO 27'r ) Yx '

this expression for assessing the source density we have neglected the

inertia coefficient for longitudinal motion which, in general terms, can be

taken as an increase of the source density.

Furthermore we have assuihed that the approximate expression for the source denSity used in the case of an ,unrestricted fluid also can be used for a ship moving

in a shallow canal.

If p is the fluid pressure on the hull at the point. , k', (') the

resistance Ha1 is given approximately by:


11,4 p (h', 0, z h,dh'


p [(rot (If, 0, c

(h', 0, f)]


A' df,


taken over the plane k' = 0 and where we have made use of (2).

Sub-stituting (13) in (14) we find that the resistance for a ship moving in a shallow canal is given by the real part of:

p *Sb 1 1


C dh' 2.4 dh h h rn, n= -CC , dh hdh c p bki k d 0c'e c_9s4 EK doSh K di

x cosh [K(4,/)]

exp [K (i tun -- d1)] dK


Scb 'di:e di S hkdh dfc d T Sn d 0 CceCCISIlL /)1

h o coSIfIC di

x cosh [K (r11 I')] [I + tanh (K0]


x cos [ (gl( tanh K exp [K (i ,7 d1)] K. dK, (15)


where: 7.

r12'= (11' h)2 (nb1)2' (/' /)2,

rn,2 (h' h)2 (nk)2 -F. [2 d17(f' f)2]2,

= (h1 h) cos 0 rib,. sin 0,

tan, =


h c (-0.dT] COS 0 nb sin 0. T.

The coefficient to is an added mass for this particular problem under the particular assumptions made, taking account of the bed, the vertical canal walls and the free surface but assuming no wave formation.


We noW assimie the motion to start from rest with a velocity c

which is then maintained constant. Obviously the terms in (15) contain-ing 'c all disappear. Carrying out the integration with respect to 4 we have that the wave resistance at any time t is given by:

== n2 0 0

-2 p gc

d 0 (U2 ± V )2 [1± 2 cos (Knb, sin0)1




1[Kc cos 0 (gK tanh K d1)1/.]t sin [Kc cos 0 (gK tanh K d01/9] t

L Kc cos 0 ± (gK ,tanh K

Kc cos 0 tanh K

K dK

x cosh2 (K d)'


U IV SS M- cosh [K (di ± z)] exp (iKx co 0) dx dz (17)

and where we have expanded the cosine, terms, dropped the suffixes,

and then substituted the,current coordinates x, z for h, f .

If we take the limiting value of (16) when t becomes infinitely large the motion reduces to the steady one and the wave resistance becomes, after we have carried out the integration with respect to K:


IT (U2 ±V) 1 ± 2 E cos (Knb, sin 0)



K sed 0 d 0, (18)

00 cosh2(K di) (1 -- Ko sec20 sech2 K di)



where U, V are given by (17), K is the real positive root of: K K, sec20 tanh K d, = 0;

K, = -ci and where 00 is given by cos-1 VK, d for K, d1 < 1 and zero

for K, d, >1. It is well known that we cannot have waves whose

speed of advance exceeds Vgck.

If the ship is -running at a higher

speed than this critical all the wavelets whose angle of direction is less

than 0, disappear, 0, being therefore the lower limit of the integral in (18). Carrying out the integration with _respect to 0 we find that (18) may be written: P

+ -2

TC b,. 4 P g[u where:

-F. Vo2 + 2 (Up2 +Yr12.)

cc . n=1 Uz, V, ==Nn SS cosh ['1<.' (d, z)] X ekp [ix (K, K' r, dx dz,



( 2n7921

d c-nsh2IZ'

n 1 41

K d

K'n.being the root of:


Ko tanh (K' di) = n re


nnyn 1)1

and Ko = It will be seen that when K, d, <1 the root corresponding to n 0 is zero, this in turn leads to U0 = Vo = 0 for K, d1 < 1.

We now consider the particular case when the depth d, of the canal is infinitely large., From (15) we have that the resistance for accelerated motion in a deep canal is given by the real part of:

c P h dh' df' cS

h rn, rnz

k v (1


, k I 'n 'cc

-6 hidh hdh cif 0cdT d0

cos [(1


exp K [i

(I + I)] K dK,


(n = 0, 1, 2, ...)



2p gc


If the motion in the deep canal started from rest with a velocity c which is then maintained constant the wave regiStance at any time t'

becomes from (20):

0 (I° ± J2) [1

± 2 E os (Knb, sin


9 . n=1

cc cc_


[sin [Kc cos 0_ + (gK)11.] tj ' siii #[cox_cs (1 _- (gIcri.] 1(1


L Kc cos 0 ± (gKri.


- Ke coS 0 (gKr/.

j K

c1K, (21)



I + iJ =

[K (ix cos 0 z)] dx dz. (22)

-This result is also" obvious from (1p) and (-17),

Taking the limiting value when t becomes infinitely large we get

from (21) the wave resistance for steady motion in a deep canal:

a result which may also be readily obtained from (18).

Carrying out the integration with respect to 0 we find that (23)

becomes: , p g


4b, [

2 n =1(1-;2+jrz2)1, (25) 4 p gKo , ,2 2


secs 0 d 0; 0)] dx dz, ' (23) (24)





I' + iJ


) 0

2 E cos (Ko nb1, sin 0 sec20) n=1

exp [K° sec 0 (ix + z sec x

r.12 = (h' h)2 + (nb1)2

rz2 = (11; (nb1)2 + + f)2,


h +

c (7) d






and: Mn2 1 + (b2inK72 2 13 1 e p [Kr, z ix (K0 &)1.] dx dz, b x







(n = 0, 1, 2, ...),

which is in agreement with the limit (19) tends to when the depth becoMes infinitely large:

We consider next the particular case when the width b, of the canal becomes infinitely large whilst its depth d, is finite, that is, shallow fluid.

By making use of Riemann-Lebesgue theorem it will be seen that all

the terms in (15), (16), (18) for which n

± 1, ± 2, ± 3, ... vanish and

only those terms for which n = 0 remain. Thus the resistance in shallow fluid is given by (15), (16), (18) for ihe particular cases mentioned in the -text where in these expressions all the terms containing finite values of II


Lastly we consider a deep unrestricted fluid. Taking the limiting value when both b, and d, becoine infinitely large we have from (15) that

the resistance for accelerated motion is given by the sum of the terms

in (20) for which n 0, a result which is readily obtained from (20) itself.

Assuming now that the motion in a deep unrestricted fluid starts from rest with a velocity c which is then maintained constant, we have sinailarly from (16) that the Wave resistance is given by (21) in which the sum of the cOsinus terms is zero. This is in agreement with the

limiting value of (21) when b, becomes infinitely large.

From (1.8) we have that the wave resistance for steady mOtion,

in an unrestricted fluid is given by (23) where again the sum of the

cosihus terms is zero. This result is also obvious from (23) by taking the limiting value.


The linearized problem orresistance of a ship in accelerated motion in a canal is formulated and Solved. From this solution expressions for tesista.nde in particular cases are deduced.




HAVELOCK, T. H. Some Cases of Wave Motion due to a Submerged

Obstacle. Proc. Roy. Soc. London, A Vol. 93, pp. 520-532, 1917.

HAVELOCK, T. H. The Wave Resistanceof a Cylinder Starting from

Rest. Quart. Journ. Mech. and Applied Math., Vol. II, pp. 325-334, 1949.

HAVELOCK, T. H. The Resistance of a Submerged Cylinder in

Accelerated Motion. Quart. Journ. Mech. and Applied Math., Vol. II, pp. 419-427, 1949.

HAVELOCK, T. H. Discussion, pp. 574-575, in the Bulletin de l'

Asso-ciation Technique Maritime et Aeronautique, Paris, Vol. 48, 1949.

HAVELOCK, T. H. Wave Resistance Theory and its Application to

Ship Problems. Trans. Soc. Naval Arch. and Marine Eng., U.S.A., Vol. 59, pp. 13-24, 1951.

SRETENSKY, L. Theoretical Investigationof Wave-making Resistance._

Joukowsky Cent. Inst. Aero-Hydrodynamics, Moscow, Rep. No. 319'



J. K.

On the Linearized Theory of Wave Resistance for

Displacement Ships in Steady and Accelerated Motion. Trans. Soc. Naval Arch. and Marine Eng., U.S.A., Vol. 59, pp. 25-85, 1951.

MICHELL, J. H. The Wave-Resistance of a Ship. Phil. Mag. (London),

Vol. 45, pp. 106-123, 1898.

HAVELOCK, T. H. The Theory of Wave Resistance. Proc. Roy. Soc.


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