# The theoretical wave resistance of a ship travelling under interfacial wave conditions

## Pełen tekst

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### NORWEGIAN SHIP MODEL EXPERIMENT TANK PUBLICATION N° 63

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In the first part of the present work a linearized theory of wave resistance is developed for a surface ship represented by a general surface density distribution of sources under interfacial conditions and the effects of the free surface and interface on the wave resistance have been studied. In the development of the wave resistance two different methods are used; namely,

Three-dimensional-extension of the Lagally's theorem, The rate of dissipation of energy.

In the second part, the linearized theories of wave resistances are developed for submerged bodies (symmetrical with respect to a horizontal plane) represented by a general surface density distribution of the sources

under following conditions:

A submerged body moving on the interface of an unbounded two fluid system,

A submerged body moving in the light fluid layer bounded by free surface and interface,

A submerged body moving in a two-layer fluid system with a common interface and bounded by horizontal planes at the top and bottom.

Various possible variations of the problem will be found at the end under the heading of "A summary of the results in formulas and figures".

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### PARTI

THE WAVE RESISTANCE OF A SHIP MOVING ON THE SURFACE OF A LIGHT FLUID LAYER WITH THE PRESENCE OF A DENSE FLUID

BENEATH PAGE

Introduction . , 1

Assumptions 3

Linearized boundary conditions 3'

The velocity potentials of steady motion

The poles of the integrals occurring in the velocity

potentials , 15

Details of the satisfying radiation condition 18

Velocity potentials of ship-shaped bodies . . . 38

Derivation of the wave resistance for steady motion

obtained by integrating pressure over the hull 42

The wave resistance for steady motion derived from the

three-dimensional extension of Lagally's theorem . . . . 43

The wave resistance for steady motion derived from the

rate of dissipation of energy 51

Deductions of the velocity potentials being the functions

of the time 65

### ?ART II

THE WAVE RESISTANCE OF A SUBMERGED BODY MOVING IN A TWO-LAYER FLUID SYSTEM

The wave resistance of a body moving under interface . . 75

The wave resistance of a body moving on the interface . 80

The wave resistance of a submerged body travelling in tne light fluid layer bounded with the free surface

and interface .

The wave resistance of a submerged slender body moving

under interface in a shallow dense 'fluid 96

5. The wave resistance of a slender body moving in the light fluid with the presence of a shallow dense fluid

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PAGE

The wave resistance of a slender body moving in the light fluid bounded by flat ice at the top and an . . .

interface at the bottom 102

The wave resistance of a submerged body moving in a two-layer fluid system bounded by horizontal planes

at the top and bottom 104

A summary of the results in formulas and figures 113

ACKNOWLEDGEMENT 123

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I.

RESISTANCE

### OF A

SHIP MOVING ON THE

### SURFACE OF

A

LIGHT FLUID LAYER

. Introduction

Near the mouths of some of the Norwegian fiords, in the Bosphorus and in the oceans the occurrence of a layer of water of smaller density on the top of denser water is quite common. Such layers may be caused by the spreading out of fresh-water from rivers, by melting of ice, by suspension of silt in the

sub-layers of the water, by the entrapping of dense water by sills (a common occurrence in fiords) and by other oceanographic processes.

Almost a century ago 'tokes gave the first elementary theory of internal waves, later Ekman studied the problem with an elementary theory and carried out

experiments with the model of the Pram. The occasion of the study was the classics.] account by Nansen of dead water experienced by the Pram during North Polar

Expedition near the coast of Siberia. Recently in 1950, attempts have been made in America to approach the problem, the aim of the work is to determine experi-mentally the heights of wave of interface caused by a cylinder moving with a constant velocity in the upper light fluid, extensive experiments are made in a small channel with various depths of layers, sizes and shapes of cylinders, and cylinder velocities.

waves in a homogeneous deep water having a common boundary surface with air are characterized by maximum vertical displacements at that surface. The vertical displacement of the water particles decreases exponentially downward.

In stratified water, considering first two different density liquids reaohing half infinite extent up and down, occur also some type of waves which have again their maximum displacement sat the common boundary surface, and vertical dis-olaCements of the liquid-particles die out exponentially upwards and downwards. From theoretical point of view the common boundary surfaces of air and water and two superposed liquids are similar conditions. The only difference in each case is the density ratios, i.e.,

e water p salt water

_ 800 _ 1.025

e air efresh water

Therefore owing to the small density ratios occurring in the superposed liquids system, the same amount of potential energy raises waves of considerable height

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### =(C COO) =

1-4-I a th 1.21.7k1

### "

for internal waves

where 4e= 121W2

and 9 is any direction of wave propagation with respect to the motion of ship.

The maximum added resistance is obtained when the speed of ship attains a value in the vicinity of

### (1-r-;597;

This speed corresponds to the propagation of the longest internal wave, The added resistance gradually decreases

for larger velocities of the ship. If the thickness of the layer is about equal to the draft of the ship, the velocity of propagation of the internal wave is small. For ships moving at such lower speeds the interaaiion is very great. Submarines, however, may come close to such a deep boundary with the result that at a certain higher speed the forces and moments are high enough to cause it to lose control.

at the interface compared with those on the free surface. If the upper light liquid is bounded with a free surface then two different type of wave system

(so-called free surface and internal wave system) are superposed in the corre-sponding liquid regions, and with the above given reasoning, waves of considerable height are produced at the interface. To this cause is ascribed the abnormal

resistance sometimes experienced by ships operating on the free surface of the fresh water layer. Corresponding to a given velocity of propagation 'C' there are two possible wave lengths which belong to the free surface and internal waves. If h is the thidkness and P1 the density of the layer on the top of deep water of density Pg then the relations between the speed and wave length for two system

of waves may be given in the following manner

C4-069=

### 21

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a. Supposing that two different density fluids are present, one lying over the other. Variables-referring to the"upper and lower fluids have subscripts 1 and 2 respectively. Because-in this section we are only concerned-with the wave resistance of a ship which is moving with a constant speed of 'C! at the

free surface of-a layer of a light fluid then it is sUitable to accept Cartesian coordinate system which is fixed to the moving body, the origin has been taken at the undisturbed interface with (0z) vettically upwards and Ox parallel to the direction of the motion. If we superpose on the whole syStem a velocity -C, the wave profile, caused by the motion of the body, and reference frame arid body itself will be at rest and fluid will be streaming from the right to the left.

Consequently, in the calm water condition, upper fluid fills the region 0<z < h

and lower fluid fills the region" -

.0 . (Fig. 1)

### ace

The fluids are incompressible, homogeneous and inviscid.

The motions are irrotatiohal and characterized by two VelocitY potentials

-0

dlt and 46 so defined that the fluid. velocity vectors a

2 -1 and q2 are given in the

--0

### 71, -L.-!!

usual notations by qi = -,gradsui:

### in

their respective fluid regions. If 1 j, k

show unit vectors parallel to the Cartesian coordinates x, y, z the fluid velocity_ parallel to these coordinates may be written

u. =

p

### 454

, respectively, i = 1, 2.

It is assumed that the ship is advancing with auniforth speed C for a Sufficient length of time consequently steady State condition has been established. Theoretically the time required will be infinitely large.

3. Linearized boundary conditions.

We shall derive the free surface and interlace boundary

conditions by linearizing them. Therefore, resulting theory is.valid only when disturbances in the fluid are small. In situations where one is dealing with waves generated by a moving body, this means that some aspect of the body or its motion

(8)

respectively

-4-restricted In a. suitable way. In the case ofsteady moving ships or submerged bpdiesy-the restrictiont fall into one of the following classes:

Thin-ships, approximating

### a

vertical plate,

Flatshios, approximating a horizontal plate on the free surface,

Bodies submerged so deeply that their influence on the surface is small, Submerged thin wings Of small angle of incidence.

We all derive the boundary conditions for linearized theory-by using a dimensionless parameter E Which may be related with the motion

### in

such a way.

that When 1-.0 the motion also approaches a state of rest. It is then assumed that the various functions entering

### into

the problem may he expanded into power series of E the series are substituted into the equations and boundary cOnditions \$3.40_ grouped according to powers of g . One may obtain in _these expansions by taking only the coefficients of £ "The First Order Theory" or "Linearized Wave Theory".

### 2

By taking the coefficients of 6- and higher Order in the perturbation procedure one may obtain a systematic procedure for

### iMproving simultaneously and step by

step the accuracy with

conditions on-

### the hull,

free surface and interface are satisfied.

.Let

### s.=-11.t9r(IY)

and 3=X.(211) be free surface and interface equations

E

1

(1) (4) (2)- 3

7( E -+

(I)

E2

### (5)

where we suppose that . s are the solutions of the (WI:dace' s equation.. If

then from (1), (2 ) (4 )1 (5) 97-7 X= .._ 0-40 and. p static. pressure. That

is With the E'vefluid motionsin ttleir respective regions are to becalmed.

If a body Is at rest in a uniform stream whobe Velocity is - C in the x-direction the velocity potentials 4% and

### A

can be written in the form

ST)

.

### -2.

2

Where Cx is the velocity pbtntial of the uniform stream and.where 41. and

are respective velocity potentials of the Perturbation caused by the body in the layer of the light fluid and denser fluid.

a. Pressure condition

(9)

sufficiently great distance upstream from the disturbance. We assume further that the deviation of any fluid particle from thestate of uniform flow is resisted by a fictitious dissipative force proportional to the perturbation velocity. This force does not inerfere with the irrotational chatacter Of the motion. Hence the components

### Eax rjrlyC-gloy) of

the external

py44

force are given by

### 2!

Oh, (4)=.

.where pis a positive coefficient which is retained in the intermediate analysis to!a degree sufficient to attain its chief pur2ose and is made zero in the final result. Since the velocity ui, vi, wi

gb. respectively, we can write also 14.

,

### no:

of the perturbation are given by - ebiat I- 43:. )

ict"

g-I

### 4

multiplying the first by dx, the second by dy, the third by dz and adding them together, we have

trof

### det,

integrating this equation we find

(fre

where now

### J2i

is a force potential..

Then the fluid motion is steady and irrotational and eternal forces are conservative, the pressure equation reduces to ernouili's equation of the

-following type

where c is he scalar resultant fluid velocity, 2. is the pressure, 4? is the

uniform density and C. is a constant which has the same value throughout the fluids considered at any time. Now at the interface Bernoulli's-equation for the two liquids

/ 2

q, - 4 --- =r

i = 1 and 2 (7)

1,

2

4)2

4'2

412

### p

{Ccor itx4 Roix)2+(ca,,A(0)2.]

fi,c2- (a)

C

2V

2

### 2

.2

If we equate Bernoulli's equation for a point on the interface and far in front

of the disturbance where both q 02:4,0 then we obtain that the constants given in (3) and (9) are equal to each other. i.e..

,

### °

where po is the uniform pressure above the free surface.

The only necessary condition at every point of the interface is the equality of the pressures on either side of the interface. That is P.= .

From (8) and (0

(10)

(1),02.) fie'

### (n)

which will be satisfied on the interface, thac is we must insert 3,---)ax

### )9

in the above given expression: Then

01: ( y, )

### Y, 1((ze ID)

expanding with respect to 1C(.2c1y)

_

y X(26,

LI

### -tin

substituting in the above given expression the power series expansions. for

.

and

### tV

from (2) and (4) we obtain

(0 - O

Y,

### LEZ429(L2.).)

1; 2

arid arranging to powers of

(0 2 (4 CO

y)

### 44]

E

L 6 (11)

putting these OtivalUe8 from (11)

(10) and

### grouping

to powers of S and

equating the coefficients of the same powers of E We have

et

4 C

64,ork = 11)213

0) UV

### (12)

&2)4CCk22) L(Cilis)21 (

'Da

### -(12),

(id) etc. are all to be satisfied On the originally undisturbed interface

### = 0.

b. Kinematical boundary condition'

The other boundary conditionWhich must be satisfied by invisCid fluids at interface is a kinematical one tihi.ch arises from the fact that the velocity relative to the surface of a particle lying in it must be wholly tangential, thus if P(x,y4W) . 0 is the eqUa.4611 of interface-4 We Must have .

### 4

(11)

These kinematical boundary conditions must be satisfied on the interface,

that

### is

on 3.77(4114),-Again insertion of the power series

and

### from

we may write (14) in the following way

(i)

or

a 3.

### a

DC

Now expression (15) gives first order

Where.

(1)

ch

43

23 (4)o) (r) (f)

CID (.xiX 0) 4.2

LAI=

### e

to the following conditions

(a) (1) ,(1) + d>

(-11Y,

### 0) = 0

(2) (4)go (4) 6) (a) (I)

(xly)-1-

0)

2/

### 25

23.

-which are alsoto.be satisfied on the undfsturbed interface Or for z

### aj.

kinematical boundary condition.

c. Linearized boundary conditions

As the analysis restricted to the linearized theory we must break off the perturbation series given in (1), (2), () and (4) after the terms which have only first powers of& . With. this stipulation the interface boundary

(12)! and (15) yield

(1.4110.1 I

2 43.

### (14)

becaUse these are Similarly related to those given in the usual wave resistance theorie's and have been deduced on many occasions.

Deduction

### of

the other boundary conditions here have been found Unnecessary conditions

for

### 3.O

By elimination of ?Ll?ettlset these two relations we obtain first order interface boundary condition.

Thus

### (18)

(12)

necessary in the firSt place to seek velocity potentials of the motion, these functions must satisfy the following conditions

A2c1,

### 24'=0

Equation of continuity

### 3:: R

Free water surface condition

-2

with izo= p

1 "

### .3=

Boundary condition at interface

qb

### o

Vertical velocities Are to be continuous at intern

### Moving

body all Sorts of fluid motions must be vanished and very far aft of the moving body wave motions must be bounded,

= 144vi -P

-Awn' cfr

or

2-,

### asI

2 a -4, 2

4. The velocity potentials of steady motien

In the presence of a uniform stream any ship-shaped closed body moving on the surface of the liquid can be represented by suitably chosen sources and sinks (or doublets). In order to build_a,closed ship-shaped body the only

. necessary condition

### is

that.the total strength

### of all

sources and sinks must be

zero If the fluid emitted from. the sources shall be annihilated by the equal strength of sinks and thus determine dividing stream lines which will be con,- .

sidered a ship-,shaped body surface. Therefore it may be convenient to consider

### at

first the velocity potentials of

### a

single source. It is clearly more simple to consider a source of strength 'm' at the point (0,0,f) -in the light fluid layer above the undisturbed interface..

Hence,

the analysis

### which follows, we

consider a source of strength situated (0,0,f) in a uniform stream Moving With a velocity C in the x - direction (Fig. 1). Keglecting for the time being the free surface of light

fluid and interface, that the layer of light fluid extends to infinity in every direction in the fixed coordinate system, the velocity potential of a source

(13)

potential

### co

-I= Cur

On the other hand if we itpose all !;he boundary Conditions given above we must then add a correcting potential

to the potential

given

### in (19)

must be a solution of Laplace equation, then.

C Jr4- -+

### "

hore usual and suitable way of solving such problems are to work out with.. integral equations. Therefore we make Use of known integrals

### +

3 If we apply equations AMMO

(Laki

### 0

-11 0

where ay= (2e44941S4161)

If we take4.4-4-5= -7-11.+4:41(aiyiv one may represent it in the integral form

as shown below n co 0 00 .

ikeD

f°1

211

### 0

..-. n 0

where l 4- h I indicates absolute value, andco =tz c 0 4 9 4. y swp] . The double Fourier transform in x and y of dli (044) by \$(k10,3) is quite - general and besides

4

in this way chosen 0(coiv satisfies Laplace equation.

Because of the perturbation caused by the source of strength 'm', there is some kind of fluid motion in the denser fluid below and associated velocity

too. That is op

2

### p

Laplace equation to

### 4(.2,0

end 4420,,z) we find the following diff.

### 4.210,0,3) =0

The solutions of these diff. equations are

2 2 where n

j

in

### a

so defined .uniform stream will be given by

et, itself

(14)

(b4

(1

### veze41e, sto R

where we have made use of condition t.,4147.?2= 0, Now we can write velocity potentials given in (21) and in (22) in the following way

'

n

n

OtO

B

-V 0

### 0

We have here three unknown functions which will be determined by

boundary conditions given on page 8. If we apply the boundary condition below to (22) and (23) namely,

we find

4

we have

2

### [A

again with the application of the free surface

and lastly application of interface condition

e,

clooci

we get

j71q(k Ro4

-3

;_k; pi I

### (27)

From these three equations, (25), (20 and (27), we can determine unknown functions A(k,9), B(k,9) and a(k,9): We use Cramer'S rule

1

...-..1

A!,

2

,

### Otod 9404 leco0Je

k It Cad 6 4404(Ito efrio) e .6

ri

cd.be

c4r40)

2 0

### (25),

condition, namely, 2 ,

- Ro+temco-30)e

2

### (23)

( 24 ) using other given

(15)

211 B 271

ize+110

4

utd

etu

citS

1

k,-%Le/mei-14h

### ,kh

1 cit:0+ 120+4uciv

( k

g

(kelt). 64 ko+

ek(khi-4

is

_

.

U114) 2

-

2

1

### kose49+iesecorek(h4) 1

ii

P (11.-12.2tC-41e31%6){g172-Prii(k44uSerA+ (Pf fr),24C1')

7,7

(16)

271

k c,

(

4/ sece )

s4c2e

271

stc261

e-km34.

??7

0

### S.tc2 4141 Seco) e

.11(f2-t-fl)(k-f1eiRce)- le

9),

Sec20

217

A) Oe

stc2c

17,

1..,h

I 1.

(17)

Inserting

### A(k',9), B.(k,(4), C(k,e)

functions into (2) and

ce velocity

2 e

1 2

7;

77

0

- -277 4

.

-.(_

k /7

-1

### k,

su29 -4-i euga9)ek1 Vg

? .

17 oo

n oo

( 1 - x )

ft

An

### c°

(k_- OA Sic' 0 -Pit Al

2148 1

it e3

411

a

1

-11

V 00

sec28

r SeCW e

.rn

'

2 2 1

1

L

g 1141

### in the calculation of the wave resistance af the surface ships. (Fig. 2)

(18)

Putting these coordinates

### in

the Above written velocity potentials and. dropping the primes, we Obtain following velocity potentials.

00

+ C1---E)

8

2D

0

IV/

A 00

4.

### cis

( k+ ke secl8 + i IA, sece) e cek

2 '

4

### -#

0 1 + Je VA kh-G&W 4-c

E)Inicisni Q9

### 4n

(k+ le, sde-i-i eu FrA----9) ekEloi

cpk+ (3-h+1)1

(o4k.h

42 _fi

t,

,

### k

4 344241+ 41/S4411.7

### ir

ek bit .4-ii-le(3- h_j,

00 2

vieln %..

--'11

Where

ler

ago

014

11 _fi

-11

R

If 0

### fk

,s* k C141-1-4Ler S) k,

)

+

3.41`,6--; ci;

1 + a t

a./

2

)

2

kbteco + A "Seen

Ole=

2

34.t81

I 2

4 2

### r

.1 M.

(19)

_LS _1\$

04 the other hand, if we take limiting valUes,:ofqi and q2 When

### fa

becomes 2

infinitely large-(or ge

### 1-=

0 ) it is found that after some reductions

### 03

r2 4

reduces to shallow water velocity potential and cps vanishes namely

Chs

4 7;

e"`LCA-k-(3-

### le=

0

-It is interesting to note here that, When the density of sublayer becomes infinitely large, fluid behaves like a solid bottom.

Lastly by taking the limiting values When h becomes infinitely large one finds that

### egives

velocity potential of a source in an infinite deep and

4

homogeneous fluid and e2vanishes. Because t coordinate of 41 varies within the interval -

in

### oe.

All these results are compatible with the

2

boundary conditions given on page 8, if we take their limiting values respectively

### For

instance, if we take. limiting value of interface boundary condition, When pt

becomes infinitely large, it reduces to free water surface condition, because of being of the source outside of the denser fluid-which is insensible to

### all

kinds

of perturbations caused by the light fluid, owing to its infinite density. Therefore it must remain in calm condition and. behave like a solid bOttom.

Surface

### ?Tier/ace

5. The poles of the integrals of the velocity potentials

It is very important for the later developments to study the poles of the yarioUs integrals of the velocity potentials given in (7) and (18).We may observe that various integrals have the

at.

1

### (40)

(20)

-qe are supposing that

### e

changes between the limits of

### 004 I

this may be obtained in the process of reducing the range of integration with respect

to 9 from VIIITOto ( Oil!) . The process will be shown in the next paragrph. The k1 = k osec29 is the root which makes

### (30

zero, that value is usually

### on

as the associated wave number which gives the characteristic wave length in the

deep fluid in the direction of la.

2T1

or C ced

### A= --=

le,Secl9

The real and positive w values making the equation (40) also zero, for each 9 value, are some type of wave numbers which give the characteristic lengths of

the so-called internal waves

### Ui

or using this we can give the speed

ill co.

and length relation for internal waves, namely

71 1

(i-.704

### 1<A:14e/34w)

Y = s 77) ptoie (Fig. 3)

Writing (40) in the following way

2

(1 ae)

h k0

### (1 +ar ihkh)

I - ae P2-e, -0 e 2 I 0 -k.och (41)

The first part of this equation shows a straight line which goes through the origin and has a slope c<t...

e

### 4 0

the other part. is the graph of

h I 2 e

### \$h

(1- a)a k h on the base of I<-kh Which goes also through the origin and has (

### t-4-ae Mkh)

a horizontal asymptote which has the value of jf= 1,11::e . Taking the derivation

\-of graph 0 -.r)t A k h we find

(

### With) [OW- 0 001

Therefore the graph has a tangent at the origin which has the value of

is smaller than the

2

value of tangent of the graph or if C2 ci09<3/10- gio we have always a real positive root within the range of integration with respect to 9. But if

C2 > Sh(1- Pv.) a real and positive root requires the range of integration to be modified and begin from 0.-- clue44(Sho- ftn1i.2

### V-71

is the

r2

(21)

highest speed for occurring longest internal progressive waves.

We shall also show that for

fixed

### e

Value, equation (40) has only one

real positive root. Complex or pure imaginary roots are not existingTo prove this we are inserting a complex variable fiv:(k.449 instead of k in the equation

(40) and separating the real and imaginary parts, we obtain

Where (k 4'H).

### ht=

given expressions. It both (42) and (4;) is not existing. If is Inserting this in (42) If

### k

From (46) one may easily observe the following results, namely

I. If

root (of

. 2

### .2

sin 141 = 0 . 2 sin M >0

s4:112.

### (hk+ mh).

takes the values within the range

if 0

11 K

cX

### < co

04: k<och a real and positive

:

equation (44) requires a negative value, for sin2 M which imposes that

(42)

(45)

may be seen that the only possible solution which satisfies

### hi (K+ 1M) = 0,

Consequently pure imaginary roots are

the real and positive root of equation (40) we may write,

d (43) we have

(c/a4af

ae

### (4- ie) P4/2t4

sin2M is solved from the equation (44) we obtain

### (46)

M Must have imaginary values. However this is contrary to our suppositions

already made. Consequently complex, roots are not existing in the range considered.

If K =c01, sin2M:= 0, this is the only positive solution of equation(40), and at, the beginning of this paragraph we have already discussed it.

If"K changes in the interval 40141,e<00 a real and positive root. of equatiOn (4) requires a positive value for sin-M. Dividing equation (45) With

(44) and putting the positive: value of sin2M being the function of variable K.

(22)

1/2 9442

(47)

### 54k+ar

s441111.1

If k takes the values within the range

i while sin N remains, ini 2 this range, between zero and unity. One may easily observe .that equation (47) is always a non-equality; namely

4/2

A-4

-I.

### sa

Thus equation (47) has no roots within the range 0 4:)C.e K. Where k:= k

is a certain value which makes sin2M m 1. For IC values changing within the range

### k'<fk4cv

, sin2M is always larger than unity. But this requires again imaginary M values. However, this contradicts the fact that

### M has a

real value. Therefore within the interval

### eih6(4:ce

there are not any real and positive roots of the equations (47) and (44). Consequently complex roots are not existing within the interval considered.

Summarizing all these results we may conclude: K = kh = 0 is always a root,

If

(1-

### 14),

there is always a real positive root of different

from zero within the range of integration with respect to A. But, if 02>p

a real root requires the range of integration to be modified and begin from

1/2

### c2`.

In the first quadrant of the complex plane

### (k4iv9;

pure imaginary and complex roots are not existing.

If we had carried out the same sequence of thoughts and calculations for the fourth quadrant of the complex plane we would obtain same results given under 1., 2. and 3.

6. Details of the satisfying radiation condition

Till now we have shown that velocity potentials given in (37) and (38) satisfy all boundary conditions except radiation condition which imposes all

sorts of fluid motions must be vanished at infinite distance upstream from the moving body. We are justified in making this assumption, at least as far as the wave motion is concerned, since group velocity in a dispersive media is never greater than the speed of propagation of the waves and hence gravity waves cannot precede the disturbance when the speed of advance is constant. No

(23)

restriction is placed at infinite distance back of the moving body except that they be bounded. Thus the mathematically Statement of the condition

°

bounded

2

co 2

### 24...e

We now have to show that velocity potentials given in (37) and

### (30

satisfy this condition.

The first two terms of the velocity potential (37) being local functions do not help the wave motion. Since these terms vanish when x-9.T.co. Therefore we will be not concerned with these terms in the subsequent analysis.

In order to take the limiting values of the various integrals when ev-woo it is necessary that sec0 should stay either positive or negative within the range of integration with respect to 0. Reducing therefore the range of

integration with respect to 0 from (-nn) to (0,1). If we separate the range of integration in the following intervals and make suitable substitutions -for 9 vakiable we get

CO 7.

11/2 n/2 n

substituting =

0

N

ge49)

Vli Do

### ik

(a CO''&pasty S414

### -

111 GI CAVI 1 aVIP)

E le

.4

6

substituting

/1/7 =

substituting

- 9I

2 '

### I

(24)

Taking one of the integrals of the expression

cp

-A ) -tilt]

14",..,

### /stet/041r

L 70 c--00 K-- k6

4-1141

### Oa

(49) 0 0

separating the real and imaginary parts we get

n/1

s4.269

0

0 0 0

0

11/2 op P/2 op

4.

I1i I

e404 (

### 0[(k-kage28)2+e 2

;14291 It may be observed from (48) that by reducing the integration range with respect to 9 from &47,17) to

### (0,10

all the ,integrals of the velocity potentials given by ()7), (38) have only their real parts. The imaginary parts

vanish. Therefore, in the following sequence of development we may take the real parts of these integrals if we should think that it is necessary.

### 4(41)

-(48) of the type

### (50

The integration of the second and fourth terms of (50) with respect to k when

### la-,

0 may be carried out by using the known results; that is e°

II/2

2 2

ko

64,,to

irl .22

,

k'z

2

### o

using these results and adding together the real and imaginary parts, we get

4/2

L

i (30/Til

## )-tu+

167i-

d- k _ lee

0 0

### (5i)

By reversing the Sign of i ih the above , we obtain

ilia-

.7112.

,

174

11)4 0

.e4444'

...

_ .11./n

k6

6

### 0

where 0, (.x004944,400)

Inserting all these results gained in (51) and (52) in (48) we may write

(25)

following relation 1 (

(3- -

iv Se C 9

o .0,.. cp .

tec'L?Ode

3c clk

.7 a)k

0

110 2

### itio ( k..7 WO cwi (

ko y See i S.4.;4 e)

### PIO 60

0

where in the last integral term we have expanded bine-and cosine

We may notice that the first integral term on the right hand side of the expression (53) is equal to the following integral

### 0

The proof may be given by reducing the range of integration of the above given

expression from (-PM to (OA) with respect to e. Therefore

co-oe

112

sa

ly 0

given by (37).

### (Al) we get

Let take second integral term of the Velocity potential Again, reducing the range of integration from (41,71) to

co

4.

functions.

11/41

114

1-4-0 t 1 1

0

### (chkh-l-aeS-kkh)Lk

1-.1(____±.22_:__Akii h,ev42,9

(26)

### 11-'0

Variables, putting

.co jaw

ea"

### 0 01-N24

where and In the 39 k ter -1,444

0

go

EAl2.464,2) d /V

### Nta-E1

limit when 110-, the

as small as We please. Thus al

### (k)

:r.,0 IV 24 IN t]. 0 where

### (44itAkh)

separating its real and imaginary parts, we have

Do op

CO

84444102

.+ tsictA-t)

### eu

integral takes the Value n , since E may be chosen

2

11

### (56)

zero, except in the narrow

oc is the real and positive

Usingthis

### result

we obtain from (56) and (55)

### IUCO

The integration of the second and fourth terms of the above expression may be performed in the following way. Let us take the integral term in k of the type

### to-to 0 [ AP4 ta]

Because of eu the limit of the integral is everywhere range of the integration where isi-E<k<044-6 with

root of N(k) = 0. Thus 0(41

oc..-£ or)

1,411 to

+ itm

1. JO

Go

0

TA17-1-141

### '

j t s 1 1.4- bu2)

Or 00

o(41

### -

eu-->o ( A/2-1- e02.1

### 0

oc-E

differentiating N with respect to k

014?

(27)

00

Pl2

N

### d 114 i 1

44hre h) 54411 (01.rudOeed

80

Pr

o/2

0

irlz op 4.

°°

,

a

,tiv;r7

oo

%.44.(ke

4.° -Po

6

2 2

### )4s4c

L 11

(28)

the real and positive root of N(k) = 0. Then

r

a' o( \$E ot

R MIL N2*-1 0411

et2.1

### iert

Using previously obtained results

I 4 : .1 1 44

444w

eg-g

tr ,

\$

0

2 z _2 ---% --=

### tt

4/ 671) 0 s4;14 cot co )

### R log) N2(of )

Because of gal' the limiting value of the last integral term of (60) is zero. Using all these results and inserting in (60) we get

_I

catt

/41

at&

eb)

ri ne

0C-E

-24-Where

### g

Let us take third integral term of the above given expreSSion. Because of poo the limit of the integrals are everywhere zero except in the narrow ranges

### of

integrations where

and

Here

### .is

Changing the sign of i in the above expression

o

-

1

( 66 )

(29)

17/j

h9)

### I(67)

where in the last two terms we have expanded sine and cosine functions. Substituting the results obtained in (54), (50 and (67) in the velocity

potentials given by (37) and (38) and arranging them in a suitable way we obtain following expressions for the velocity potentials for a source.

0-1= 27-1

2

740

-

A - ff

il

### ,

( f _ Al /ad" /(edik-l-k311/2: aoStCs,h)ke

41/

__1

(

### nje

1+ a° tA le h ;fa

P 0:' 2_, A

049

211

_,_

7

411

-# 00

411

hh

(..

oc,

E ....04 A

(

2

A

.1-90 i.t;

go

where CO=

### (41CC406q4S44469

Inserting the expressions (65) and (66) into (59) we get

n

.

stcti

0

7112Ito

2 I

### Az. awl 7 )

co- d ( k ;we; g6+4

-0

(30)

11/

2

4.

(oc

(ec

C2

11 00

Ap

0

02

0

77/2

G.

2

90

DC --s -+ CO we

Ass= 14 1

1

2 2LU.0

(31)

A S

Where 97 and

### lr.

denote the velocity potentials obtained without

### ry0

introducinge coefficients and these build up the first parts of the velocity potentials given by (68) and (69). Now we will show that the velocity potentials

,

given by (68) and (69) satisfy'the radiation condition, consequently the terms *obtained in the cource of vanishing process should be considered as the additional

terms to the velocity potentials gained without using e4.s-method. In order to

show the evidence that the velocity potentials given by (68) and (69) or their identical expressions given by (37) and (38) satisfy the radiation condition we should obtain the values of these velocity potentials at a very long distance upstream and dowstream. That is, We have to calculate the limiting values of the first parts of the velocity potentials of (68) and (69) when x value tends to infinity on either side of the x-axis.

We may notice here that the integral terms in k of the velocity potentials are not definite since the various integrands have the poles on the real axis at k = 0, k =ot and k - kosec29. Therefore, in the following analysis we

### shall

interpret them by taking their principal values.

Taking the first integral term in (68) and reducing the range of inte-gration with respect to 0 from

to(0,1) we get

o

### (7o)

In order to define the above given integrals we are considering contour inte-gral of the following type

i

(t...

### kesec20

.P CbT

(71)

where is a complex variable equal to (k

### 4

im) and (z = f)4. 0. The integrand

2

has a simple pole at (k =

kosec

### e)

on the real axis. Let us first suppose that

. . A

### a=

(oect49s0181;1q therefore we integrate round the contour shown on (Fig. 4) consisting of positive part of real axis k, a circular arc

### r

of a radius R which tends to infinity and positive part of imaginary axis. Since the integrand has the pole at

### k0sec28

on the real axis we

### cut

out the part within a small circlet of radius r about the pole and then make r tends to

zero thus taking the principal value in the usual sence. Because of letting the pole outside of so defined integration contour, On

anywhere inside of this contour the integrand

is finite and single valued therefore r (vie-Pim

### plane

according to the Cauchy theorem

### frood1.6

Applying this theorem to (71) we have

(32)

A 19( ,

co

Is-f))

1 m 44

)1.

4,

o

Iki

"4-hi ]

Ii

0

Ph O. .

8

ice. *. [t4

1cl

R

### (72)

Because of accepting 13:1= (

C4467-

### Y3449)>0

the integral taken over the

circular arc

### r

vanishes since with the increasing radius R corresponding integrand diminishes exponentially to sero. Thus we get

OP ,

1

tr)02.1

4

0 00

### 0

Reversing the sign of i in the above expression

op ,

rn

dk ,....

.

(74.:)

'

### 0

where 4D may be taken either.- (x.ccs9 + y.sin9) or x.cos9 - y.sin9) inserting

the results-. of

### (73) 'and (74)

in (70) We obtain

ceihb

7111

eoti00

0

11

0

C

### e

For 010 = (x.cos9 - y.sin9).< 0 the contour we choose shown in (Fig. 5)

consisting of a circular arc r of radius R which tends to infinity, positive half of the real and negative half of imaginary axis since the integrand has the pole at k = k0sec2 9 on the positive patt of the real axis we cut the part

within a small cirlet of radius r about the pole and then make r tends to zero thus taking the principal value in the usual meaning. Because of letting

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