THE TECNICAL UNIVERSITY OF NORWAY
THE THEORETICAL WAVE RESISTANCE
OF A SHIP TRAVELLING UNDER
INTERFACIAL WAVE CONDITIONS
BY
TARIK SABUNCU
ON LEAVE FROM DEPT OF NAVAL ARCHITECTURE
TECHNICAL UNIVERSITY
IN ISTANBUL
NORWEGIAN SHIP MODEL EXPERIMENT TANK PUBLICATION N° 63
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,
Threedimensionalextension 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 twolayer 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".
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 shipshaped 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
threedimensional 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 TWOLAYER 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
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 twolayer fluid system bounded by horizontal planes
at the top and bottom 104
A summary of the results in formulas and figures 113
ACKNOWLEDGEMENT 123
I.
THE WAVE
RESISTANCEOF A
SHIP MOVING ON THESURFACE OF
ALIGHT 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 freshwater from rivers, by melting of ice, by suspension of silt in the
sublayers 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 experimentally 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 disolaCements of the liquidparticles 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
=(C COO) =
14I a th 1.21.7k1
a"
ii+Xth(2gbil
_{"}
for internal waveswhere 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
Cr=
(1r;597;
This speed corresponds to the propagation of the longest internal wave, The added resistance gradually decreasesfor 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
(socalled free surface and internal wave system) are superposed in the corresponding 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
C'=
C4069=21
a. Supposing that two different density fluids are present, one lying over the other. Variablesreferring to the"upper and lower fluids have subscripts 1 and 2 respectively. Becausein this section we are only concernedwith the wave resistance of a ship which is moving with a constant speed of 'C! at the
free surface ofa 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" 
co4z.
.0 . (Fig. 1)Tree waier sur
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, kshow unit vectors parallel to the Cartesian coordinates x, y, z the fluid velocity_ parallel to these coordinates may be written
u. =
cka
pvi =
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
respectively
4restricted In a. suitable way. In the case ofsteady moving ships or submerged bpdiesythe restrictiont fall into one of the following classes:
Thinships, 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 theoryby 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 withiththe bounar.jf
conditions onthe hull,
free surface and interface are satisfied..Let
s.=11.t9r(IY)
and 3=X.(211) be free surface and interface equationsE
4)
16242)*
(1) (4) (2) 3(3)
7( E +X 4 & X
(2).
(I)(2)
P
P
E2'P 4
_{(3)}
(0
2 12/
3 pi
et
e
4 6
4
'(4)
to
_{2(2)}
2(31
02= 642tro60246014
(5)
where we suppose that . s are the solutions of the (WI:dace' s equation.. If
then from (1), (2 ) (4 )1 (5) 977 X= .._ 040 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 xdirection the velocity potentials 4% and
A
can be written in the formST)
=
+ C a
+ C
.(6)
2.
2Where 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
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
(IY,z )
=
Eax rjrlyCgloy) of
the externalpy44
force are given by
X
 by v.
_{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:. )
= 14i
ict"gI
ly
eY1
4
multiplying the first by dx, the second by dy, the third by dz and adding them together, we have
S2.6i = g el;

trofdet,
integrating this equation we find
ail
3 3
(frewhere 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'sequation for the two liquids
/ 2
.52
fe
q,  4  =r
C'
i = 1 and 2 (7)3
1,2
e
i(c +co+
ct,y).
(0,1)
j
c,
p+
)24
(
4)2vf+
4'2tJ
÷$
412= C2
p
{Ccor itx4 Roix)2+(ca,,A(0)2.]= art
fi,c2 (a)qqb
_{la 2}
Ucb
(1) )4 (4)
)z.l +3K64lck
CC22X
2V2a
22 2 2.
2
.2If 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..
c 
4ô 02= c
_{,}
_{z}
e c2 P sh
_{°}
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
fig XC4L4[(qt,i,)2+ (40,024
(
(1),02.) fie'col =
?131+ezeoz1
(+
(4102+
(02)2.)
flfv1472
(n)
which will be satisfied on the interface, thac is we must insert 3,)ax)9
in the above given expression: Then
01: ( y, )
I
Y, 1((ze ID)
expanding with respect to _{1C(.2c1y)}
X2
+
_y X(26,
03
LIfX44+
tin
substituting in the above given expression the power series expansions. for
X,
.and
tV
from (2) and (4) we obtain(0  O
0) eez
Y,
LEO;
+
e Cck,]+[x.+6x.4)LE
+eel) 3+1
LEZ429(L2.).)
1; 2
arid arranging to powers of
e
(0 2 (4 CO
63
= 604;
y)+ 5
4,s
44]
EL 6 (11)
putting these OtivalUe8 from (11)
into
(10) andgrouping
to powers of S andequating the coefficients of the same powers of E We have
et
Ey x
_{4 C}(Pix
64,ork = 11)213C 412+ elt
0) UV
(12)
&2)4CCk22) L(Cilis_{)21} _{(}
4;4
et42)
'Da
is
c
4:4
C(
d:14Ccii))2+
(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 interface4 We Must have .
o
In Our case
dt
(Ct
+
y
e
or
(c+ 4)141(2p
0,0(1 Ole 0
(c+
tz)e)(x+ tyxy
_{4}
These kinematical boundary conditions must be satisfied on the interface,
that
is
on 3.77(4114),Again insertion of the power series1(
andfrom
we may write (14) in the following way
(i)
13_ a
orb
a 3.a
DCNow expression (15) gives first order
Where.
the above expreSsion lead
(1)
i
chcay,s9),= 0.
43po
(xly10)=
23 (4)o) (r) (f)(x, y, 0)4
X
CID (.xiX 0) 4.2Y
lY
LAI=e
to the following conditions
(a) (1) ,(1) + d>
_{.x)Y, 0)X(111)4,}
_{(11Y,}0) = 0
(2) (4)go (4) 6) (a) (I)10,
(xly)1xT
x
0)+9 (alY10))C(ad)(P2VIIY,0)=70
2/_{25}
_{23.}which are alsoto.be satisfied on the undfsturbed interface Or for z
2)04= a ot
a a
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
+4tj4)}
=
pfo
(1.4110.1 I4X c
c
2 43.v6.
(15)
(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 conditionsfor
3.O
By elimination of ?Ll?ettlset these two relations we obtain first order interface boundary condition.
Thus
_{FI tOiax÷}
ko 013
_{CI]}
e[txtke 413
4"102'A
(18)
necessary in the firSt place to seek velocity potentials of the motion, these functions must satisfy the following conditions
A2c1,
m. 0
24'=0
Equation of continuityU.
cla.+ kc,"3 eut..1.=
0
3:: R
Free water surface conditione
+k
eu0 j= e.,10
k
2
eck e4)
with izo= p1 "
2.22
23
x
.3=
Boundary condition at interface
iv.
qb_{o}
Vertical velocities Are to be continuous at internrad dr
0
for
y2.
Radiaiion condition: Very fat ahead of the_{Moving}
body all Sorts of fluid motions must be vanished and very far aft of the moving body wave motions must be bounded,.14.1
= 144vi P64,
Awn' cfrbounded
or
2,
asI
2 a 4, 24. The velocity potentials of steady motien
In the presence of a uniform stream any shipshaped 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 shipshaped body the only
. necessary condition
is
that.the total strengthof all
sources and sinks must bezero 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 ofa
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,
in
the analysiswhich 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 lightfluid 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
potential
co
I= CurOn the other hand if we itpose all !;he boundary Conditions given above we must then add a correcting potential
A
to the potential0
givenin (19)
must be a solution of Laplace equation, then.
C Jr4 +
(..% y )
"
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 AMMOfoofk
(Laki0
ciak(1ic0+3.)
2n
folefe
k
for 3o
0
11 0where ay= (2e44941S4161)
If we take4.445= 711.+4:41(aiyiv one may represent it in the integral form
as shown below n co 0 00 .
s=
k[na13_41)
13.
ikeDf°1
"
Pelf
8
0 (k, ex e d le
_{(21)}
211
I;
0
... n 0where 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
ikar
Oalyip=p0143720218,pe
2
p
Laplace equation to
4(.2,0
end 4420,,z) we find the following diff.2
k2ii(fr,9,3).=
0
4.210,0,3) =0
The solutions of these diff. equations areEP; (kie,
Mk,
ek3+ i3(10)ek3.
toe,e,
c hie)
ek
Li
r
2 2 where n2+y+ (3.f)
jfor 3>0
(19)
(22)
in
a
so defined .uniform stream will be given byet, itself
(b4
_
elk
as
(1
Az elle (ik,,
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
'
13411
ngist+ g]
n
_{kaie}
011191eld:al
OtOA (AM g ale
fda
B(9112)e ock
4 2
V 0
n 0
hCio144J
=
ole
(i)e
ock
#
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
+10.__Lwa.)(0
=
612
a3
efx
4we have
h
2(kalekweceie)e 4 8(ieeede+120+40,
col
[A
again with the application of the free surface
and lastly application of interface condition
e,
(1, 4 !eel
tua)0
axe.
ag
cloociwe get
j71q(k Ro4
codoi+ B kR &PIO+ v
_ (k
3ko+ euce00)C
;_k; pi IeZe
efornie
(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
I
.....1A!,
.
2,A!,,
,I
Otod 9404 leco0Je
k It Cad 6 4404(Ito efrio) e .6ri
_{(k}
cd.be
k+ kJ &I'M
P4 (kola 4464, Li el &roe) 6

P (keld9k4:1
c4r40)2 0
hi
(25),
condition, namely, 2 ,
k(h i)
C d
 Ro+temco30)e(26)
zn
e
_{2 al}
44.
03
a
20
(23)
( 24 ) using other given211 B 271
( ode +
ize+110049j
gh(h4)
f,(keriok6iitu
ctie)eki
4,
kh
(kW8
44ease)e
p(k eia 9 ko+ eve
6)
utd
9 leo+
etucop) e
(k
citS
ha+
&roe)
1
:o
9
k,%Le/mei14h(k c
( k die+ko kriocirm)
,kh
1 cit:0+ 120+4ucivo
f; (kel4e1A+ i eletdo)
( k
ko+ild co 9)

Itf

kor4)
(kale+ ti:6ucule)e
o
( k
_lay+ Tema)ekg
 ez(k
9ko+ ;to eye)
(29)
(30)
kf
g
(kelt). 64 ko+
arai
ek(khi4kbivI tamale
(31)
4 .
in
(k+ke
RIO
4
istuSLAIRPC rACkze (e2411)02+11uSeahek( I)
Pct)(kkat4104ir Sge)e
_J
k0441 7
277(k  ke
Wei 1 fu 34110 fgeefiKk 444 SIc9)(tri'dkoSietaie(ecoo, )2.1074.14e
_{sea) ikhl}
(32)
.
,
U114) 2.
.4044) /
{02
ko sec2ef IP WV e   (k+kosus4114) stce)e
)
f3= 1ff PO

'
L(e2+1)4)(124 ea sme) (Fr f.) ke We] eiel (P
2 f
1)(k+ tt seci241E" sta)e. kb
k(h ,e)
C
r
(k 4o)
to sae)
((kk0selP
4 ild gee )e

(k
kose49+iesecorek(h4) 1
ii
P (11.12.2tC41e31%6){g172Prii(k44uSerA+ (Pf fr),24C1')
el. (ecrsk+k sec+1 fusers)) e"kh }
(54)
se2e + 1 tv sice)
f
2 k sez2B
(k
 ko
geeze+
Sece)

ko Sicet9
lei Ace)
Using this identity and splitting up small fractions
( 5)
CA
7,7Ai? e
_{271}l
(2hf)
k c,uoe
(
k
gel +
4/ sece )k (3h4)
(
el) ke
sec20 e
itagc9)(f2
)ke
step]
eh*(e,)(ki k
s4c2egeco)
+
(ft"
( k÷ lecSeti 9 +1141
)e k
(311+4)
271
fi(ft+e)(k+
sec69
(f2.
)
stc261ekil+ (f 2e ,)(ki ke
stti94
ace)
ekm34.1e(114)
??7
(92' Pi) ( k + k
0S.tc2 4141 Seco) e
.11(f2tfl)(kf1eiRce) le
tc2e] elett(pei)( 4190%09 4LçU
skee)e(e P)
k: doe
"
+ 2
k
+le Seel pfefi)(katuSeCe)(Pe 91)k SIC
e
(pzF;)(korasele 4trosee6)

k 0141
9),
( Pt
k
 ko
Sec20;
eg sea) e
217
(k4
We) (PrfAS/c2P]
A) OesecV+ Lege"'
k(h1)
_
tri
(40 (k+ko ucze+ifuseci9e
217 if f24c)(kf ituser9)(ece)fresu2oJ
eh+ (e2
)(It + vto We 4 itu sea
C=2f
(k+7e. See9)
k (hf)J
77 tip"
k+
Rai (f2f,)
Rag
et
cfc.e )(k+k,
Ireee+ jesece)
k+;
geo)
stc2cik(111)
17,
,
(kksea+seatc,e)(1z*
X.)01213t sit/jet(P2 Pt)(k +fr. sect 4 t
setajE khi
Now we may write denominator in a different form, namely
fl(feF.)(2i iesea) (ezft)lea
(c r)(k
)26 stz2e + 1 to sace)j
"
=
2 s (d4h4zs4kh)[t (11)tAkh
kdere
_{gele4esece]}
_{1..,h}o+ar ak A)
at=
Li
I 1.
Inserting
in this way found
A(k',9), B.(k,(4), C(k,e)
functions into (2) and,(24) we obtain OS and
ce velocitypotentials.
2 e
a
tr
k.m
pies
1 2di
1 ktice
l
4(3.2h+f)
7;
77LI  ko ;WV + i
im fte0
J+
if
0,t
Liz_aj. k,
viede
_e
dk
 277 4i 00
_{.}_{k El}
tb,413 244in
4
(eAkh
+ae skkh) [ k A.. /
.(_n_i_ir
k /7k°
sd04./e., mei
47
1 4crti, kh
1h
")Lk+
( 1 x ) in
oi,g/
(
(4+
k,
su29 4i euga9)ek1 Vgth +8)]
2n
dikhtarsiik
ClallSk
? .17 oo
I8
0
1+ithkA
le. secs4 II v
gee]
n oo
_
( 1  x )di
(k+k.seepie.slce)ekr1::(33h41)}
4n
_{ft}J0 (4 "th 4' x s411219Ek .
(4i24g/et kw's 4
ziusecef
+
(44 ,yri
I
An
c°
(k_ OA Sic' 0 Pit Allice) ekti dw IT (3 hi ' $ )]
.1
. (cAith+ a° Mk h)L k il a) tfirk.h ko
sta4.1.641
 n
0
4 + ee thiall
2148 1( 1%/i) m ete
(k+ko pit) 4 itar Rai
ek ( iciaki (34 ItIn
it e3
411
a
(aidah+ae g i thh) ik cl glib k h le sec?, 4 Z ea' se.cd
+
1+zelikh
a1
a oo
Lt4
(tit ith ail
1
(edil kh 14 ghki9De k. seee+ibusecejik _
(1x)th kh
e
kiicroe+kcj..3h+en
1+ Xalk 17
110
9
gee,' + ea ;we]
(35)
V 00_{kf(ca43.1}
(4 +MO
shkchf) oik
jib(ciikh4 xsitkii) tk
sec28h
itusec.e)
a, 74
1
f
11 0
4421tiikii
71Go
kLico+(3.11+j)]
(ke&'
r SeCW edk
.rn
j
do
'
(c,iikh 4 at AkbiLk
Sie04/eusic8102 (ix)thkh 120
sod
r
2 2 1
Lx+Y1(34)2,/
1r
L304
(
g 1141)21
(36)
Now we may change coordinate system by Shifting the origin from
interface to the free surface, we are doing so because it shows some convenience
in the calculation of the wave resistance af the surface ships. (Fig. 2)
Putting these coordinates
in
the Above written velocity potentials and. dropping the primes, we Obtain following velocity potentials.00
kjico+(lhnj
+ C1E)
k.11isileolDf_
8AO
2D
_{Calais}
4 at Oithgk
kb
ft
0/ +X
kh
Mb
IV/MI
A 00
k II
0*(3. hf))
4.
(iallni
cis
( k+ ke secl8 + i IA, sece) e cek4n
(cAkh+X Oki))
I k
(12eit#46
b
2 '4
#
0 1 + Je VA khG&W 4cewe]
E)Inicisni Q9
4n
(k+ le, sdeii eu FrA9) ekEloi
LL
cpk+ (3h+1)1(o4k.h
4 cr
Okh)fk (1"kilk.V19+.164/ sec.],
it
0
_{142 ffikh}
,,,nTo
+aAin
fa
(k ke Peg + i & seri) e
k Lelle: (34 h
I))
42 _fi
8
(Cilkil+arki+a,tockit kopett+ .ei,
Seal+
im
1
dB
.I
t,
( k+ koteee +ie41M69 e
k (LI c 3.1. h+ 4
))
4,
akb4 at 94khgk (1a1 t A kh
w
,tr
a
i+avt*kh
k
4 344241+ 41/S4411.7kIi Real!
ir
ek bit .4iile(3 h_j,)3
n
g
00 2n 4
(ak h
a' 0 khgk kebee0+6. esti:011k
(10)akh
vieln %..'11
a
44 jetio0+940.,
+levjetai
(37 )
Where
k C;(84(341)
ler


k.§.11sti
agof
e
01411 _fi
(k ko gee ie
stun+
o,
110
R+4711170f
(Chkh
If 0ft
fk
,s* k C14114Ler S) k,e
kh) (k ke Sec
441 steal it_
kris+
)(
+tece)
3.41`,6; ci;+21 slikh)Lik
(10)6,7 hh
_{1 + a t}_{Ackiimo}
44.6.
a./I
2rt [32+,4 (Pi)
)r =
2cs+
(iz)t4kh
_{kbteco + A "Seen}4+Zakh
Aiwa]
(38)
If we put
4211;
or
Ole=Pret=4/
in thesabove_written expressions of
ite
it is immediately obtained that
t
and
If
each give velocity
in an infinite and homogeneous fluid. Namely
Jr
24%110)443h)]
wi Rada
dk
[
k Pea
i
34.t81N
_{0}
and
potential of a source
oft 4)
I 2(b5= OS=
4 _{2}r
r
.1 M._LS _1$
04 the other hand, if we take limiting valUes,:ofqi and q2 When
fa
becomes 2infinitely large(or ge
1=
0 ) it is found that after some reductions03
r2 4
reduces to shallow water velocity potential and cps vanishes namely
Chs
=
+ M
4 7;iR
a
de
f60
e"`LCAk(3
k ko sfie+;
gzei Mk
)
r ;
eAkh Lk
th(kh)
koSta +4414fa]
le=
0It 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 and4
homogeneous fluid and e2vanishes. Because t coordinate of 41 varies within the interval 
0044
<:_ h
inoe.
All these results are compatible with the2
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 ptbecomes infinitely large, it reduces to free water surface condition, because of being of the source outside of the denser fluidwhich is insensible to
all
kindsof perturbations caused by the light fluid, owing to its infinite density. Therefore it must remain in calm condition and. behave like a solid bOttom.
Free waier
SurfaceSoarce esirengthin
ai (0.0,f)
?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
poles
at.f

k6sci9J
p
'I)
1(Fig. 2)
4=0
k.(141t4kh
k. ?zee
]
11xt4kh
(40)
qe are supposing that
e
changes between the limits of004 I
this may be obtained in the process of reducing the range of integration with respectto 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 usuallyon
as the associated wave number which gives the characteristic wave length in thedeep fluid in the direction of la.
2 n
2T1or 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 socalled internal waves
_{=}
Ui
or using this we can give the speedill co.
and length relation for internal waves, namely
(14)8 A
.64(24")
(c
=
2
71 14
(
2.1e)
(i.7041<A:14e/34w)
Y = s 77) ptoie (Fig. 3)Writing (40) in the following way
2
"h"
(1 ae)k h
h k0_{(1 +ar ihkh)}
I  ae P2e, 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...
.th_L9
0
e4 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} (
t4ae 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
CO
(q)
(
1 + aetA kh)
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
C=
V71
is ther2
highest speed for occurring longest internal progressive waves.
We shall also show that for
a
fixede
Value, equation (40) has only onereal 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
(ale+ ao S.61) SA+ at SWN
(i at)
koh stc28
_{(c.}
s41<)2 ( 12'2) S411
(ii)k0h Stczil
Where (k 4'H).ht=
given expressions. It both (42) and (4;) is not existing. If is Inserting this in (42) Ifct(cA a pa Oa)
14 24«
a
(e444at
Via)
(eik4 ak)((cak,z)
(caafae)a
ae a (
otA a+ae)
(i1)
k
From (46) one may easily observe the following results, namely
I. If
K
root (ofk pea
. 2sin M <0
.2
sin 141 = 0 . 2 sin M >014
s4:112.(aie_ot
s4k)2_
?°,i_atz) saiv
(hk+ mh).
takes the values within the range
if 0
et=och
11 K
a.odh
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 arethe real and positive root of equation (40) we may write,
(1a)
ho hstc2B._
$44h
sAct
d (43) we have
(dee +
gAgiVor
(c/a4af
SAP(
k4at k4k) ak4a0
sam
(cA e
aesigle)2 (1a2) g&VM
4/2 s4;14 2
( clie4
k k )2
(4 ie) P4/2t4
sin2M is solved from the equation (44) we obtain
(44)
(45)
(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 sinM. Dividing equation (45) With
(44) and putting the positive: value of sin2M being the function of variable K.
M(k)
1/2 94422M(k)
(47)
k+a :4
54k+ar
_{s441111.1}If k takes the values within the range
_{adh4;k4nl}
i while sin N remains, ini 2 this range, between zero and unity. One may easily observe .that equation (47) is always a nonequality; namelyMOO
4/2g.412/Y(k)
s42
ti(k)
'"
( a
A4a! g i()
ie '" Ue A
I
e
I.a 'S 10
k
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 thatM has a
real value. Therefore within the intervaleih6(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
024=0
(114),
there is always a real positive root of differentfrom 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
90= oaccod [
(4...
1/2c2`.
In the first quadrant of the complex plane
I=
(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
restriction is placed at infinite distance back of the moving body except that they be bounded. Thus the mathematically Statement of the condition
tifr
°=0
4:
bounded2
a
co 224...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 x9.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 evwoo 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
0
nh
Ni
CO 7.I
de
] de
ft
dsfc
J do
11/2 n/2 nft J°=fc JdO
substituting =91n
_{$4.e}
0j(L.
ci
_
]
=
0
N°°12[1(24(g1)7
def
e
h h. ste8 +
ge49)0
0
Vli Dok (1f)
l' k (1 dee ty ple) i k(a WO yang) 
ik
(a CO''&pasty S414
111 GI CAVI 1 aVIP)f
Sin di E
E le+ g
.4e
+ e
_{}}
_{dk}
6(k.:44 + lestc07
f k k. *4249

i
tu me
j
(48)
substituting9: VI
/1/7 =_fc
],69, =
fc
d91
substitutinge
 9I#
2 'I
Taking one of the integrals of the expression
on
cpekg s
A ) tilt]I 4
14",..,/stet/041r
L 70 c00 K k6sec
41141Sea]
Oa
_{(49)} 0 0separating the real and imaginary parts we get
n/1
k(3/).
44,;
ve3
e do
e
_{svii(kce)atk}
(kk.
s4.269e" , a
[(k
_{k}
_{0}_{stc29)2+ iv' sec's]}
0 _{0} _{0}
0
11/2 op P/2 _{op}
4(3f)
k (34)
4.
iipzipalle
SAi(kee) ca_
i'
A
I1i Ieigdedef
i
e404 (if as ) oa,
0
_{0}
(11 k6
tea)
t°4 8
_{0}
0[(kkage28)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 partsvanish. Therefore, in the following sequence of development we may take the real parts of these integrals if we should think that it is necessary.
s
ee4(kar)
cik+
in"
eze
1
,
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
.44.01
pc111239 di
ek04
'2. me (34)
2 2e gim
kosl4 6) SZC de
0
ONkg gee )24.60 f sec?, 9 j
o
64,,to
irl .22
4° ks4)
,,
k'z2
h.,stc 9 (gS)
fAvisfirec9
cal e(
utd(kit
) out
,.
e,.10
[0a.. kget.30 24. e2sec2 e i
=. n
E e°1 (k. or
Size) stc2ede
o
0
o
using these results and adding together the real and imaginary parts, we get
4/2
Oo ,
11"1342810
eka")) dkiicaj
=
fgel0 dal e
L
i (30/Til
)tu+
167iiff
ekdscla°29" ' + ; "1 '
1f
161
d k _ leegee +i ed Jere] Joo 0 e
w2e)
0
_{0} 0(5i)
By reversing the Sign of i ih the above , we obtain
ilia
,
kRA.4 'jell
.7112.IL[44'hi'ali
,kaSeeet"(411
174
l
11)4 0
.e4444'
f
We de
e
A
...j/r24,64 e
_dk
_ .11./ne
steo ca.
[ k

k6si rtecel
ec
J( b k 3,(0]
0
_{d}
60
_{0}
where 0, (.x004944,400)
Inserting all these results gained in (51) and (52) in (48) we may write
following relation 1 (
i
i
°2ek [
(3 i ) +
a
4 4 + cr
ilme 9 n
E k ko Stc 20
iv Se C 9)
o .0,.. cp .tec'L?Ode
PSI)
I
Pc4114+6Y t ia)k4 ;(xr9
44.15 4°)+?
eisdeft"i
(ja
I
(
_{3c} clk)k11
.7 a)kI
(k ke sec26)
00
110 2k ;e4 0 (j,f)
I
+ 4ft
e s
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 bineand cosine
We may notice that the first integral term on the right hand side of the expression (53) is equal to the following integral
ka3f)4 i
cave 4ys4401
J114c28a2yr e
(k

itth9)
olk
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
w
(3s)4 i(
cooeg4iie9)3
f e
eu,.
_{k kd P470 4 ida Sect .1}
oek
0
1122.gdolek["+i(lud86)1
k
sa_{+"0}
eks..4'.'24(2,1.48))emoseleszdoeloco
ly 0t '61)
_{(54)}
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
_{k}
_{(3=1}
_{171))}
fitjeedel
(c440140 s4 kh
e
k_ (iz)skii
_{k,}
aa
4.30
_{fusta J}
 ll
functions.0
_{14zakh}
004111(..teode+asepi)i
h,f
kti(.x
y$4.10)+(aii4 ))
vied, _e_
fe
(53)
11/41
( a AA +at' S A kh ) [ k_ ( 1 21) Mit h
_{ho gel) 44 Iv ma]}
114
a
140 t 1 1
kh4
stc286/4
eelt 11(xeltdetY$46418)+(3115)3+
'&4u 4,4 o 1$4.410)4(a hni
JIC
0
(chkhlaeSkkh)Lk
1.1(____±.22_:__Akii h,ev42,9I', we]
dk
0
_{i+aetAkh}
(55)
Jfr(hc4i(k)dk4 14n
11'0
Variables, puttingx
.co jaw01 F0)90
ea"0 01N24
where and In the 39 k ter 1,444I
(h) e czik
N
06
goGeE M.<42( +
/V(4+6)
EAl2.464,2) d /VNtaE1
limit when 110, the
as small as We please. Thus al
Vem
r
(k)
:r.,0 IV 24 IN t]. 0 where(k_ (1A9t41211
(44itAkh)
separating its real and imaginary parts, we have
Do op
f(k) p%)1(kal)C14
I
CO
f
44; (k at)
"Asie
ro) eAto(k ea) ea
r
f N24 e2.1
0
'
instead of k we obtain
(40 (c4 AA
84444102[(04ch
ita.k)L (ix)hieget 3EN
f
.+ tsictAt)/
NfrE)
eu
integral takes the Value n , since E may be chosen
2
(cia
÷
gic h) r (a)
11
[( 04P(h + ar got 11)2  (1 ae) A k. stc'e
(56)
zero, except in the narrow
oc _{is the real and positive}
Usingthis
result
we obtain from (56) and (55)_{(57)}
where
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
toto 0 [ AP4 ta]
Because of eu the limit of the integral is everywhere range of the integration where isiE<k<0446 with
root of N(k) = 0. Thus 0(41
oc..£ or)
1,411 to
1.11kLc_ikpoz )di2
_{= tu}
_{+ itm}fr(h)dle 1 r02) di,
_{1. JO}Go
64""
0(10414/1.1
I AP # tu2]
TA171141'
j t s 1 1.4 bu2)Or 00
o
o(41"E
ew
4,f P 0) dh
"4
f Fa) dk
14. , o [N+ pE23

eu>o ( A/21 e02.10
ocEdifferentiating N with respect to k
dN ,..
j (akA 4 x
s4k 1)2 (1x)hk,
sici7
014?I akh +aes4OhJ2
.ejwi fied0
del
tts"'
(col k h
+ ao
Okh)lh
(")g"
Je Itc20 + i to sea]
00
f+agitAith
0
o
eillfhj)+;(2coe+ysieileg
IVA,1
dti
(cAkii4x0101)L ie_ (io)akh
le, seciq
 it
0
4 4 z a lath
Pl2
I
4 A
N*
( c
d 114 i 1
44hre h) 54411 (01.rudOeedCs('14
54:8) gee; de
I? cid h I ay ;4402 (iX)h lee Ste28 ]
80The rest
term given.
from (41, q.)
Prfpre
de
o/2I0
0
44 di
00
irlz op 4.1 ph del
e
ea=
jh
w)
90= extol
)f,ifl
>
04
(58)
of the integrals in (37) and (38) are somewhat similarto those we
have already dealt with, except the last one. Now we take the last integral
by (37). Again reducing the range of integration with respect to 9
to (0, I) we get
°°
k Rs 111)4 I (x e4o1244
ssie)1
I
,e
R [ tf+ i ev me] i Al 4 tem we
j
c"
Where
if
P'4113yf)+ 1(.2. cove k y
k[(3A1)÷1'
e
R(Pf+
tic8J(N4 ifm meg
X Cod 80V $416))
hd')
k[(43h.1) i(ave44,;4e)j
+ e
_{olk}
REM itogrecoJLIVi iA/Peciq
1k
(1.706 kh k fr&26,7
where 11= I
ko %LAI
,14Z6ikh
a41R=(ckh4g,bhJ
Considering the integral tert in k of the following type
,tiv;r7
_{3'}
_{ekE(37/14111/°)}
_{die}
11'oJorti+leiPaHNPitateceJR
separating its real and imaginary parts we have
ook(3_1,_1)
I 11(3, hcf)
.2.1' e
a eel ke0)44 4 I
e
%.44.(kes) g44. + 24:14, ty
01
+Plie.k
ai)
ot
NM R
PIN R
4.° PoR Ltittlfgr N441
0
0
61 414 6
.13
(114/V)e u(k)d44;,,
it(3iie)
4,
e ret4(ko,(Axes,,afk

2 22/
k 0  h  I )
_{)4s4c}
L 11the real and positive root of N(k) = 0. Then
(3hf)
tot
k(11n
.43111t: f 04+
e
szn(km) olA
_{=}
e
com)
R r pr244021[N2.4.e,:21
rk
m1
NI
0
a' o( $E otheti
_{h(3. hf)}
Al3=
e
ko2)
k
we
kw)dk
R MIL N2*1 0411er
+44
R(et
et2.1N
iert
Using previously obtained results
k.rek(3,111)
k.seC29(3hj')
it'll
Ad,p
le
4444(km) dk
?e
szei (kat?) WO ( k stc2e ksec2f)
a
.eu,o`'
_{IZCH.10.11%1}
_{R Cit. Sec269N(ke sawte)}
k, E
I 4 : .1 1 44
c
.14. (k a) ) ct k
= a
g (c4igh+ ws49 thi
Au:,i(kar)
rtEio h 1)
003 h  I)
,
,
444w
AMC N2+ tej
t(c.faA42, 0,411)2 (4 ar) hkes(4))(3c kw))
egg
ki+s
ko StC18 (3_A_ i)
k lahe)
4
n ty
I
0
4.41it...(k.v.ecitiv)
( k) ea
tr ,e
,
$444°
R j1121. 14/2] N
id Cif k
0Sea) )4 (2X I)S4(A le I
.
sea ).1 k. sec?*
kis
ae.+1 4 (3_ h.4.1
rx( hI)
al
tr(
N e 441( k to) ds
2 z _2 % =tt
4/ 671) 0 s4;14 cot co )ei..0
a.: H [ it( *If J
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
.0 A4(3_44)4.10]
_{ik[(3hzil+}
_I
A,;,4
e
cattevad
_{pf.i.atust49.1(Nflid SiajR}
/41
R
at&0
eb)g
(
_{mach)}
in
,

ri nekleciP re4(
secle).4.(2aeI) c(hkesecie)]
acbthi.
Opth)a (14 hk,seep] (91k, $eg
(65)
0CE
24Where
ef.,
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
kosta4k<k.vas4E
andcxEzkzac.t.E .
Here.is
Changing the sign of i in the above expression
&jwIe°
*a J h,f)

c° 4113
tb)
f[
e
f
e
_{ock}
ti 
Nap' IV1 L Pal R
o
/4NR
140 sea L(Sh1)
_{etr(1 h1)}
_{}
_{its]}
e
sitad.
az4i
k
_{(Jh1)}
e
u.4(kez)olk
it
L Nti.euzj
E
1r(k
jaq
4n
since N(.) = 0
( 66 )17/j
oc(3.114)
tip_i
doeh+Jeag
h &44 (10(.2'
h9)ofi (P4
she)
welt
de
077koseci9t(ekd
h+
ay gho(hg 0a9hkosec261]
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.
01= 271

M

kep
2f
[ k  k.
7i 1762+ (3i)J
74077i
Pc 0 de, . 
de
see9J
A  ff0
4 Lie174(1;)]
,./1.
il,
( f _ Al /ad" /(ediklk311/2: aoStCs,h)ke
41/
__1
tAkiiko
_4

0
(nje
1+ a° tA le h ;faP 0:' 2_, A
k fi ce h]

(ia9.in
049(k+ ka.tecw) e
c4k(3.14) cek
4.
211
(eilkh fat ihkA)[ 4 (1a9thkh kugeti9.1
_a
0
I +Er mk4
4_ a. g
fm foie
kk. gezw) e
alk
_,_n f 02
(
7igt ab+(g+hD1
411
fit
0
(cM2h4311.4 kh) Lk_ (41...af)
a k.h ba ?AA 7
'
atm kil
# 00
klieg 4 (3._ iiD3
dk
4
(1:.X)k2Inpeadef,
4
e
411
°
(cii kir+ Z Akh)[ le (  1 ale )
a le
hh
ke frec203 I k k0 gee
 P
o
1+YtAk
4
I6
R/2 ek see° (31.414) we) coo ( Loa pie sal 0 ) Sec26 cier
r a A (i ka seci;)+ (2 2P1) SA Chkd see)]
I°
%a(
h
4. (14)
i
(..(04 1 3 ka SU'  $ 9 ) kcffec A +.1, Six h) S441 0(..7 coal) C41(04 S4419) cle+
f(cAc. fit +aes40(h)2 (ix)hie. seo]
oc,
E ....04 A
(
I
2
ni
jiff
A.190 i.t;
e ',not (14r)10(4fro SIC vj ti..nociv+ 9 t
Alt) 3416(1C449) cedk y sin; 0) di;
j$A(ahfa'S,40(h)2 (130 h ko sizici
4P/20e(jhS)
+ (1x)
e
(c4h er S4 hoc)
Co4S)
(c( g Vim 9 )
ko secig)ReA
h kaeklicc h )2 ( at) ide uzi
go
where CO=
(41CC406q4S44469
Inserting the expressions (65) and (66) into (59) we get
n
1
k ukh.111ito]
if
e
de
1113h.11410]
Amy
rya;
do
664
.f 4
RC tftitolf.A1+ tui
e
dk
stctie
ea
R PIN
4ft
0if.
7112Ito(1 hf)gc28
i
2 I. ste2_
+ n
e
sat (
Az. awl 7 )
co d ( k ;we; g6+4o)
0de+
11/
2
2
ar(3h1)
4.
4 (ix)kern
e
(age
h+ar
shoch)
(oc x c#49) C
(ocgbii, 0)410
(ec
ko Sec2.9) [(chat,
_{$40(h )2 (I at) h ke WO]}
where
ao= 0
C2<4(ia9
67or: Cr)lieCi4/41")
(68)
11 00
_{k[1.1124(j+h))}
As_
ApIdol
k e
sAki)
_{ciA,}
+
5
li
(e(
k 17+ X frolkhnk_ 114th kh je,
sage)
A
0i
14arth
kh
o
02ekiliv+(j+hf))
0i k
kwiskzsciel
+
it
_{(akh+ aosAkh)fk leases] [ k (1x)}
_{Ago}
_{ko Raj}
II
0
_{1+ NM kh}
77/2
kd see° (3+1, .I)
t
t
4X?ri
Al.1
e
S441( /cox Stc6)e#1(keY &we 1419) seats 49 do
G.
I cis (O. sea) 4(2.z /) IA (hke stee)J
h +z
%a ra+h)
2`4
a ft
/
iron!
e
sea) e
 ockocs e_loc(cAtx
9404h)sal(gacodo) 6,d( eq 164).20
(0(  ko sec2 6 91040( h+
_{a° Me cA)2 (1apP1 4 . gee? 0 ]}
90
We may observe that already obtained velocity
_{potentials do not contain}
values.
_{Because they have served their chief}
_{purpose by contributing some}
new terms to the velocity potentials in the course of vanishing
_{process. On the}
other hand if the fictitious dissipative coefficient
_{td, were not introduced at}
the beginning we would obtain with the same
_{sequence of calculations similar}
expressions to the velocity potentials given by (37) and. (38).
_{But only difference}
is the omission of all Lhos in the latter. That is, in order to
_{obtain these}
velocity potentials we should put e=o in (37) and (38). .ks
we may expect these
new velocity potentials satisfy all boundary conditions given
_{on page 8, except}
the radiation condition. That is, in the limiting
process when
DC s + CO wefind that the velocity potentials do not vanish. It is
_{therefore necessary to}
superpose some new terms to the velocity potentials in such a way as to cancel
Out each others at a Very long distance upstream from the perturbating
body.
However, from a close inspection One may nOtice that the ve] °city
_{potentials}
of (68) and (69) are built up of two parts. The first parts of these are identical
to the
velocity potentials obtained without using the
 method, and. Second
parts were deducted in the course of vanishing process of
_{ets. We can show}
this consideration symbolically with the following notations
Ass= 14 1
0
if
ex:0
_{1 }to4141,,,a Os= 03
÷
2 2LU.02 u.r
(69)
> h(1x)
A S
s
Where 97 and
lr.
denote the velocity potentials obtained withoutertp
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.smethod. 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 xaxis.
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 integration with respect to 0 from
(am
to(0,1) we getelk
=
6 e
e
in,
to,kr(34)+1.02) fs; look(34){. i k(ace.la 4. y sen 9) ik (xesoF y $410)
ikke seca93
_{Oa ke See]}
.fl
0
_{0}
_{o}_{(7o)}
In order to define the above given integrals we are considering contour integral of the following type
i
0,1[0:/)1ite]
1.
(t...kesec20
.P _{CbT}
(71)
where is a complex variable equal to (k
4
im) and (z = f)4. 0. The integrand2
has a simple pole at (k =
kosec
e)
on the real axis. Let us first suppose that. . A
a=
(oect49s0181;1q therefore we integrate round the contour shown on (Fig. 4) consisting of positive part of real axis k, a circular arcr
of a radius R which tends to infinity and positive part of imaginary axis. Since the integrand has the pole atk =
k0sec28
on the real axis wecut
out the part within a small circlet of radius r about the pole and then make r tends tozero thus taking the principal value in the usual sence. Because of letting the pole outside of so defined integration contour, On
anywhere inside of this contour the integrand
is finite and single valued therefore r (viePim
plane
according to the Cauchy theorem_{frood1.6}
Applying this theorem to (71) we have
NPole al
imk
k, pea
A 19( ,
co
rka
rk rim
+(
Isf))
,
I kt+re
1 m 444
)1.kilo,
4,(54n
.1027(k.
otrace]
'lb 
r4to
Iki4 r e
"4hi ]t r
e doc .4
g
Iidet
t k
 k.
Pei ]
0ktir
Ph O. .44"
8im i I (secede z x
si"..40)4(3,in
ercideiei$Zs'fadLigo÷(3h)]
ice. *. [t4 ko
Will]
,Reid
1cl4.
I
pm
12.1Pc.)
lam= o
I'
Rel
(72)
Because of accepting 13:1= (
a
C4467Y3449)>0
the integral taken over thecircular arc
r
vanishes since with the increasing radius R corresponding integrand diminishes exponentially to sero. Thus we getOP ,
j ,.k i
1de 4, am]
lim (Jf )
tr)02.1SeciPLOD 4 ;033
( IL k see)
e
ca
e_____L ___Im
+ in e
I'm
41 k kee I
(73)
0 00
0
Reversing the sign of i in the above expression
op ,
rele&i. 04 win f
[ .4 ,r, (A47_
rngo
4
i
evj
dk ,....
e
.thn
...1 II e
.1 11 141 seep]
6;7:77,7iiiroj
(74.:)0
'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 obtainceihb
41 Sii )
7111fril
eoti00di = 4 gado f en
ORYS4dgeolal(31)#41$14 pm tn (.1.1)1 Of*"
[ k Ala Reg]
i 1,121 k? We]
0
0
0
11
/14540ii
I
0
e $41 oho( sarAo ad (As g
seci sim i9 116 Ca
(75)
44"
'Kilda
L k As Stet]
fapf
ice* 4.173
fa= 4/7
CMO.= sea) eel
(kysicie i4:s f ) ,Pci dp
,,
§ 0f)
nit
(76 )
0
0
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