3hip Form Improvernet
by Use of
Wave Pattern Analysis
In the recent years techniques of direct measurement of the wave-making resistance of ships, the so-called wave pattern analysis, have been greatly advanced and have become routine in the towing tank.
Eggers, Sharma and Ward (1967) reviewed thoroughly the current methods of wave pattern analysis('). Ikehata (1969) also reviewed the recent works in this field done by Japanese
investigators (2)
It has been confinned that, in spite of the differences of the methods of measurement and analysis, the longitudinal cut, transverse cut and X-Y methods give consistent values of wave making resistance Thus it may be said that the method of direct measurement of wave-making resistance has been established.
As far as the application of wave pattern measurements to hip form improvement is concerned, Inui (1962), before establishment of the above techniques, developed a method to obtain a ship fOrm of least wave-making resistance by
cancellation of waves created by a bow bulb and a main hull form (3) To determine the best location of the bulb and its size he used photogrammetrical analysis of wave patterns and wave profile measurements on the hull side.
In 1966 Sharma tried to apply wave pattern analysis to bow wave reduction() and demonstrated later a successful design
of multihulled forms5. In 1970 Everest suggested that
analytical studies of measured wave data indicate regions on the hull where modifications are effective in reducing wave-making resistance (6), At the bow region of a high speed liner form, he calculated
the effect of the addition of small
displacement volume on wave-making resistance of a basic hull. The principle of Evetest's method has been known as Hogner's influence lines (1938) (15) But it differs from Hogner's in that Everest used measured data to obtain the
Dr Engr., Resistance and Propulsion Reseaich Laboratory, Nagasaki Technical Institute, Technical Headquarters
TECIIIIiSCHE UNIVRSflET Laboratorlum voor Scheapshydromechanlca ft,rchief Mekelweg 2,2628 CD Deift TeLO15..788873.Fa O15781e Elichi Baba*
A method to apply wave pattern analysis to the ship form improvement is proposed. The principleofthe method is to find a thin ship which is added to a given basic hull form so as to reduce wave-making resistance. The wave pattern of the thin ship is calculated theoretically and superposed linearly on the measured wave pattern of the basic hull form To find the wave making characteristics of the basic hull form Hogner s idea of influence function is employed The influence functions are calculated from measured wave data
instead of theoretical calculation Therefore they inform more realistic values of modifications than those by a pure theory only
Further, a method of empirical correction for the finite breadth effect of the given ship on the thin ship has been developed. Taking an exampleofship form improvement, it is dernoijstrated how the present method works ef.ctively in reducing wave-making resistance under appropriate desIgn constraints such as constant beam and displacement.
1. Introduction
wave-making characteristics of a basic hull forms while Hogner uses theoretical values.
Since 1965, in the Mitsubishi Experimental Tank the wave pattern analysis has been a routine procedure applied to ship form improvement. The method developed by the present author is also based on Hogner's idea in essentials. The differences from Hogner's are the following two points.
Measured wave pattefns are used to calculate the in-fluence lines of a basic hull form, viz., the more tealistic information of modifications to the basic hull form can be obtained than by a pure theory only.
Optimum modifications to a basic hull form are deter-mined by means of calculus of variations. This method can be applied not only to the optimization of sectional area distributiOn but also to the bow bulb design and to the partial modification of ship forms. From a large numbe? of applications to high speed liners and container ship forms, it has been confirmed that the preseiit method is quite powerful in reducing wave-making resistance under given design constraints.
2. Theory
- The linearized wave-making resistance theory developed by
Michell (l898)() and Havelock (l932)(8) are usually valid only for thin ships of large length-beam ratio, say, 38 or 20 as
shown by Weinblum, Kendrick and Todd (l952)(), and
Sharma (l969)('°)
It is known that for conventional ship forms which have the length-beam ratio from 6 to 8, the Michell-Havélock theory gives a poor esthnate of wave-making resistance. On the other hand, if we observe a wave pattern far behind a slip, it may be safely said that the wave height is sufficiently small enough in proportion to the wave length for the linearized theory of surface waves to be applied Therefore in thi paper it is assumed
that wave patterns
far behind aship can be
superposed linearly. In fact,
the higher order theory of
Iwave-making resistance developed by Maruo (1966) suggests that the velocity potential far behind a ship has an asymptotic form which satisfies the linearized free surface condition(h1)
The non-linear effects appear only in the terms of wave
amplitude. Then the asymptotic expression of wave pattern does not change the form from that of the linearize4 theory. Therefore it may be said that wave patterns far behind a ship can be superposed linearly and wave-making resistance de-términed from the measured wave data includes the non-linear effects. Thus for conventional ships, the wave pattern analysis is more useful to determine the wave-making resistaiçe than
direct application of Micheil Havelock theory
Next, let us consider a slight modification to a basic hull fc.rtn to reduce wave-making resistance, viz., a very thin ship; for instance, of length-breadth ratio 150 with negative or positive ordinates added to the basic hull form. The wave pattern of the thin ship is calculated by Michell-Havelock
theory while the wave pattern of the basic hull form is
determined by experiment. When adding this thin ship to ordinary ships, one must notice that there exists a certain degree of velocity deviation around the basic hull surface from the uniform flow since L/B ratio is not large enough That
is the velocity deviation àrrnind the basic hull form would
affect the thin ship in such a way thit the amplitude and phase of the wave pattern of the thin ship wOuld differ from those calculated by Michell-Havelock theory. On the other hand it may be assumed that the similar effect Of the thin ship on the basic fOrm is negliglbly small, since the LIB ratio of the thin
silip
is so large. In the section 4 a method of empirical
crrectio
for the phase and amplitude of the wave pattern of tIe thin ship is described In this section however for the simplicity of discussion, the wave pattern of the thin ship is calculated by Michell-Havelock theory and superposed linearly on the measured wave pattern of the basic hull form.It is assumed that the fluid is incompressible and ideal. A ship is floating in a unifOrm stream U. Fig. 1 shows the
coordinate system with the ongin at F P of the ship in still water. The xaxis is in the direction of the uniform streamand
z.axis directs vertically upwards.
Far behind a ship the wave pattern can be
expressed asymptotically( by,r12
o
(xy)J
S( 9)sin(x,y) + C( O)cos(x,y) dO...(1
where
x,y): wave height Of a basic hull form measured at the point (x,y)
SC O),C( 9): amplitude functions determined by wave patteri analysis
sin(x,y) sinxosec29(xcosO+ sin9)
cos(x,y)=cos xosec2O(xcOsO+sinO)I
x0=g/U2
g :acceleration due to gravity
Then the wave-making resistance is calculated by
R,,0 rpVf[S(0)2+. IC(0)12]cos39d0
(2)
where
wavemahng resistance of a basic hull form
-LI
0z
Fig. 1 Coordinate system
p density of water
On the other hand the wave height of a thin ship is calculated by Michell-Havelock theory, vz..
r'2
, (x,y)=/
a(9)sin(xy)+b(0)cos(xy)}d8 ...(3)
,r12
where
1(x,y)wave height of a thin ship,
a(9) )
2x
L ro 2b( 9)5
OJdxJ 77( x,z ) exp( x0z sec 0)X
x{C?S(xoxsec8)dz
(4)
17(x,z) half breadth of a thin ship at the point (x,z) which
corresponds approximately to the doublet
distri-bution with the axis directing opposite to the
dirCction of the uniform stream
L : ship length
d : ship draft
By the superposition of on , the total wave system far behind a ship becomes
(x,)=lo(±;y)+
t(x,y)1 [IS(9)+a(8)}sin(x,y)+ {C(9)+b(0)}COS(x,Y)]d0
Then the wave-making resistance of the total wavesystem is expressed by
Rw=,rpU2fHS(9)+a(9)l24(9)(8)2]C0S39d0 (6)
By the substitution of (4) into (6), the alternative expression of wave-making resistance becomes
R,= R, +P-1-2°f dxA( x)f dx'A(x' :f( 14 zds&0)z sec9X xcos1x0(x-x')secOl dO dxA (
x)f '
dsec 0) cos 9[s( 0) cos(x0xsecO) C(0)sin(x0xsec9)]d8(7)
where AC x) = 2 17( x) dIn this expression It is assumed that the sectional area of aship plays an important role on wave-miking resistance and the infuence of the frame line form can be neglected(i3)(14) In other words the thin ship is assUmed to be a wall-sided ship,
i.e. 17(x,z) = '7( x).
For convenience of calculation the following non-dimen-sional expressions are introduced
C =R,,o/4pU2L2
A() =A(x)/A
F: Froude mumber = UI JL midship area
beam ofa ship
Cm: midship area coefficient
Then the wave-making resistance in non-dimensional form can be expressed by
c =c,,
fdeAcfAe'Ke_nde'
+4Cm (-)f1eGe)de
(8)
where
K(e_)=f'(1_e_isec
)2 secO cos {-(-e')secO
(9)
)cosO[(0)cos(secO)
(8)sin (secO)] dO
(10)C,, , the first term of (8), is the wave-making resistance
coefficient
of a basic hull form and the second is the
wave-making resistance of a thin ship itself, and the third is the interaction term between them.
It is the main objective of the present investigation to find the optimum value of A( ) which gives the least wave-making resistance with proper constraints such as constant displace-ment and/or constant beam.
Hogner (1938) neglected the second term of(8) as a higher order quantity when improving hull forms and called G(e) "influence lines" (15) After his work several papers related to the idea of influence lines have been published (16)(20)
Here, however, all the terms of (8) are treated and the method of calculus of variations are applied. It is needless to say that G( which is called hereafter the influence function according to Bessho (18) is calculated from the measured wave data, while Hogner and others calculated it
in a purely
theoretical way.
To understand the nature of a thin ship which reduces wave-making resistance, a specific problem is considered here, viz., the problem to optimize the thin ship with the
displace-ment of a basic hull fOrm unchanged. The condition of
constant displacement is expressed by
Using Lagrange's multiplier A, let
Setting the variation of ,. due to small change of A() be
zero, an integral equation for A(c)is obtained.
(f)2fA(
e' )K(ç -' )d
MTB85 May 1973
(11)
At first, existence and uniqueness of the solution of this integral equation must be examined. The kernel function
K( e - c ) is rewritten as follows.
K( -
f/2
secOcos{ F2 )secO }o+f2e_8( seC'8
2)secOcos {F2 (- e' )sec8}dO
=fY. (--2i-eI) +L2e'° (e"'
2) x
x
secOcos{(e-')secO}d9
This kernel function includes terms of Bessel function Y0 and a regular function. Therefore the kernel function
K(e-)
becomes singular at = since y,, ( e-e'i) has alogarith-mic singularity at = '. The Fredholn type integral equation of the first kind with kernel function Y. has been proved to have a unique solution by Dörr (l95l)(21). Bessho (1963) studied the characteristics of the solution in details('8). A numerical method to solve the equation was shown by Karp, Kotick and Lurye (1960) (22) Although the present kernel includes a regular function besides Y,, the singular property does not change. Therefore the author considered that the integral equation has a unique solution.
Next, it is worthwhile to study the nature of the solution for understanding the mechanism of optimization.
Let A(e)=A1(e)+A2(e)
where A1 ( ) and A2 ( c ) satisfy the following integral equations respectively.
(12)
,r
"'\L) F,'
fA2(e)K(e-e')de'=1 G()
Cm ()
(3)
The right hand side of the equation (12) is constant and
independent of the measured wave data. This integral equation
is equivalent to that of minimum wave-making resistance problem with constant displacement (18) The solution of the problem gives the optimum sectional area distribution with cons tant displacement. Denoting this by (e), then
(14)
On the other hand the right hand side of the equation (13) is expressed by the influence function G(e) which is calculated from the measured wave data. Then the solution of (13) has a direct relation to the wave-making characteristics of a basic hull form. Let A () be an effective sectional area curve of the basic hull form which can be obtained from measured wave
data.
Siniilary to formula (4) which shows the relationship between a ship form and amplitude functions, the effective amplitude functions of the basic hull form can be expressed by
(o)) 1
sec8 1 'B
r'
O(1-e
)2(jjCmJ Aex
(cos
X
By the substitution of 3(0) and (0) into (10), the alternative
0.0 15 0.019 0.005 0.0 014 C, = JPU2V 0.16 0.18' 0.20. 0.22
expression of the influence function is obtained as
)de' (15)
After the substitution of.( 15) into (13), the integral equation becomes
f2(e' )K(c4' )d
-j'A(
e' ),K(e- e)de'Since this integral equation has a unique solution,
A2(e)---Ae(e) (16)
Thus the solution of the original integral equation (11) is determined, viz.,
(17)
This means that the optimum thin ship
is the difference between the sectional area curve of the minimum wave-making resistance with constant displacement and the effective sec-tional area curve of a basic hull form as obviously expected from the physical meaning of the problem. In other words, the thin ship is added in such a way that the effective sectional area of a basic hull from becomes the optimurn one.F,,
Fig 3 Results of towing tests and wave analysis of M25, M27 and M29 M29 M 27
---M 25 0.24 0 Lb 0 __D-__C,,, from wave analyw
0.26 0 M2c ..' 0.28
7
0.30 0323. Numerical examples of infitience functions
To understand the nature of the influence function vhich plays an important role in the optimization problem, the numëribal calculation of the influence functions are carried
out for thtee ship forms selected from Maruo's series of
minimum wave-making resistance ship forms(23) The selCcted ship forms have the same values of prismatic coefficient 0.60 and are designed at F = 0.250, 0.267 and 0.289 respectively. The sectional area curves of the three fOrms are shown in Fig. 2. Some part of the aft ends are modified to suit the conventional ship forms, although the theoretical values are symmetrical to the midship The frame line forms are also
made to resemble the conventional ships.
-Fig. 3 shows the towing test results of those ship models denoted by M25, M27 and M29 respectively. The particulars of the models are shown in Table 1. The residual resistance was calculated by use of Prandtl-Schlichting's formula of frictional resistance. The stud type turbulence stimulator is used at the square station 9½ ( = 0.05). Comparison of the residual resistance coefficient curves shows that M25 designed at F,, = 0.250 arid M27 designed at F,, = 0.267 .are actually giving small values at the design speed. However, the hollow
3.0 2.0 1.0 0.0 X iO F.P- 0.1
0(E) Influence functions at F0267
M25
---Fig. 4 Comparison of influence functions of M25, M27 and M29 at F = 0.267
Table 1 Particulars of models used
pOint of M29 is shifted a bit to the lower side of the Froude number.
In this figure the wave-making resistance curves obtained by wave pattern analyses are also shown. The equivalent singulari-ty method developed by the author (24) was used for the analysis. It is quite interesting to notice that each ship fOrm derived by the minimum wave-making resistance theory gives actually the least value of wave-making resistance at the corresponding designed speed. It is evident that the results of wave pattern analyses tell the wave-making characteristics more clearly than that of the towing tests.
Next, the influence functions of the three models are
calculàtedat F = 0.267 and compared in Fig.4. The influence function of M27 designed at F 0.267 is locating in the middle of those of M25 and M29 and gives small values in the region 0.2<c<0.7. Taking into account that the three models have the conventional stern forms, then excluding this part, it may be said that M27 has almost approached to the optimum form, since
the magnitude of the influence function is
small(18). At the fore end part, however, M27 has rather large values. This implies that Maruo's optimum forms still have some room for improvement.
The present numerical examples indicate that the influence function can be used as a method to judge the wave-making property of ship forms.
4. An example of ship form improvement
In this section an example of ship form improvement by the method mentioned in the previous sections is shown.
.0
As described iii the section
2, phase and amplitude
correctiOns for the effect of the non-uniform velocity distribu-tion around the basic hull surface, which may come mainly from the effect of finite breadth of ships and from viscous effect as secondary, on the wave pattern of a thin ship are considered in the followings.
Instead of the expressions (3) and (4), the expression of the wave system of the thin ship is taken as f011ows.
,
(x,)=f[ai(0)sin(x,Y)+bi(0)cos(x2/)]dO
(18)where
ai(0) )
-2x
1L fo??(xz) uuuen'Or= -sec4O jdxj
-. e Xb1(8)i + r
Ju J-a h(x)x?s
[x0lx(x')}secO]dz' ...(19)
By the expressions (18) and (19) it is assumed that the effect of addition of a small amount of volume (x,z)dxdz
is reduced by 1/h(x) and shifted toward F.P. by 6(x) because of the non-uniform velocity distribution around the basic hull form. Although h(x) is determined from the ratiO of both original and effective sectional area curves of a thin ship by solving ititegral equations corresponding to (15), it has been confirmed that in practice the ratio of both theoretical and experimental values of influence functions of the thin ship gives good approximation for h( x), where the experimental
value of the influence function is determined from the
difference between the influence functions of the basic hull form and a hull form which is modified by adding an arbitrary thin ship to the basic hull form. 6(x) is also determined from the comparisons of both theoretical and experimental values of influence functions of thin ship. In our experimental tank-h( x) and 8(x) have been prepared for the routine practice with respect to varous ship fOrms.
Then the total wave-making resistance coefficient corre-sponding to (8) is written as 2
2/B\°1
I.'r'
Cw=Ce+CruJ)
jdje
J d h(e') XxK[e-5(e)-e' +8ç')]
+4Cf--G[eo(e)]de
(20) LIF,ij,,h(e)For the clear demonstration of the effectiveness of the
present method, M29 which was originally
designed atF,, 0.289 was chosen as a basic form and the optimum
sectional area curve of a thin ship was determined under the
MTB85 May 1973
5
M25 M27 M29 M29B
Load condition Full Full Full Full
Lpp (m) 4.200 4.200 4.200 4.200 LWL=L (m) 4.284 4.284 4.284 4.284 B (mm) 626.86 626.86 626.86 626.86 d (mm) 250.74 250.74 250.74 250.74 (m°) 0.3892 0.3828 0.3807 0.3802 ç'i (m2) 0.5331 0.5272 0.5253 0.5248 S (rn°) 3.330 3.305 3.297 3.305 Cb 0.5896 0.5798 0.5766 0.5760 0.6078 0.5977 0.5944 0.5938 C,,, 0.9700 0.9700 0.9700 0.9700 Trim 0 0 0 0
1.0
0.8
0.6
0.4
02
Fig. 5 Comparison of sectional area curves of M29 (Basic-form) and M29B (Modifiedform)
0.3 0.4 0.5
following design conditiOnS. Designed speed: F =0.250 ConStant beam
Constant displacement volume
Instead of solving integral equations directly A( ) is expanded
by the limited number of cosine series
with unknowncoefficients
A()=ccosie
.. (21)The conditions of constant beam and displacement are written by
cs(O.5i)=0, ate-=O.5"
(22)csjn(i)/i= 0
(23)Letting the variation of C. be zero after the substitution of (21) into (20) with the constraints (22) and (23), a simultane-ous equation for c1 (i = 1, 2,. .. , N) is obtained. Besshoand Maruo showed that although the solution of minimum
wave-making resistance forms under the condition of a
constant beam does not exist, an asymptotic solution does exist for conventionalspeed range(F <0.35) when expanding A( e ) by a finite number of series of eigen functions (18)(23)
It is also known that the solution of the integral equation
022 024 026
--Fig. 6 Results of towing tests of M29 and M29B
treated in the section 2 diverges at the fore and aft end with the order °(e (18) Here, however the error due to the expansion (21) is assumed to be small, since it can be thought that the singular property is already included in the basic hu form. By the numerical studies it was found that a stable solution is obtained when N is around 11.
Fig. 5 shows the optimum modification to the basic hull M29 at F 0.25 0.
Towing tests and wave pattern analysis of the new ship form called M29B were carried out. The residual resistance coefficient curve is compared with those of the basic hull form M29 and M25 which is designed at F' = 0.250 by Maruo's theory. It is
observed in Fig. 6 that at the design speed
(F = 0.250) a large amount of reduction is obtained by the present method. And M29B gives a little better result than M25 in the region below F = 0.250.
In Fig. 7 the comparison of measured wave patterns is shown. It is evident that the wave heights of M29B are reduced in the range 4m-< x<12m. Fig8 shows the results of wave
pattern analysis. A large reduction of amplitude spectrum is attained by M29B. Around C = 50° M29B gives smaller values of amplitude than M25. C,, valiAes of M29, M29B and M2-5 are
shown in Table .2. From the table
itis found that 68
0.015
0.0 10
0.005
00 028 0.30 0.32
3
2
0
40
8
Fig. 8 Comparison of amplitude spectra of M29, M29B
and M25 at F = 0.250
reduction of wave-making resistance is attained by M29B. And M29B gives a little better result than M25.
Fig. 9 shows a comparison of the influence functions of M29, M29B and M25 at F = 0.250. It is observed that the influence function of M29B gives the smallest values in the
2 0 3 E 0 F P.
0(r,,) Measured wave patterns
x G (E) Influence functions at F=0250
M 29 M296
w
V5"
'-
V
x Y l.5m 10 at F,=o.250Fig. 7 Comparison of measured wave patterns of M29 and M29B at F = 0.250
60 70 80
Fig. 9 Comparison of influence functions of M29, M29B and M25 at F = 0.250
Fig. 10 Convergence of optimum modifications
15 - 20
Table 2 Results of wave pattern analyses of M29, M29B and M25
MTB85 May 1973
whole range.
Further, for the examination of convergence of hull form, the optimum thin ship was calculated again taking M29B as a
basic hull form. Fig. 10 shows the comparison of the optimum thin ships for the both cases of M29 and M29B. It is found that the amount of remaining modification is quite small. From the practical standpoint it may be said that M29B has already been converged. From the author's experiences of application to high speed liners and container ship forms, it
has been confirmed that the convergence of ship form
improvement in the present method is fairly well.
From this example of ship form improvement onecan see how the present method works effectively in reducing
wave-making resistance under the given design constraints. 5. Conclusion
The studies in the previous sections are summarized as
follows.
M29 1.104 x 10-k 0.00380
M29B 0.35 1 0.00123
M25 0.366 0.00126
A method to apply wave pattern analysis to ship form improvement is proposed. The principle of the method is to find a thin ship which is added to a given basic hull form so as to reduce wave-making resistance under the assumption of the linear superposition of wave systems. The wave pattern of the thin ship is calculated by Michell-Havelock theory while the wave pattern of the basic hull form is determined by experiment.
To find wave-makirig characteristics of the basic hull forms, Hogner's idea of influence function is employed. But the influence functions are calculated from the results of wave pattern analysis. Therefore more realistic informatiOns
on the magnitude and location of modifications are
obtained than by a pure theory only.
A method which corrects the effect of the non-uniform velocity distribution around the basic hull form on the phase and amplitude of the wave system of the thin ship has been developed. Employing this correction method in calculus of variations, it is demonstrated how the present method of ship form improvemnt works effectively in
K.W.H. Eggers, S.D. Sharrna and L.W. Ward, An Assessment of Some Experimental Methods for Determirung the Wavemaking Characteristics of a Ship Form, Trans. SNAME, Vol.. 75 (1967) p.
112.
M. Ikehata, On Experimental Determination of Wave-Making
Resistance of A Ship Japan Shipbuilding & Marine Engineering
Vol. 4, No.5 (1969) p5.
T lnui Wave Making Resistance of Ships Trans SNAME Vol 70
(1962) p.283.
S.D. Sharma, An Attempted Application of Wave Analysis
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S D Sharma Der Wellenwiderstand eines fiach getauchten Korpers
und seine BeeinflussUng durch einen aus dem Wasser
het-austragenden Turmaufbau, Schiffstechnik Bd. 15-Heft 78 (1968) p. 88.
J.T. Everest, Suggestions for modifications to a high speed liner form based upon wave pattern measurements, N.P.L. Ship Report
137 (1970).
J.H. Michell, The Wave-Resistance of a Ship, Phil. Mg. Vol. 45 (1898) P. 106.
T.H. Havelook, The Theory of Wave Resistance, Proc. Roy. Soc. A, Vol. 138 (1932) P. 339.
G.P Wemblum I I Kendnck and M A Todd Investigation of wave effects produced by a thin bodyTMB Model 4125, TMB Rpt. 840 (1952).
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8
References
reducing the wave-making resistance at a desined spec with proper constraints such asconstant beam and displace-ment.
The principle of the present method can b applied to the design of bow bulbs, multi-hulled ships and to partial modification of hull surfaces.
6. Acknowledgment
The author wishes to express his deep apprciation to Dr. Kaname Taniguchi, Director and Manager of the Nagasaki Technical institute of Mitsubishi Heavy lndusties, Ltd., for his suggestion on the possibility of optimizing the longitudInal distribution of displacement volume by means of wave pattern analysis. In 1965 the author started to develoj, the present method. Since then, he has been encouraged and Stimulated through valuable discussions with Dr. Taiiiguchi and Dr. Kyoji Watanabe, Vice Manager of the Nagasaki Technical Institute. The. author also wishes to express his appre4iation to all
members of the Mitsubishi Experimental Tank for their
cooperation in carrying out this investigation.
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RINA, Vol.111(1969) p. 487.
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Phys..Bd. 3(1952) p.427.
S Karp J Kotick and J Lurye On the problem of minimum
wave resistance for struts and strut-like dipole distributions 3rd Symp. Naval Hydrodynamics (1960).
H Maruo Problems relating to the Ship Form of Minimum Wave Rasistanôe, 5thSymp. Naval Hydrodynamics, (1964).
E. Baba, Study on Separation of Ship Resistance Cdmponents,
NumeriaI Evaluation of a Singular Integral
Where
f( ) is a regular function
K(P)jYo(IpI)+R(p) ...(A-2)
R( p) =f_.seC9 (jSec8_2)cos (#2sec o)sec 0 d8 (A-3)
Determining the optimum thin ship by the method mentioned in the section 4, one must carry out the numerical evaluation of the singular integral (A-I). In this appendix a method of numerical evaluation employed by the author is shown.
By the substitution of (A-2) into (A-i), the following
itegral is obtained
I(e)=
fJ(e')Y0[2Ie_5(e)_e +o(e')I]d'
±f')R[e-()' +(e')}]de' ..(A-4)
Since the R is a regular function, the second term of (A-4) can be calculated numerically by Simpson's nile without any special treatment. Then our problem becomes how to evaluate the first term of (A-4), since Y,, has a logarithmic singularity
at
_o'()_' +&(' )= 0.
The method employed by theauthor is as follows.
First, the integral range [0,1] is divided into(N-1)parts,
viz.,
= 1 (A-5)
For the routine practice N = 51 is chosen. Then let us consider the integral
II(K)=
'
) YO[IcK-(K)-e'
+ )IK=1,2,3, N
...(A-6)
(1) When K =1, (A-6) is rewritten as
, 1
L1ea(1)c'8(E)
]de' (A-7) Since the integrand of the second term has no singularity in the range e 2 ' , this term can be calculated by meansOf Simpson's rule. For the first term of (A-7) the following approximation is employed. d c
f(e'
Then we have Appendix X-'1(E2-H1-a
OL F,+(ç )Id
(A-8) Where Yi'(p) =fPy0 (t)d.When K= N, the same approximation can be applied, vi
H(eN) 1f( CNI ) +f( e I 1::n2(
)
xy;i[( Ev-i)Ii_'(Cwi)I]
tN-I
1
f(e' ) Y[J-5(CN)-C'+(E')I]dC' (A-9)
When K=$= N, K* 1, the integral is divided into three parts as
II ( C) = (ftK_f
th*f
tN)
ti tx-i tK-f-i
-x v0[-2Ie-&ce)_e'+ (
For the first and the third integrals Simpson's rule can be applied. For the second integral J(C) is approximated first by
Lagrange's expression as
(CK_i-CK)CKI-EK+i)
f()
(CK-CK-i)CCK-CK+I)
(CKl-CK-i) ( CK+I-CK)
and then the function Y, is approximated as before, viz., I]- y [i_5"(Cic) ICKCI] After these approximations the analytical integration is carried out in the range C< Then the following evaluation for the integral II () is finally obtained.
/ fEx-i N - i +
I
)fe')
YO[-2K_o(eK)_e'o(e')I]de' +2f(C_)S3(K) f(C) [-4s3(K) +S1(K)] +2f(C+1)S(K) (A-il) where S1(K)=i-'(E) '[
F, JrF,1[d1-'(e)I
S3(K)=-411)1I2
{..y;l[d1_5(CKfl MTB85 May 1973 (A-b)10
y0 and Y1 are Bessel functions.
