The linearized theory of wave resistance and its application to ship shaped bodies in motion on the surface of a deep, previously undisturbed fluid

The Linearized Theory of Wave Resistance

and Its Application to Ship-Shaped Bodies

in Motion

on

the Surface of

a

Deep,

Previously Undisturbed Fluid

by

J. K. Lunde

Skipsmodelltanken

Norges Tekniske Hogskole Trondheim

Skipsmodelltanken Medellelse Nr. 23

Mars 1953

English translation published in July 1957, by

The Society of Naval Architects and Marine Engineers

74 Trinity Place, New York 6, N. Y.

SHIP-SHAPED BODIES IN MOTION ON THE SURFACE OF A DEEP, PREVIOUSLY UNDISTURBED FLUID

1. Introduction The mathematical-hydrodynamical

considera-tions on the wave resistance of ship-shaped bodies present, in this writer's opinion, the most interesting problems met in theoretical Naval Architecture. The first experimental towing tanks were designed primarily for research on

this aspect of ship resistance. A tremendous

amount of experimental work has been done, the data on which are available in the transactions of the many engineering institutions. The goal, to express the wave resistance of a ship as a function of its shape, has nevertheless not yet been attained by the experimental method. On

the other hand it cannot be denied that system-atic model testing is the most important method of approach when considering improvements in the resistance characteristics of a ship.

It is not the author's intention to treat in de-tail the mathematical-hydrodynamical theory of the wave resistance experienced by a body

mov-ing into still water. This has recently been

done

ft',

however, and anyone interested

should refer to this publication. We shall rather first limit ourselves to discussing some of the physical considerations which have S led to the development of the linear mathematical theory. It will be necessary in the course of this dis-cussion to give some of the mathematical ex-pressions, but without developing them. Certain conclusions to be drawn from these expressions will then be discussed. Theoretical computations will thereafter be compared with the correspond-ing experimental results. In this the writer will not make use of all the available publications but will quote _results which he believes to be typical.-, As it is not the author's intention to

write a paper on the form of merchant ships, only some of the more _obvious conclusions will be mentioned in connection with the usual type of ship". It should be noted, however, that cer-tain conclusions must be somewhat modified

iSee the list of references at the end of the paper.

3

when

the ship is

propellor-driven.

This is

particularly true for single screw ships. The in-fluence of the boundary layer on the wave resist- )

ance will finally be considered, with a discus-sion of some results on the calculated resistance

to accelerated motion.

Although. it has not been the author's intention

to discuss the wave profile arising from the

motion of a ship, it could not be avoided at

cer-tain points in the following text. A general

discussion of the wave profile, however, has not been the author's primary aim in this paper. In a classical gaper over 50 years ago, Michell presented an approximate hydrodynamical solu-tion for a slender ship form in rectilinear, steady motion on the surface of an ideal2 fluid [2]. For.

many years this paper was unfortunately over-looked and forgotten. From 1919 on, however, Havelock reconsidered the whole problem after having published many papers- on wave profile and wave resistance for a pressure point, sub-merged cylinder, sphere etc. Some years later Wigley began his work on a comparison between measured and computed wave resistance for

ship-shaped bodies. This led to a number of valuable publications. Weinblum also tackled the problem and has contributed a series of in-teresting papers since his first publication in

1929. Hogner's publications, in which he treats the theory itself for both wave profile and wave resistance, should also be mentioned. Somewhat later Sretensky began his work on the theory, and still later others, e.g., Guilloton. It also appears that the Japanese have worked in this field, particularly during the last War. Most of these studies have not been published, or at least they are not available outside Japan. It is essentially, however, the work of the hydrody-namic scientists mentioned here, and particularly

2An ideal fluid is here understood to be a fluid without

internal friction. Such a fluid cannot, e.g., transmit

Havelock's, which has led to the comparatively clear understanding of the properties and defi-ciencies of this theory which we have today. It is not to be denied that the theory is based on

2. The Velo

In the following we shall limit ourselves to

motion in deep water, as the same problem in

shallow water is much more complicated [1], [2], [3], r4], [5], [6], [7], [8], A solution of the shal-low water problem, however, must be considered the more general, as it is possible to deduce

from this the corresponding solution for deep water or a canal where the water may be either shallow or deep [1], [5], [9], [10]. We shall also only consider a fluid which is inviscid and in-compressible. A body moving through such a fluid will then only experience resistance through the formation of waves. In Section 12, however, the effect of the boundary layer will be discussed.

Since the motion of a body in a fluid must begin at a certain point and at a certain time it is natural to consider this problem first, although it is not the easiest [1], [5], [11], [12], [13].

A slight reflection on the problem we are try-ing to solve theoretically will convince us it is not simple, but that it does present many in-teresting, mathematical difficulties. We are asked to find a potential function which satisfies, among others, the conditions (a) on the wetted surface of the ship, and (b) on the free surface of the fluid on which the constant atmospheric pressure acts.

Let us consider (a) first. Instead of conceiv-ingthat the ship moves rectilinearly into still water, we assume it to be stationary with the water flowing past it. This does not alter the

problem. The condition to be satisfied on the

wetted surface of the ship is that the normal component of the absolute velocity of the fluid must be zero everywhere on this surface, i. e.,

the velocity of the fluid is tangential to the

wetted surface. To satisfy (a) it is therefore necessary to know the shape of the wetted sur-face. Yet this varies with the speed because both sinkage and trim, e.g., are functions of the speed. Condition (a) is thereby considerably complicated. It is thus not quite correct to use the shape of the wetted surface when the ship and fluid are at rest to satisfy condition (a) for the speeds which are of interest, although this is usually done in practice when computing the wave resistance.

Let us now consider (b). This condition is given by a non-linear, partial differential equa-tion which has to be satisfied at the free surface

many assumptions. Theoretical and experimental work must necessarily be tombined to supple-ment each other, testing in practice the influence of these approximations as far as this is possible. city Potential

whose shape is not known but must be deter-mined as part of the solution. Because of this it is necessary, at least for the present, to line-arize the problem. As the partial differential equation (2.14), however, for which we are re-quired to find a particular solution, is linear, it

is the condition (b) at the free surface of the

fluid which has to be linearized. The problem may now be solved, at least approximately.

The complete velocity potential Oswill be built up from the velocity potential due to sources and sinks': A source is a point at which fluid is created and from which it flows radially out-ward into space. A sink is a point at which the

fluid vanishes, i.e., a negative source. The

stream lines for a source and a sink at rest in a fluid are shown on Figs. 1 and 2, respectively.

If the strength of a source is designated m, the

rate at which fluid volume streams from the

Streamlines Streamlines

FIG. 1. Source FIG. 2. Sink

3Symbols used in this paper are defined in the Nomen-clature on page 62 et seq.

'The velocity potential 'D is a scalar function which

describes the motion of the fluid. (Scalar functions are

exclusively determined by stating their magnitude, i.e., with real numbers. Mass, volume, temperature, density et al. are examples of scalar quantities. When a function has sense (direction in space) and location (line of action) in addition to magnitude, it is a vector quantity. Velocity, acceleration,

force et al. are examples of vector quantities.) The vele&

ty components of the fluid, u, v and w, at point P are there-fore given by = w =-1)z in the x,, 37,- and

z-directions, respectively, where loz =a/äx et al., the

coordinates of P and the time at which the components are to be measured, being inserted in 4D. By using the minus

sign indicated, p becomes the impulse which instantly

sets up the fluid motion from a condition of rest. Here p is

the density of the fluid. For a potential motion ,D must

satisfy the so-called equation of continuity, which for an

incompressible fluid may be written l'zz + 1Dzz = 0

where lbxx =aq/ax2 et_al. This equation may be

inter-preted to express that matter may neither be created nor

Streamlines

FIG. 3. A source in a unifrom stream.

source is here made equal to 4irm. Similarly.

the strength of a sink is designated -m. The

velocity potential at point (x,y,z) due to a source at the origin is m/r, and for a sink -m/r, where

r2 x 2 + y2 + z2. _{If the source is now placed in}

a uniform stream, the stream lines of the source itself, will be deflected as shown on Fig. 3. If

a sink is located behind the source on Fig. 3, and these are of equal strengths, +m and

m, all

the water streaming from the source is again

absorbed and the stream lines are as indicated on Fig. 4. We note on this figure that there is one completely closed surface of stream lines. As already mentioned, the condition must be satisfied on the surface of a body at rest in a uniform stream that all fluid motion be tangential to the surface of the body, i.e., with no compo-nent of the velocity normal to the surface. This same, condition the stream lines must satisfy, to wit, that all motion be tangential to them. We may therefore conceive of an arbitrarily chosen surface of stream lines as replacing the outer

contour of a body in an inviscid fluid. The

Sink distribution

FIGURE 5.

- _

FIGURE 4.

closed surface of stream lines in Fig. 4 could thus be the outer contour of a symmetrical body. By using several sources and sinks and/or by varying their strengths, it is possible to vary the closed stream line contour, i.e., we can bring out a number of bodies of different shapes, all located in a uniform stream. To illustrate this

Fig. 5 shows the stream lines about _{a body of}

rotation created by using a continuous distribu-tion of sources in the "entrance" and a

corre-sponding distribution of sinks

in the "run".

That the surface of the stream lines shall be closed, the sum of the strengths of the sources and sinks must always be equal to zero. Without this condition the water streaming from thesources would not be absorbed again by the sinks, if these are the weaker.

An obvious next step is to assume that a ship may be replaced by a distribution of sources and sinks located on the wetted surface of the ship,

Source distribution

/.

7

or, for a slender ship, on its vertical centreline plane. Since the problem itself has been line-arized, it is thus only necessary to first find the velocity potential of a source in rectilinear motion under the surface of the fluid. The velocity po-tential of the ship as a whole is obtained by integrating or summing up, over the wetted sur-face

of the ship, the potential for a source.

Such a summation or integration over a closed surface in space will, when the strength of the sources is assumed to be a function of, e.g., the slope of the surface at every point, automatically lead to a total strength equal to zero of the source distribution. A closed surface of stream lines coinciding with the wetted surface of the ship is thus formed. In an ideal fluid, on the other hand, there will never be the component of the wake corresponding to that due the viscosity in a real fluid. A source distribution such as implied, however, will approximate the two components of the wake which are due to potential flow and wave formation, respectively. This approximation is improved upon when we also consider the effect of the boundary layer on these two components of the wake.

We will use a rectangular coordinate system with the origin in the free surface of the fluid, the x-axis positive in the direction of motion,

the z-axis positive vertically upward and the

y-axis normal to these in a righthand coordinate system.

Assume that a source of strength m is suddenly formed at the point (0,0,f) in the fluid at time t = 0 and is active during the time interval an This is an impulse, but no impulse forces act on the free surface of the fluid, only the constant atmospheric pressure. The velocity potential during the short interval of time Sr at the

arbi-FIGURE 6.

trarily selected point P must consequently be

given by m (2.10) r, r, where x2 + y 2 + z + 02 and r22= x2 + + (z f)2 (see Fig. 6)5.

This impulse leads to an immediate elevation of the surface, which, as the source vanishes, spreads as unrestrained ring-waves over the free surface of the fluid under the action of gravity. This is similar to a stone falling vertically dowr on the surface of a fluid at rest'.

At the arbitrarily selected time t the velocity potential of these ring-waves will be given by [1], [12] 77 00 - /2 (2.11) Br

Lao

f

sin [(gK)1/2t] x o

x cos [K(x cos 0 + y sin Mel' f)K1/2dK

It may be shown that (2.11) represents ring-shaped waves which spread over the surface of the fluid, gradually decreasing in amplitude and increasing in length.

Before we proceed further let us discuss the classical problem of a pressure point in recti-linear motion over the surface of a fluid along the x-axis. Instead of conceiving of this as a continuous process we may assume that the mo-tion of the fluid has come about by impulses act-ing on the surface of the fluid at points xi, x

x3, ... at the times t ta,

t3, ..., where x, <

<

< .. and t, < t, < t, <.... Each of these

impulses causes a series of ring-waves. Due to mutual interference they form a resulting wave

sFrom the footnote on page 4 we have that may be

interpreted as the impulse force which brings about the

fluid motion from a position of rest. We have meanwhile

contended that no impulse force pc2 acts on the free surface

of the fluid given by z = 0 during the time interval Er the

source is active. In the interval dr, for z =0, consequently,

We see that this condition is satisfied by (2.10)

when z = 0. The second term in (2.10), furthermore, is a

sink above the free surface of the fluid, i.e., the fluid

according to this term extends above and beyond that which

is to be the free surface. This presents no difficulties, as

Cauchy and Poisson discovered that every plane consisting

of fluid particles and which is horizontal when the fluid is at rest, satisfies the condition for the free surface. Every

surface of stream lines in mathematical hydrodynamics may therefore be presumed to represent the free surface. This is analogous with the above, where we replaced the surface of a body with a surface of stream lines.

°In many respects the motion occurring when a source is suddenly formed in an infinite fluid and is active for a short period of time, may be compared to the motion caused by a gas bubble suddenly expanding to a certain size.

system consisting of transverse and diverging components. These wave components have much in common with those we see astern a ship mov-ing on the surface of deep water. If more and more impulses now act, at the same time that the distance and the time interval between two suc-ceeding impulses diminishes, we obtain the same fluid motion occurring when one individual pres-sure impulse moves rectilinearly over the surface of the fluid. A specific velocity potential cor-responds to each impulse and hence also a spe-cific fluid motion. The velocity potential of the continuous motion at time t is then given by the sum of the velocity potentials of the individual impulses which acted before the time t. The same is true for fluid motion [146].

This method, which was used by Kelvin, may

also be applied in the present case. We there-fore 'assume that the fluid motion set up when a

source moves rectilinearly under the surface, may be composed of an infinite number of sources acting instantaneously and successively in the direction of motion. The fluid motion caused by the motion of our source is then obtained by summing up all the impulses before the time t at which we wish to consider the motion. This argu-ment may be expressed mathematically, and we find that the velocity potential due the source is given by, as a first approximation, [1], [5], [12]

t IT m(t) m(t) g112 (2.12) cb =

-

+

f

m(r)dr

f

dO 71 72 97 0 77" cos (Kji,)eK('f)Kil2dK where 7), x + c(r)dr cos 0 + y sin 0 1.12 x2 + y2 + (z f)2, r22 = + + (z - f)2

and where x, y, z are measured relative to an

origin which lies in the free surface of the fluid above the source and which follows the source in its motion. In (2.12) c(t) is the rectilinear ve-locity of the source, assumed here to be a

func-tion of the time t. We may therefore say that

(2.12) is a general equation since we have only assumed that the motion began at time t = 0 with-out establishing the velocity c or the accelera-tion of the moaccelera-tion. In this equaaccelera-tion the last two

sin [/FT(t - r)]

7

terms are the mirror image of the source, in the free surface of the fluid.

It is quite easy to show that (2.12) satisfies the conditions of the free surface of the fluid which for accelerated motion is given by

(2.13) ciSt, +gy5. + c201, + 2cc1t- = 0

also satisfies the equation of continuity for an for z = 0. The velocity potential given by (2.12) incompressible fluid, to wit,

(2.14) _{Ckxx cky}

C6Z = °

In both of these linear, homogeneous, partial dif-ferential equations ck = &Wax and ck_{.x X X} 020/8x2, for example, while c = dc/dt.

Condition (2.13) is found by assuming that the atmospheric pressure which acts on the free sur-face of the fluid is constant, and by assuming that the square of the absolute fluid velocity may be disregarded compared with the other terms which occur.

Steady motion signifies that the motion is in-dependent of time. In the present problem we have such a motion only after the ship has been moving with constant velocity a-very long time. If we conceive of the motion of the source having started with the velocity c, and that this there-after is its constant velocity, we should be able

to _{find the velocity potential for rectilinear,}

steady motion from (2.12) by taking 'c constant, integrating and letting t We then obtain

77/2 4mK 171 911 0 sec20d0 x (2.15) = T2 77 0 x

fK

Ko sec' 0

cos (Ky sin 0) eK('DdK -cos (Kx -cos 0)

0

"al 2

- timKo

fsin

(Ko x sec 0) X

0

x cos (Koy see() gin 6) elco(z-n8"28sec1t9d0 where r, and r, have the same significance as

previously and where Ko = g/c2.

This expression agrees with that which we ob-tain by assuming steady motion from the start and solving the problem directly [1], [6]. From

(2.15) we see that the first two terms in the ve-locity potential for a source in steady, rectilinear motion under the surface of the fluid may be in-terpreted as the velocity potential of the source in an infinite, uniform stream and the velocity

potential of an equally strong sink in an infinite, uniform stream. The source and the sink repre-sented by the first two terms in (2.15) are lo-cated symmetrically with respect to the x-axis. The last two terms in the equation are the mirror image of the source, in the free surface of the fluid.

As given by (2.15) the velocity potential sat-isfies the condition at the free surface for steady motion, to wit,

(2.16)

Orx5z' °

for z = 0 and the equation of continuity (2.14). The surface condition expressed by the linear, homogeneous differential equation (2.16), and which may be derived directly, may also be found from (2.13), as all the terms which are derived with regard to time are equal to zero for steady motion.' Terms we disregarded in (2.16) cor-respond to those we neglected in (2d3). In con-nection with these terms Wigley has noted [6] that the square of the ratio of the absolute fluid velocity (which is then due the wave motion) to the ship's velocity will, for ships with a length. beam ratio of 8, vary from about 1/100 at high velocities to about 1/40 at low velocities. This should suggest the magnitude of the terms we have disregarded.

Assume that a is the surface density of the source strength at point (h,k, f) on the wetted

surface S of the ship. The velocity potential

due the ship may now be expressed as (2.17) (1:0= fqS(a,xh,yk, z+ f)dS

where irk is given by (2.12) for accelerated motion and by (2.15) for steady motion. In both of these expressions m must be replaced with a, x with x h and y with y k.

There are probably those who have questioned why a train of waves astern a ship, which began to move at a given instant, becomes longer and longer although the waves appear to follow the

ship with the same velocity. This constant

lengthening of the train of waves is due the group velocity.8 Let us conceive that the fluid surface, which is at rest, is everywhere hori-zontal except over a very limited area where it assumes a wave-shaped character (see A on Fig. 7). We shall assume that these waves are

en-tirely straight, and quite long in the direction normal to the wave length. Such a surface

pro-2If x =Lx', z = and 0=Lc4,' are inserted in (2.16),

tiC,V-i-F-acb'zi= 0 is obtained. This is now the surface

condition. From this expression we see immediately that Froude's number F = c110 appearss,. as a deciding

param-eter in the problem for steady motion L180].

file may, at least approximately, be formed by taking a long plank with the necessary number of grooves and lowering it onto the surface of the fluid and introducing a vacuum in the grooves. If the surface is now set free by the sudden re-moval of the plank at time t= 0, the original dis-turbance, hitherto contained at rest, begins sud-denly to divide into two wave groups moving in opposite directions from the axis of symmetry. At B and C on Fig. 7 the surface profiles at time t = 41Fr and t = 80-7, respectively, are shown. Let us examine point F, at A, B, and C and f at B and C. On A, F, is the visible, leading part of the wave profile moving to the right. On B and C, F, and f show the corresponding point at the times 41/71 and WI., respectively, if it moves to the right with a velocity equal to half the wave

'It is to be noted that when an isolated group of waves, all of about the same length, move on deep water, the

ad-vancing velocity of the group as a whole is less than the ve-locity of the individual waves comprising the group. If we

trace a particular wave we note that this advances through the group, slowly dying out as it approaches the lead. Its

previous place in the group is then taken over in succession by other waves. These all move forward from the after part

of the group.

The simplest analysis of such a group may be based on a combination of two wave systems.

The profile t of a sine wave advancing in the positive

direction of the x-axis may be written as

C= a sin(Koz- nt)= a sin K0(x- ct)

where a is the amplitude of the wave, c = n/ K0 its velocity of propagation, and X= 27t/K0 the wave length. From the condition of the free surface we find that c2= g/Ko.

Conceive now that two sine waves with the same ampli-tude, but with different velocities of propagation (and hence

different wave lengths), advance in the same direction and

are separately given by

a sin(Kox - nt), C2 = a sin[(Ko SK0)x - (n + Sii)t]

The resulting wave profile is given as the sum of ti and

=e=4L-Ft2 2acos[-1 (x8Ko-t8n] x

i.e.,

2

x sin[Kox- nt)++(x81(.0-- tSri] .

If 8K0 and Sn are small quantities, 1(x81(0- tSri) will he a small quantity compared to (Koz-nt); and we may quite ap-proximately set

A sin(Kox- nt) 1

where A = 2a cos -2-(x81(0- an). The resulting wave profile is an advancing sine wave whose amplitude is not constant but varies slowly with both x and t, since SKo and Sn are as-sumed to be small quantities. For constant t, consequently,

but for various values of x, there will be long stretches

where the amplitude A is almost equal to zero or to 2a. On the fluid surface we note a series of groups which are

sepa-rated by virtually still water. The individual waves have a

piopagation velocity equal to c = n/Ko, while the maximum

value of A, and hence the group centre, has a propagation

velocity Sn/SK0, or, when the limiting value is taken,

dn/dKo= c/2, where we have made use of the expression

c= n/Ko and c2' g/Ko. For deep water the group velocity

velocity (i.e., the group velocity for deep water) or equal to the wave velocity. We note that there is a visible wave motion far ahead of F, on both B and C, but only a small, although visible, wave motion ahead of f. The visible wave front thus moves away from the axis of symmetry with a ye--locity greater than the wave veye--locity. Gradually this velocity increases. Observe now the visi-ble, tracing part of the wave profile designated F, on A, Fig. 7 for those waves which move to the right. (It is obviously not correct to assume that the steady wave profile shown on A divides on the axis of symmetry when the fluid surface is released, and that the part to the right of this axis moves to the right and the other part to the left) The location of F, after a time lapse 131,/zr is shown on C, if it has moved to the right with

Assume that the motion is steady and that we have two infinitely large planes, A and B, fixed in space normal to the direction of the ship, A afore and B astern. Let us consider the increase in energy of that part of the fluid which is con-tained between these planes, the free siirface and the wetted surface of the ship. Assume now that E(B) is the total energy, both kinetic and potential, flowing into this space per unit of time through B. Let E(A), in a similar manner, be the total energy flowing out per unit of time through

A. If p is the fluid pressure and u the horizontal

FIGURE 7.

3. The Wave Resistance and Source Distribution

a velocity equal to half the wave velocity, i.e., the group velocity for deep water. We note that F, on C coincides with visible, tracing part of the wave profile [80], [157]. It is thus apparent that the tracing part of the wave profile moves forward with a velocity equal to the group ve-locity. A similar phenomenon occurs, for ex-ample, when a stone is dropped onto a water sur-face

or when a wave generator in a tank is

actuated. For a ship beginning its motion at a given instant we obtain, in a corresponding man-ner, a train of waves where the leading part ad-vances with the same velocity as the ship and the tracing part with a velocity equal to the group velocity. As a result the train of waves stretches out astern the ship more and more as time goes on [115], [149], [161].

velocity of the fluid at an arbitrary point on B, the work which the fluid to the left of B does on

the fluid to the right of B is given by pu per unit time and unit area. Summing up pu for the entire plane B, we may say that the work done at B per unit of time, on the fluid limited by our planes, is W(B). Similarly W(A) is the work done per unit of time at A. If R. is now the wave resistance of the ship, and c its constant,. rectilinear ve-locity, the ship does an amount of work on the fluid given by R.c per unit of time. But the to-tal work done per unit of time on the fluid under - LINEARIZED THEORY OF WAVE RESISTANCE

consideration must equal its increase in total energy per unit of time, and we have [1], [8], [14] (3.10) Rwc +W(B) -W(A) = E(B)E(A)

Plane A is afore the ship and plane B astern, but are otherwise arbitrarily located. We may therefore assume that A is located infinitely far ahead of the ship. Here the fluid is undisturbed by the motion of the ship, and E(A) and W(A) will approach zero. Infinitely far behind the ship the fluid motion goes more and more over to free, un-restrained waves. A long time back these were formed by the ship, but they now advance ex-clusively under the action of gravity. If we there-fore conceive of B being located infinitely far behind the ship and A infinitely far ahead of it, (3.10) gives us

(3.11) = - (E - W )1

where E and W are computed from the wave mo-tion or the velocity potential infinitely far behind the ship, where there are free waves [1], [4], [8], [131.'

The quantities E and W are now computed from (2.15) and (2.17) remembering that we may disre-gard all terms which approach zero when x ---) From.(3.11) we thus find that the wave resistance Rw for a ship in steady, rectilinear motion is given by 1,(2 (3.12) Rw = 167TPK02

J

(P2 + (Y) sec' 0d0 where K,, = g/c2 and

f cos

Q a sin {K,(x cos 0 + y sin 0) Beef)} x

(3.13) X SC° Z "Ca °as

This is the general equation for wave resist-ance when the ship is replaced by a continuous distribution of sources and sinks, and .when it moves with a constant velocity c on deep water. We have then assumed that the motion has been going on for a long time.

For numerical computations it is an advantage to set sec 0 = cosh u, such that (3.12) and (3.13)

'This method has long been known for the two-dimensional

problem. Kelvin, far example, used it for two-dimensional waves on running water (waves in a canal) [154], also [132]. [155], [156], [160], [161]. The corresponding, but considera-bly more complicated three-dimensional theory was developed by Havelock [4].

may now be written

00

(3.14) Rw =16spica f(P + 12) cosh' udu where

[KO (x cosh u + y sinh u cosh u) x

X.eKOZ COSI? uds

If we assume that a ship with a high speed-length ratio may be replaced by a limited number' of sources and sinks, the wave resistance for steady, rectilinear motion, in the same manner as indicated above, is expressed by

(3.16) R. =16u fpKs2 (I za +J) cosh' u du CO where COS_x ms sin

=I cos

I

sin (3.15) (3.17)

x iKo(hs cosh u + sinh u cosh u) x X -K_{e s}.COSh2 u

Here the arbitrarily selected source s, of strength m is located at the point (A5, k3, [1], [15],

[134,[17].

There are also other methods by which the ex-pressions for the Wave resistance may be found, but ultimately they all lead to the result given by (3.14) and (3.15). [11. I-21. [3], [4], [5], [6], [7], [8], [1]], [18], [19], [20], [21], [113], [117], [118], [120], [1431, [159].

If we conceive of the ship as relatively slen-der, we may assume as a first approximation that the source distribution coincides with the verti-cal centreline plane of the ship. In that event dS in (3.13) or (3.15) is to be replaced with dxdz. It may then be shown that the density of the

source strength, as a first approximation, is given by

c ay

a = -

F =

-(3.18)

277 X 2tr

where y = F (x,z) is the equation for the wetted surface of the ship [1], [6], [8], [18], [191, [201, [120], [143]. In reality Equation (3.18) applies only when uF and wFz are small quantities (u and w are the components of the absolute

ye-locity of the fluid in the x- and z-directions, re-spectively). For slender ships F and u will be small quantities. The product uPx is therefore a small quantity compared to the terms in(3.18) and it is justifiable to completely disregard this term. Along the bottom of most ships, however,

F. will ordinarily not be a small quantity (a

small F z is contingent on a large rise of the floor). We have nevertheless disregarded tuF in

(3.18). To ascertain what effect this approxima-tion had on the agreement between computed and "measured" wave resistance, Wigley and Lunde conducted theoretical and experimental studies on two models with relatively moderate rises of the floor and with Cz = 0.909 [22]. The greatest value of F2 for these models was equal to 10. These studies showed that the agreement was no poorer because of the flat bottom. Although only two models were tested it could reasonably be concluded from these studies that even if F. is a relatively large quantity along the bottom, of the more common merchant ship types, the verti-cal velocity of the fluid, to, is so much less that the product tvF may be disregarded [37], [117]. When (3.18) is used, y must be taken as zero in (3.13) and (3.15). By now setting X = sec 0 or = cosh u, this leads to Michell's formula for the wave resistance [2].

It may also be shown thet P and Q (or I and are functions of the free wave profile infinitely far astern the ship, and may consequently be found from this [1], [4], [8], [14].

If, on the other hand, we use only a finite num-ber of sources and sinks, the ship must be di-vided into "compartments" by means of hori-zontal and vertical planes longitudinal with the ship, and vertical planes transverse it. A source is then located in each such compartment. Let S, and S2 (if .52 > S and the compartment lies in the entrance, S, for ordinary ships will be nearer midship than Si) be the cross-section areas of the end walls in such a compartment whose vol-ume is V1_2 and whose length is x12. It may then be shown that the strength of the source

cated in this compartment is given by [1], [8], [15], [16]

(3.19) m = -- S1)

477

If S, > the expression for the strength is neg-ative, indicating that we have a sink instead of a source. We assume that the origin of our coordi-nate system lies amidship on the L.W.L. and that the axes have the same position in space as pre-viously. Consider a compartment in the entrance of the ship, assuming that X is the horizontal

LINEARIZED THEORY OF WAVE RESISTANCE

distance from the origin to S,. If (x + X, Tc, f) are

the coordinates of the source located in the "compartment" being considered,

it may be

shown that [1], [8], [15], [16]

V12

Tz

S2 S,

It may also be shown that Te,

Tare the same as

the coordinates for the centroid of the area 52 S,

in the y- and z-directions [1], [8], [16].

This method offers certain advantages com-pared to the first. The strength of the sources and sinks, and their location, is relatively easy to find even if the lines of the ship cannot be expressed by mathematical formulae. On the other hand the method is characterized by large approximations. The number of sources and sinks necessary, moreover, to achieve a relatively high accuracy compared with the first method, in-creases rapidly with decreasing Froude's num-bers. Computations then become more time-consuming, the smaller Froude's number. It should be possible, however, to tabulate once and for all the functions which occur, markedly reducing the labour with the numerical calcula-tions. At relatively large Froude's numbers only a limited number of sources and sinks are neces-sary to achieve high accuracy compared with the first method. The method is therefore quite ap-plicable to what we may call the destroyer range, the computation work for a limited number of sources and sinks not being unreasonably large, relatively speaking.

The method with a finite number of sources and sinks clearly shows that the parts of the ship's wetted surface lying deepest under the L.W.L. have the least effect on the wave resist-ance. From (3.17) we have that the I and J.-functions contain the factor eK f se o sh2 u where

f

is the depth of the source or sink under the

surface of the fluid. The greater the fs, the

smaller the functions Is and and the smaller the contribution to the total wave resistance of this source or sink. It has meanwhile been indi-cated that f corresponds to the vertical distance under L.W.I. of the centroid of the area given by the difference between the cross-sections (.32 S1) in the "compartment" under consideration. Deeper lying parts of the hull have consequently

the least effect on the wave resistance. This

should partly explain why Wigley and Lunde [22] found that a flat bottom did not markedly impair the agreement between theoretical and experi-mental results. We also note that the contribu-tion of the individual water lines to the wave

resistance, as a first rough approximation, will

decrease as e2, where f is the depth of the

water line under the surface of the fluid and c

the velocity of the ship. From this we could

easily draw the general conclusion that for the least possible wave resistance the displacement should be located as far as possible under the L.W.L., i.e., as much U-shaped sections as pos-sible should be used. This will ordinarily be in accord with experimental results, but in certain cases where V-shaped sections are shown to be desirable, is this due a favourable interference between the wave systems from the entrance and

the run of the ship. U-shaped sections would

partly nullify this favourable interference. In the pressure measurements made by Eggert anti others on models [23], [24], [25], [26], [Mu', we also find confirmation for the great significance on the wave resistance of the parts of the hull lying closest to the L.W.L. Already in the last century hydrodynamicists were aware of this. Kelvin, for example, concluded his lecture on "Ship Waves" (1887) by proposing that the dis-placement should be located as far as possible below the L.W.L. for the least possible resist-ance, as the wave motion was primarily a phe-nomenon of the free surface. He probably pre-sumed that thus shifting the displacement would not lead to an increase in the other components of resistance. In this connection it is of interest to recall R. E. Froude's rule: U-shaped sections in the entrance and V-shaped sections in the run, the latter to reduce the "separation resist-ance."" [134].

Weinblum contends that a rule-of-thumb method is useful when shifting the displacement verti-cally [28]. If, for example, a certain part of the displacement is shifted from the region near the water line a distance d, vertically down, the ef-fectiveness of such a shift will increase by the ratio

d, d,

A. L

where A. = (2rt/ g)c2 is the wave length of the free, deep water waves corresponding to the velocity of the ship c and where F is Fronde's number. Effectiveness thus decreases with increasing Froude's number.

thThe relationship between the computed Wave resist-ance, the Lid-ratio and F for a full ship with rectangular cross-section is shown in [180]. Similar curves are also

found in [59].

IlTlie various resistance components are defined in Sec-tion 5.

It can be mentioned as a curiosity that Have-lock [29] computed the wave resistance R. for a model whose surface was given by

y = b

--) (1

_d2

Results of these computations are shown as curve A on Fig. 8, on the basis of Froude's number F. He then computed the wave resistance for the same model with the bottom up, such that the keel lay in the surface of the fluid_ when the model was at rest. The model surface was now given by

y = b

x2)( z

2 d

Curve B on the same figure shows the result of this computation. We shall later come back to the pronounced crests and troughs on the curves for computed resistance, but we note that there is a decided reduction in the wave resistance from moderate to high Froude's numbers when the model moves with the keel up. At yet higher Fronde's numbers than those shown on Fig. 8, however, the two curves will finally coincide.

42 t/3.9c9 O./ dai 0.2

as

F cb/417" FIGURE 8.

We also note from the figure that, practically speaking, the crests and troughs for the two re-sistance curves occur at the same Froude's num-ber. In many respects this concurs with both theoretical and practical observations. A reduc-tion in draught for one and the same model does not essentially change the location of the crests and troughs on the resistance curve, but the ab-solute value of the resistance will change and the interference will be reduced with decreasing draught [28], [59], [89].

From (3.12), (3.13) and (3.18) we also note that the wave resistance, when the velocity in-creases, finally begins to decrease again after having reached an absolute maximum. We see

that when K.= g/c2, eK°z 1 when c

Although a varies with c, the ff expression must be multiplied by Ic2 (g/c2) resulting in the factor 1/c3 before the integral in (3.12). This factor causes the wave resistance to approach zero when c [2], [167]. This may naturally be proven in a more satisfactory mathematical manner, but it is only of academic interest. Other phenomena which we have not considered occur with large, but finite, velocities, but we then no longer deal with displacement ships.

It should be mentioned here that if (3.18) is used for the density of the source strength for a continuous distribution along the vertical centre-line of the ship, or if (3.19) is used for the source strength when a finite number of sources and sinks are used, we have in reality neglected the inertia coefficient C for the fluid mass carried along with motion in the longitudinal direction

of the ship." The effect of the inertia coefficient may ordinarily be interpreted as an increase in the source strength above that found by means of (3.18) or (3.19). We arrive at this conclusion if we consider an ellipsoid or a spheroid in steady, rectilinear motion (along its longitudinal axis) in an inviscicl, infinite fluid. In such cases the ve-locity potential may be found without any

ap-proximations and may be considered as a con-tinuous distribution of sources and sinks. The kinetic energy of the fluid mass carried along may also be computed from these exact velocity potentials. The inertia coefficient is thereby also obtained. If, on the other hand, we use (3.18) for the approximate density of the source strength and compute the kinetic energy from the approximate velocity potential, we find that the inertia coefficient is now equal to zero [1], [15].

Actually we should multiply, for example,

(3.18) by (1 + k), where the coefficient k varies from point to point on the ship, but which for bodies such as ellipsoids and spheroids is con-stant and equal to the inertia coefficient C. To show the magnitude of C for varying slenderness ratios L/B, some values for a spheroid will be given. If L/B = 1.0, 4.99, 9.97; C is 0.5, 0.059 and 0.021, respectively. We have fewer and partly less accurate data available for ships, but

12For a completely submerged body we may express the

fluid mass carried along as CpV where p is the density of the inviscid fluid,' the volume of the body, and where C

is here defined as the inertia coefficient of the fluid or,

simply, the inertia coefficient. If we conceive of the beam of the ship as approaching zero, C also approaches zero.

Froude investigated this coefficient in his fa-mous experiments with H.M.S. "Greyhound" [15], [153]. For retarding motion he found that C was about 0.20 for the loaded ship and about 0.16 for the ship in ballast. For the loaded ship in ac-celerated motion it was about 0.07. Because of experimental difficulties these values are rather uncertain, and actually apply only to this ship in shallow water [172]. On the other hand, for his computations, Lackenby [90] used C 0.08, de-rived by Tupper from experiments conducted by von den Steinen [152]. Whatever now the value of C for motion in the longitudinal direction of the ship, in the following we shall disregard this coefficient for steady motion as there is reason

to believe that the smallest values indicated

above are the most correct. With increasing

length/beam-ratio, furthermore, C decreases. We should meanwhile note this correction for later studies and evaluations of the theory of wave resistance."

If we wish to use (2.12) and (2.17) to find the resistance for accelerated motion, it may ap-parently only be done by computing the resultant pressure which acts horizontally on the ship. For a slender ship the horizontal component of the pressure acting on the element of surface dh'df will be approximately given by

ak'

-p(h', 0, ,r)

diidr

where k' = F (h', 11 is the equation for the wetted

surface of the ship. The resistance,

conse-quently, is given by ale (3.20) R

-2ffp(h',0,-r)

= ah'

pd_

ffalc"

dh'd 77- (-3-T,

f

ff

i[w

-

(f. ak h[(h'-h)2+(f+

,02]-2i=d df+

dh + Pg

ff

alc' dh'd x

ae

ak f5-fidhdf c(r)dr

f

cla x

xll

0 -Tr

13Another method for determining the density of the source

strength by means of successive approximations is indicated

in [181]. This method, utilizing Taylor's hydrodynamic

method for the design of water lines [182], gives (3.18) as the first approximation.

Do

fcos W(4 -r)] cos

(Krode-K(C41)KdK

fc(r)dr cos 0, [11, [12].

0

where iTh=

h'.-(3.22)

In (3.20) we note that the resistance of the in-viscis fluid to the accelerated motion of the ship

consists of two terms, to wit, the inertia resist-ance of the fluid and the wave resistresist-ance, the first of these terms corresponding to the one in (3.20) containing C. In the form that the coeffi-cient for 6 occurs in (3.20) it may be interpreted as the fluid mass carried along for a slender, ship-shaped body (virtual mass). To the degree of accuracy with which we operate, this coeffi-cient takes into account the free surface, but with the limitation that no waves are formed [1], [12], [131, [1091. From the form of this term we see that, strictly speaking, it applies only to high velocities, a constant fluid mass is then be-ing carried along. In (3.20) the inertia term is consequently only a first approximation. If the surface conditions of the ship are satisfied with greater accuracy, we find that the mass carried along varies somewhat with the velocity [13], (see also Section 13).

We note further that (3.20) gives the resistance of the inviscid fluid to the accelerated motion of the ship, and not the resultant force which is necessary during accelerated motion. The ship's own mass does not then appear in (3.20).

We may also assume that the ship started its motion with velocity c and that this thereafter is its constant, rectilinear velocity. From (3.20) we now obtain

t 2

(3.21) R

4gpc f dr

r dO (P2 (r) X

172

0 0 0

x cos (Kc(t-r) cos 0] cosRIXt-r)]KdK

where

Kx cos 0}eKz dxdz

and where we have replaced the coordinates h, -f with x, ,z [1], [121.

From (3.21) we obtain steady motion by

in-tegrating for time and letting t We then

find that (3.21) gives (3.12), with a given by (3.18). The term for wave resistance in(3.20) is therefore apparently expressed with the same ac-curacy as the customary expression for wave re-sistance in steady motion.

pl -ff

axay

Q cosIsin

Guilloton was the first to propose and develop an approximate method which may be used in computing both the wave profile, the stream lines and the wave resistance for all types of slender ships in steady, rectilinear motion [32], [33], [34], [35], [36], [166]. He conceived of the hull being approximately divided into wedge-shaped elements of volume of a type similar to that

shown on Fig. 9. These elements of volume,

which have plane surfaces and are semi-infinite in length are given by

y .-kn(x+ b.) (a. - z)

where k = tcznatanf3. For a picture of how the un-derwater shape of a model may be replaced by

FIGURE 9.

wedge-shaped elements of volume, we consider Fig. 10, the water line for an infinitely deep

model. The first wedge-shaped, semi-infinitely long element of volume is in this case AOB, and

is positive. Next is CDB, which is negative.

These two elements thus form the semi-infinitely long body AODC. The third element is the nega-tive EFC, while the fourth, EGA, is posinega-tive. Together these four elements form the half-waterline ODFGO. Let us now take. that part of the hull whieh is indicated by the solid lines on Fig. 11, where we assume that ABMN is a part of the vertical, longitudinal plane of symmetry of

the ship. This part of the hull is further

sub-divided by the section FDGH and a waterline CGIP. The ordinate OG is the basis for the first element, which is ABDGEF and is semi-infinitely long in the AN-direction. At F, meanwhile, the

FIGURE 10.

width of the waterline will he FII, not FE. We therefore subtract the element ACGEH which is also semi-infinitely long. Thereby the original part of the hull ABDGHF approximately appears, but the outer contour of the section through NM is now MIL instead of MIK. To bring out the part between the sections FDGH and NMIK we must therefore subtract the wedge-shaped ele-ment of volume based on the ordinates If and KL. By continuing in this manneradding and sub-tracting elements of volumethe entire hull is finally delineated, the bi-curved surface of which is now replaced by plane surface elements. Later Guilloton modified the wedge-shaped elements of volume such that these now replace the continu-ally varying waterlines with parabolic arcs, while the sections continue to be replaced by straight line segments.

In his classical publication Michell developed the velocity potential for a slender body in steady motion as a function of the shape of the body, and it is this velocity potential that Gnilloton uses for each of his elements of volume." As the problem has been linearized, the velocity poten-tial, and the wave profile, for each separate ele-ment of volume may be added directly, giving approximately the velocity potential and the wave profile for the ship as a whole. Guilloton gives tables and curves which may be used to find the wave profile formed by each of the elements of volume, and the vertical shift of the iso-pressure lines in the fluid along the hull from their initial, horizontal position when the hull is at rest. The wave profile,

naturally, is also such an

iso-pressure line. Its initial, horizontal position

was the free surface of the fluid. In order to

compute the wave resistance in an ideal fluid it

"Michell's velocity potential for slender ships in

recti-linear, steady motion should be the same as Havelock's

given by (2.15), (2.17) and (3.19), the latter with the source

distribution on the vertical centreline plane of the ship. It has meanwhile never been proven that they actually agree

with each other everywhere in the fluid, although Wigley has attempted the proof [6]. The wave profile along a plane

co-inciding with the vertical centreline plane of the ship; and the expression for the wave resistance,

are' however, the

same for the two velocity potentials. (Note: The statement

in this foot-note is no longer true, see R. Timman & G.

Vossers in International Shipbuilding Progress, Vol. 2, No. 6, 1955. J. K. Lunde.)

15

0

sure along the hull due to the motion, since theis only necessary to know the variation in pres-hydrostatic pressure for rest integrated over the surface of the ship does not give the resultant

force. Guilloton therefore integrates the vertical shift of the iso-pressure lines over the surface of the ship, finding thereby the resultant component of the resistance to the rectilinear motion of the ship.

It may be shown that there exists a relation-ship between the expression for the wave profile for an isolated source and Guilloton's wedge-shaped elements [36] (see also Guilloton's dis-cussion of [17] and Havelock's disdis-cussion of

[33]). It may further be shown that the wedge-shaped elements will give the same expression

for the, wave resistance and the wave profile

along the centreline plane of the ship, as the complete method described above, and where the shape is then replaced with a continuous dis-tribution of sources and sinks. The condition, however, is that in both cases we replace the water lines between the individual vertical sec-tions with parabolic arcs and the secsec-tions with straight-line segments corresponding to the shape of Guilloton's wedge-shaped elements.

Guilloton has also, as mentioned above, made an attempt to use his wedge-shaped elements of volume to compute the stream lines along the hull [35]. _{From the velocity potential for the}

elements of volume it is possible to compute

the angle that the tangent to the stream lines

forms with the hull at various points. If the

computation is made for a sufficient number of points on the hull, these tangentsmay be

im-posed on the body plan and the stream lines

drawn.

As in the method where a limited number of sources and sinks are used, the computation work

with Guilloton's procedure increases with in-creasing number of elements of volume. In Guil-loton's tables the underwater shape of the ship is therefore divided into five parts, with water line planes located one-fifth the ship's draught from each other. The tables simplify the com-putation work itself, but this writer has not used the method and is not acquainted with the pre-cautions to be observed. Neither can he comment on the accuracy which may be obtained in prac-tice compared with the complete method.

It will be quite natural for engineers to conceive of the resistance being computed by an

integra-tion of the horizontal fluid pressure over the

wetted surface of the ship. One may therefore ask why this method was not used in the first example where we had steady motion and where the complete velocity potential was used. To this is to be said that integrating pressure over the wetted surface necessitates satisfying the condition at this surface with great accuracy. If

this is not done, we have no assurance afore-hand that the formula found for the wave

resist-ance will agree with that found by considering

the energy in the waves infinitely far behind the ship. When the pressure method is used, the velocity potential must consequently be ex-pressed with greater accuracy than with the energy method. It appears that the expression for the velocity potential becomes considerably more complicated when we attempt to introduce greater accuracy [11], [13], [30], [31], [141], [168], [169], [170].

Similar objections may also be made to Guil-loton's' method described above.

The advantage of the energy method compared to other known approaches is that we need only know the velocity potential in the form it has infinitely far astern the ship. The potential thereby "purges" itself of most of the super-fluous terms, and only those remain which are essential to the wave resistance. In other methods it is not always clear, without a more thorough analysis, which terms effect the final result and which may be disregarded from the beginning [1]. It may be of interest to mention that an almost analogous method may be used to find the in-duced resistance for flight.

We note that the theory developed above on

the basis of sources and sinks does not take

into account that part of the ship lying above the water line on which it floats at the velocity for which we compute the wave resistance. It should be quite natural to assume that also that part of

the hull lying closest to, but always above, the water line will have a certain effect on the wave

resistance. Eggert's pressure measurements, in any case, appear to indicate this [23], [24]. Ex-periments on two models with the same under-water hull, but with different shapes above the L.W.L., meanwhile, showed no essential

differ-ence in the resistance [37]. We must show

caution, however, in drawing hasty conclusions from results with only two models. On the other hand, these results agree in many respects with the conclusions possible from Havelock's theo-retical investigations on the reflection of straight, transverse waves [8], [38], [39]. He assumed that these waves met a body whose water line',

for the sake of simplicity, was composed of

straight-line segments. The studies showed that the body had an insignificant effect on the height of the passing waves, an effect which could be disregarded as a first approximation.

Nor may it be said that Guilloton's method actually considers the hull above the water ad-jacent to the water line on which the ship floats during motion. It is true that the pressure is integrated over the entire wetted surface, but the velocity potential for the individual elements of volume is nevertheless Michell's, and this is not modified because of the effect of the waves on the motion. Guilloton's method, meanwhile, made possible for the first time, without great diffi-culty, the finding of the wave resistance and the profile for displacement ships whose shape

could not easily be expressed by mathematical formulae. It should be added, however, that for such ships Weinblum used approximate mathe-matical formulae in the form of polynomials and thus found the wave resistance for the corre-sponding distribution of sources and sinks. We have seen above that the method using a finite number of sources and sinks, also is suitable for such ships. Nor do the computations for the wave profile by the latter method offer any particular difficulty [6], [17], [68] (see also page 18).

From the formulae it would appear that the wave resistance is apparently symmetrical, i.e., they will yield the same wave resistance whether the ship now moves forward or astern with the same velocity c. A ship which is not

symmetri-cal about its midship, however, will assume

various positions when it moves forward or astern with the same velocity. As a result the equation y = F (x, z) for the ship's surface in the position it assumes during motion will be different for the two directions, except possibly for the smallest velocities. From (3.18) and/or (3.19) we note that the source strength is then also different, and we may conclude that the integral for the wave resistance will not give the same

resist-ance for forward and astern motion. This does not apply to completely submerged bodies, and we have here an example of the "reversibility paradox" well-known in aerodynamics [40], [41], [42], [43], [44], [45], [46]. We shall come back to this in Section 12, as there is reason to be-lieve that the boundary layer also plays a sig-nificant role in this regard. It is not unusual in

modal testing to find models which show no

essential difference in resistance regardless of the direction they are towed.

Naval architects occasionally assume that it is necessary to disregard the sinkage and trim of a ship in motion in order to arrive at the ex-pressions given above for the wave resistance. This is not true. No such assumption is neces-sary, but the equation y = F (x, y) for the wetted surface of the ship must be given for the position it assumes at the velocity for which we wish to compute the wave resistance. When we so often disregard sinkage and trim in using the formulae, is this due exclusively to the difficulty and labour connected with the computations of such sinkage and trim [47]15 If we therefore disregard sinkage and trim, y = F (x, z) must be the equation for the wetted surface of the ship when it floats at rest in a fluid.

All who work at an experiment tank will soon note that even experienced designers often ex-press their astonishment and occasionally their doubt that the wave resistance will vary so much for apparently small changes in the lines of the ship, changes which do not effect the form pa-rameters which are deemed to play a deciding role, while in other cases showing no essential variation. Theoretically this appears to have its natural explanation. From (3.14), (3.15) and (3.18) we see that the wave resistance is a com-plicated function of the squares of the water line angles measured at every point on the wetted surface. Integrals also occur in the formula for the wave resistance, and these may, even when they give approximately the same final result, permit of quite a variation in the water line angle

measured at all points on the hull.

From the above it is quite clear that the main problem mentioned in the introduction, to wit, to express the wave resistance as a function of the shape, is unsolvable as long as the shape of the

13At a session of the Sixth International Congress of

Applied Mechanics in Paris in 1948, Havelock gave a

lec-ture in which he presented a theory for computations of

trim, with some numerical results for a model. Most of the

papers at the meeting were unfortunately not published,

among them Havelock's. This writer has however had an opportunity to see the results of Havelock's computations, and these agree well with the measured variations of trim.

LINEARIZED THEORY OF WAVE RESISTANCE

ship is not expressed by means of mathematical formulae. It will meanwhile be necessary to re-call Weinblum's [28] recent statement:

"Accord-ing to D. W. Taylor's own statement [48] he

developed 'Mathematical formulae, not with the idea that they give lines of least resistance but simply to obtain lines possessing desired shape.' This statement is important; contrary to some attempts to ascribe magic properties to certain analytically defined curves like trochoids, sine

curves, etc., the principle of systemization is

put forward as the decisive argument for their adoption." The difficulty is rather mainly due all the variables which occur in the problem. Many of the common quantities which occur in the design of the lines of a ship, and which are used for practical reasons, are not independently variable. To establish an acceptable theory it is absolutely necessary to make theoretical com-putations for models with mathematical lines, not because, as many presume, it is thenpossible to perform the computations, but because the

form is now exactly determined by a limited

number of independent variables and because

every change in the form is then precisely de-fined. This appears to be the only way in which a theory may be built up. As has been done in practice, one should first employ as simple mathematical forms as possible, to develop the computation method itself and to find the strength and weaknesses of the theory in general. With

this in mind Weinblum developed equations for the surface of different models [121], [122], [125]. These should then be suitable for a sys-tematic study of the wave resistance and possibly lead to shapes of ships with the least resistance. He also found that the apparent advantage of using trigonometric expressions for the surface of a model instead of algebraic polynomials was misleading.

With some simplified assumptions it has been possible to establish a linearized theory for the wave resistance, as indicated above. We must nevertheless be constantly on watch against

oversimplification in hydrodynamics. We must likewise be extremely careful about physical arguments, which from time to time may lead us astray. There are examples of this in the argu-ments: "Small causes, small effects" and "Sym-metrical causes, symmetrical effects." As a striking example of how erroneous the last as-sumption may be, should be mentioned the experiment with a small gas bubble rising in a viscous, infinite fluid. If the bubble is quite small, and the fluid at rest (except for velocities induced by the bubble due to its motion), we

might assume that the bubble is spherical because of the surface tension, at least we might assume

that _{it is symmetrical about a vertical axis}

through its centre. We could therefore assume that the bubble would rise along a straight, ver-tical path. But it may be shown that this is not the case for Reynold's numbers greater than 50.

The path it then follows is actually a vertical

spiral [49]!

Oseen has been aware that arbitrary, small causes may influence the final result [50]. He

pointed out that the presence in the differential equations of arbitrary, small terms of a higher order may change the character of the solution. It need not follow, for example, that the solution, when the coefficient of certain terms tends to zero in a differential equation, approaches the solution when these coefficients are made equal to zero. It must not be assumed, however, that this applies generally. As nunierous, minute in-fluences effect the experiments, we would not be able to compute the results aforehand if Oseen's observations were generally applicable. Results of many experiments may be predicted with cer-tainty, and we must conclude that the observations of Oseen probably are limited to certain types of partial differential equations.

There are many theoretical difficulties in the mathematical development of the theory of wave resistance, discussed in the foregoing, which are neither mentioned nor indicated. We do not, for example, treat the difficulties arising when a body cleaves the free surface of the fluid, which is of course the case with a ship, and where we naturally carry the continuous distribution of sources and sinks up to this surface. This has probably something to do with the known mathe-matical infiniteness of the vertical component of the fluid velocity at the bow and stern contours. The theoretical wave height is nevertheless everywhere finite, and agreeing relatively well with that measured. Somewhat speedy ships have often more or less of a vertical sprout of water at the bow contour, and it is conceivable that it is this which identifies itself in the theory with a mathematical infiniteness here. It is known that by replacing the water lines at the bow contour by, for example, part of a parabolic arc, such that the angle of the water line is now

in reality zero, this theoretical difficulty dis-appears without the modification causing any change in the theoretical wave resistance [34], [36] (see also Wigley's discussion of [17]). In

itself this would indicate that the difficulty is due the use of sources and sinks, which is of

course an artificial phenomenon without any

natural analogy whatsoever." At the stern the boundary layer makes its influence felt, and if we take this into account it is reasonable to assume _{that the theoretical infiniteness here} may be surmountable. We have also not mentioned the modifications of the source distribution nec-essary because of the wave motion on the surface of the fluid, [11], [30], [141]. In most of the cases where the theory has been applied this modification has been disregarded.

It should perhaps be mentioned that a source of finite strength, in motion along the surface of the fluid (or immediately under this) theoretically gives a wave profile which has an infinite height vertically above the source, but which is other-wise everywhere finite. The same occurs if we

consider a continuous distribution of sources

along a vertical line ending in the free surface. On the other hand, if we consider a continuous distribution _{of sources over a vertical plane} whose upper edge, for example, lies in the sur-face of the fluid, the theoretical wave profile is everywhere finite, as mentioned above.

If a

finite number of sources and sinks is then used instead of a continuous distribution, the vertical sub-division of the hull mentioned on page 11. must not be such that sources and sinks with finite strength are located on or directly under the free surface of the fluid, if the theoretically computed wave profile along the hull is to be

compared with that measured. Any lack of agreement between the two profiles may mean-while be reduced by using more sources and sinks in the longitudinal direction (see Havelock's and Wigley's discussions of [171). Apparently we need not take the same precaution when we

compute the wave resistance, since we have

shown that this may be found from the motion far astern the hull where the wave profile from the various distributions more and more approach each other.

From the above the various steps in the theo-retical development may be summed up for steady motion as follows: First, find the fluid motion occurring because of the motion of the ship, al-though completely disregarding the formation of waves. Next, find the waves formed by this fluid motion, disregarding now the ship's presence in the fluid. Then, find the effect of the ship itself on the formation of the waves in question. In

this manner, at leant theoretically, the velocity

As mentioned on page 4 a source is a point where a

fluid is created, and from which it streams radially, while a

sink is a point where the fluid is annihilated. As a finite

volume of fluid flows from this point per unit of time

(source), the velocity at the point itself must be infinitely

potential may be found to the degree of accuracy desirable for the problem, in that the approxima-tion procedure is assumed to converge. In practice

we find that the third step already offers

con-siderable difficulty. In most cases, therefore, we are satisfied with the first two steps, leading

to the integral form in accordance with (3.12)

and (3.13). On the other hand, it is possible to obtain a concept of the effect of the third step on the wave resistance. Havelock has

investi-gated, as previously mentioned, the effect of a slender ship-shaped body on the passing waves [8], [38]. He found that the change in height

due the presence of the body was completely

insignificant. For a slender ship it does not therefore appear as if the third step will play a deciding role, at least not on the wave resist-ance, which of course is dependent on the wave motion infinitely far astern the ship.

The results we have arrived at may only be used for computing the wave resistance for dis-placement ships and not for speedy motor boats

or similar vessels. Because of their special shapes these latter give rise to a water sprout at, for example, the forward shoulder." Neither may the source density be expressed by means of (3.18) when the ship has a square stern, since (3.18) will here be infinitely large. In reality it

is doubtful if sources and sinks may at all be

used for ships with square sterns. The same is true for Guilloton's method for sub-dividing the hull into wedge-shaped elements of volume.

In the foregoing we have only discussed the theory of wave resistance in connection with displacement ships. It should meanwhile be apparent that a general expression for the wave resistance for steady motion also applies to a ship with quite a small draught and almostflat bottomed, provided only the sections and the water lines are continuous. The density of the source strength will in this case not be given by (3.18) but by, as a first approximation,

c az

a =

--2ff ax

where we not insert dx dy in (3.13) or (3.15) in-stead of dS and where z (x, y) is the equa-tion for the wetted surface of the ship [191, [117], [118]. If the draught of the ship is so shallow that it may be everywhere disregarded (z made equal to zero in (3.13) or (3.15)), we find that

"A speedy motor boat may, as a first approximation, be

replaced by a pressure distribution acting on a limited part

of the free surface of the fluid and in motion on this

sur-face. Consult [79], where a relatively complete literature

reference is also even.

LINEARIZED THEORY OF WAVE RESISTANCE

the expression for the wave resistance is now the same as that found for a pressure distribution acting on a limited area of the surface of the fluid when the relationship between the source strength a and the pressure density p is given by [19], [79], [13], [116]

277g pa = cPx

A very narrow ship, and a ship with a very small draught, may be considered as the two limiting cases for the theory. The similarity be-tween the equations for wave resistance for these two limiting cases for steady motion led Bogner to attempt finding synthetically the

general expression for the wave resistance whose limiting value yielded these two expressions for resistance, and which also gave the formula for wave resistance for a completely submerged ellipsoid, as this formula has much in common with the other two. Hogner has therefore pro-posed a so-called "interpolation formula" which gives the correct expression for the wave resist-ance in these three cases, but which in itself is empirical. We shall not dwell further on his argument, but only refer to [19], [112], [117], [118], and Hogner's discussion of [112].

It is often an advantage to introduce dimen-sionless coordinates in the expressions for the wave resistance and in the equation for the wet-ted surface of the ship. If we set e= x/1, n =

y/b, C

z/d, where 21, 2b, d are the length, width and draught of the ship, respectively, we find for a slender ship that (3.12) and (3.13) may now be written 7r/2 4

R, =

Kgpl&?

f

77 (3.23) 0 0 (3.24)

P'}.

Q

ff

dc sec2 Ode X edKoe

where 77= F() is the dimensionless equation for the wetted surface of the ship. We have here assumed that the source distribution is located on the centreline plane of the ship.

From (3.23) and (3.24), here applicable to steady motion, we see that the width 2b does not occur in the integrand, but among the constants EIS 1,2 in front of the integral sign. The wave re-sistance varies with the square of the beam of the ship. if Ri is therefore the wave resistance for a ship I, whose water line ordinates B1

mess-COB{e

sin /Koesec 0} x