On the fundamental function in the theory of the wave-making resistance of ships

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ÁJCH1EF

MEMOIRS

OF THE

DEFENSE ACADEMY

(Mechanical Engineering)

Vol. IV,

No. 2

December, 1964

PUBLICATION OF THE DEFENSE ACADEMY YOKOSUKA, JAPAN

Lab. y. Scheepsbouwkunde

Technische

Hogeschool

Dem

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Chairnan:

Prof. T. SHINGO :Civil Engineering

Members:

Prof. T. OKuno :Mathematics

Prof. T. MIYATA :Electrical Engineering Prof. K. HÂTA :Mechanical Engineering

Prof. T. FUJIWARA :Chemistry Prof. T. KUYAMA :Applied Physics

Assist. Prof. K. TAKAO :Aeronautical Engineering

Communication are to be addressed to the

Chairman of the Committee

THE DEFENSE ACADEMY Yokosuka, Japan

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Memoirs

of the Defense Academy,

Japan, Vol. IV, No. 2,

pp. 99-119, 1964

On the Fundamental Function in the Theory

of the Wave-Making Resistance of Ships

By Masatoshi BESSHO*

(Received June 24, 1964)

Abstract

The author introduces the functions which play important roles in the theory

of the wave-making resistance of ships, converts their double integral to a

simple one, obtains various integral representations and analyzes their properties especially in the relations of the well-known simple functions.

Then finally, he gives the list of the available tables of his or similar functions.

1. Introduction; Definition and_Differentiation

It is difficult but necessary to compute the fundamental function in the theory of

the wave-making resistance of ships for the development of this theory.

The author has tried to analyze this function and finds its double integral to be

convertible to a simple one, making use of the velocity potential of T. H. Havelock16, but the preceding works2356 were limited to the case of two variables x and t (see the

definition below).

The functions are defined as follows6:

y, t)= hrn

(i)' Ç

Ç' exp.[kt+ik(x cos u±í sin u)] udkdu , (1.1)

4ir jjo

kcos2u-1+picosu

O(2)(x, y, t)= hm

L" "

Ç' exp. [kt+ik(x cos u+u sin dkd , (1.2)

47r

kcos2u-1.ticosu

P,(x, y, t)=--[O,'(x, y, t)O,2(x, y, t)],

(1.3)

Q,(x, y, t)=--[O,»>(x, y, t)+O2'(x, y, t)], (1.4)

where z, y and t are assumed real positive and n integer greater than 2.

All functions are real and they have usually oscillatory parts but O(1) is monotonie for the positive x.

From the definition (1.3), we can easily find that

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}

P2(x, y, t)=(l)"e

sin (x sec u) cos (y sec2 u sin u) cos2 udu

f/2 2

P2,5i(x, y, t)=(l)'1

e sec cos (x sec u) cos (y sec2 u sin u) cos2''1 udu

These are the dircect generalizations of Havelock's P,5 function12'3.

For the negative x, we can easily find also that

O'(x, y, t)=()5O,5(2)(x, y, t)

P,5(x, y, t)(l)"1P,5(x, y, t)

Q5(x, y, t)=(l)"Q(x, y, t)

and then

O,'(x, y, t)=(l)"[O,,'1(x, y, t)-2P,.(x, y, t)] . (1.7) Accordingly, we don't consider the function Q,5(2 in the following, because it is the same one as 0(1) when the sign of x is reversed.

For thenegative y, we have also

y, t)=O1(x, y, t)

P(x, y,t)=P,(x, y, t) ,

(1.8)

Q,,(x, y, t)=Q(x, y, t).

Now, the velocity potential of the unit source in the uniform flow of the unit

velocity flowing from the positive x direction down to the negative at the point (x', y', z')

under the water surface where the z-axis is taken positive vertically upwards, is written

by T. H. Have1ock'6 as follows.

S(x, y, z; x', y', z')=1/(X_x')2+(y_y')2+(Z_Z')2 (x_x')2_(y_y)z+(z±z)2 exp. [k(z+z')+ik(xx')cos O+ik(yy') sin O]dkdo,

'r j.,,jo kcos2 0g+pi cosO

where g means the gravity constant in this unit system. This is written by making use of the definition (1.1) as S(x, y, z; x', y',

z')-i/(xx )2±(yy')2+(zz')

+4gO2[g(xx'), g(yy'), g(z+z')J,

(1.9)

Hence, the most fundamental functions are O and its derivatives.

In this respect, the definition (1.1) is not very convenient but we follow it merely because the author has used it to the present.

Differentiating the definition formula partially, we have

J.

(1.5)

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where

On the Fundamental Function in the Theory of the Wave-Making Resistance of Ships 101

Cr Coo

qzn(x, y, t)= et cos (kx cos u) cos (ky sin u) cos22 u dkdu

47r Jr Jo

q2r1(x, y,t)=

(1

e_letsin (kx cos u)cos (ky sin u) cos2° u dkdu

47r -r o

and we learn from this definition that

a a

qn= --qn--i

Ox Ot

a a a

qn+tqn

at

+ xqn + yqn 0,

ax ay

and especially by integration

tx q3(x, y, t)= 2rp2 -

i

q-2(x, y, t)==

-i-;:'

q,(x, y, t)= 2r(r+t) p' + rt qo(x, y, t)=2r(r+t)2

Here we are to notice that the restriction for the order

introduction of the relations (1.10) and (1.12).

Lastly, we can verify the next differential relations.

And then, combining two equations, we have

p'=t'+y', r'=x'+y'+t',

}

j'

J

n may be rejected by the

}

(1.10) (1.12) (1.13) 02 02 0'

(ax2+ay) Pr

(1.14) r)r1) P

=qn,

Ox (\Ox'

fl-)'

at (1.15) a

-Ox a Ot

O)

Q, j O°) Qn j= O4'L1)

Q')

Qz

-+qi

+qU-2

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(02 02 0

o,»'

-+ 0t2

+

)

Q J = ox (02 \0y2 01' Fig. 1. v-plane / 0 02 02 1

O'

02 0 0

0x4+0x0y2)t

Q

=(+)q,

/0

02 0z 0y2, 2. Integral Representation I.

Let us consider deforming the integral (1.1) in the simplest case n==-1. Considering the integration with respect to u in (1.1), we can write

also itas

y, t)= (o

ikzcosu

cos u du et+kU e . dk

472J,r/z k cos2 u-1+i cos u kcos2u-1+pcosu

If we change the variable k to m as m=kcosu and then u to y as

secu=coshv,

this becomes

emz

O((x, y, t) =-

dv e" cos h (viafl

4rJ-

jo

m+uicoshv mpcoshv

where p__/t2±y2 and a=tan'(y/t).

Now let us consider the integration

with respect to y in the complex v-plane

(see Fig. 1) and deform the path of

in-tegration A to the line B for the above

first term and to B' for the second term. Then, this integral on B and B' goes to

i

du . . . . dm,

47rij_,,

j_m+isinh(u+u)i

and then integrate this with respect to m

in the complex m-plane.

This is a well known integral and vanishes except O<u<ß, and equals

slobti)slob where ß=sinh°(x/p) B

COSti

]dm

Other parts of the integral are residues at the poles lying between the line A and

B and between A and B'.

(1.16)

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On the Fundamental Function in the Theory of the Wave-Making Resistance of Ships 103 Calculating these residues and changing the variable slightly, we have finally

O'(x, y, t)=--

exp. [{p sinh (uilE)x} sinh u] du,

2 L1+L2

where L1 and L2 are the paths of integration shown as in Fig.T2.

but

Oo(x, y, t)=

Fig. 2. u-plane

In the same way, we have generally

O(x, y, t)=

emz

+

mpicosh y

exp. [{p sinh (uia)x} sinh u] sinh 'u du, for n1, (2.2)

(_ ])v-1Ç

2

ji±2

The equivalent formula for the case n=2 and 11=0 was obtained by R. Guillotonl) who introduced it from Mitchell's integral, so that we may see the identity of his one with HaveIock's".

For the case nO, the integrand diverges near the origin, so that an artificial

technique must be used.

Let us consider the case n=0 for example; we can write its integral as:

i

f dv re_mt cosh

vcosh(my sinh v)[+ih

y

47rj_coshvjo

]dm

i

°

siflhVdÇ0_,,,tcosh,(ih)[th

same as the above]dm=O,

4j_coshv

jo

as easily from the symmetric character.

Hence, adding the latter integral to the former and integrating in the same way as the above, we have:

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Oo(x, y, t)=---

ir

exp. [p sinh(uia)x sinh

i(icoSh u)

2 JL1z, sình u

du,

(2.3)

and moreover

i

(icosh u)

O11(x, y, t)=

exp. [the same as the above] .

du,

(2.4)

2 JL1+L2 sinh2 u

O'(x, y, t)=---Ç

exp. [the same as the u+ cosh u sinh° u)du (2.5)

2 JL1+L2 sinh3 u

All these formulas contain imaginary parts but only their real parts are to be

taken.

Now, since we have already the integral representation of P as (1.5), subtracting it from the above formula, we have Q, by the definition (1.3) and (1.4) as follows:

Q_ZU(x, y, t)=(_1)_1'ze_tSin2vcosh (y sin u cos u) cos (x sin u) sin°''u du

rß+i

+--

exp [p sinh (uia)x sinh u] sinh°'' u du, for ni

Q_2_1(x, y, t)_(_1) '2_tsir "cosh (y sin ucos u) sin (x sin u) sin°' u du

(2.6)

----

exp. [the same as the above] sinhsUudu, for n0,

and

Qo(x, y, t)= - in ¿1 cosh (y sin u cos u) cos (xsinu).

(i cos u)

. du

Jo sinu

1 -'-

(icosh u)

+-

exp.[{psinh(uia)x}sinhu]

. du

2.j sinhu

+

CO2u cos (x cosh u) cos (y cosh u sinh u) tanh u du, ... (2.7)

The last one is equivalent to the next formula which is obtained by the integration

of (2.6) making use of the relation (1.10).

r+t \

Qo(x, y, t)=Q0(0, y, t)---log (P+t

Z du

+\ e cosh (y sin u cos u)[1 cos (ísin u)]

Jo sinu

du

[iexp. [{p sinh(uia)x} sinh u.]

h u'

(2.8)

2jo sin

3. IntegraI Representation II.

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On the Fundamentad Function in the Theory of the Wave-Making Resistance of Ships 105

that their integrands converge very slowly in the range for small t and y, especially in the formula (1.5) for the P, function.

For such a range, the following analysis will be convenient for numerical works. For convenience' sake, let us consider the function P_i and other functions will be

deduced by the differentiation and integration.

At first, we have from (1.5) after the substitution of the variable in its integral,

P_(x, y, t)=--exp. (_---t ± [ix cosh up cosh2( u_i--)du (3.1)

where its real part is to be taken.

Since we have the integral

e c0sh2

()__

1_Ç° exp.

2VIrpj_ we can rewrite it as

exP(_fldv

exp.

[ix cosh uiv cosh (u_i-)]du

If we introduce here the new variables

as shown in Fig. 3, that is,

Rcosç'=xvcos----, RsinçL'=vsin--,

2

a

or R=x2±v 2xv cos -,

then we may write

x cosh uv cosh ( u_i--)=R cosh (u+içb).

Thence, shifting the path of u-integration in parallel by J' and carrying out the

integration which gives the Bessel function of the second species, we have finally

Pi(x, y, t)=, e4

exp. (-i) Yo(R)dv, (3.2)

This is suitable for numerical computations when p is small.

When y vanishes, this becomes

P_i(x,0, (3.3)

r

y2

/

[---iv

cosh Ç

ui--)jdv

4p 2 2

which was given by E. T. Goodwin10, and we see that the right hand side satisfies the differential equation (1.15), that is, of the heat conduction.

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Moreover, taking care of this point, we have generally

P(x, y, t)=2 exp.(v2/4t)P(xv, y, 0)dv, (3.4)

From the stand point that these function can be represented in the form of the

solution of the partial differential eqation, we can apply this method to the equations

(1.14) and (1.16).

In the former case, Laplace's equation, there are many formulas but we will not

consider them here.

In the latter case, we can change it to the next equation

72

2

1\

+ d2

+)[P(X y, t)e2]z0 ,

(3.5)

This is analogous to the equation of the diffraction of the wave, but its wave number

is imaginary, so that it will be treated in the same way'.

4. Expansion of P, Function

Let us consider the function P_1 once more in the form (3.1), which equals also

1 t e_+Pc0th (2l_t)

cos (x cosh u)du

P_i(x, y, t)e

Remembering the expansion 23I2)

cos (x cosh u)== (-1)'EJ2,(x) cosh 2nu),

where & means 1 for n=0, 2 for n1, and the definition

du,

we can integrate (4.1) term by term as follows,

P_1(x, y, cos na,

This series is convergent but its convergence is very slow for large value of the

argument (x2/4p), and so for such range the next asymptotic expansion may be suitable.

Namely, we proceed as the above expanding exp. cosh (2u_ia)] in the Bessel

function with the imaginary argument, and can obtain the expansion,

P_i(x, y, cos na. (4.3)

(4.1)

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On the Fundamental Function in the Theory of the Wave-Making Resistance of Ships 107

In general, we can not have such an elegant expression as this, but we can expand it in the Taylor series by the definition (1.10) as follows,

P,(x, y, t) Xtm

Pn_m(0, y, t),

m.

P(x, y, t)=

P2(x, y, 0),

,, m!

The former is convergent but the latter is divergent and an asymptotic expansion,

and the functions in the right hand sides are expressible by known functions in the

next three cases.

P,(x, 0, 0) is the same one as defined by T. H. Havelock'2, that is,

P(x, O, O)=---

Yo(x)dx1, for n0,

P_,(x, 0, Yo(x),

for n1

2 dx

P(0, y, t) is zero for even n because it is odd in x by its definition, namely,

P2(O, y, t)=0 , (4.7)

When n is odd, the simplest case is given by (4.2) putting in it x to zero, that is,

P_1(O, y, , (4.8)

and moreover by the definition

P_2_1(0, y, t)=---(

for n0,

(4.9)

P2_1(0, y, t)=-- e_t/2Ko(-_p)dt ,

for n0

(4.10)

When y vanishes too, the above relations become simpler and we can obtain the next recurrence formula by partial integration of the definition formula.

(n+---)Pzn+i(O 0, t)(tn)P2,_j(O, 0, t)+tP2_3(o, o, t), (4.11)

In another way, we have from (4.8) and (4.9), especially,

P_1(0, O, P_3(O, 0, (4.6) } (4.12) (4.4) (4.5)

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Then we can get all functions by arithmetic.

Finally, it is easy to see that

P21(0, 0,

0)=(_l)1)

(4.13)

5. O,'(x, O, O) and O'(O, y, t)

The functions O'(x, y, t) or Q(x, y, t) can not have such simple series expansions

as in the preceding case, because they have a part expressed by indefinite integral as we see in §2.

Then let us consider here their degenerate cases only. Firstly, getting to zero y and t in (2.1), we have

O(x, 0, 0)= SfILO"du= --[Ho(x)

Yo(s)] , (5.1)

where H0 means Struve's function2328.

By the integration and differentiation, we have directly

Oo'>(x, 0, O)=_- [Ho(z)_ Yo(x)---ldx,

7rxj

O(x, 0, 0)=--+---{H1(x) Yi(x)J

2x

and the recurrence formula

nOu)(x, 0, 0)+(n-1)O2(x, 0, 0)

=x[O1(x, 0, 0)+O(x, 0, 0)+q_i(x, 0, 0)+q%a(x, 0, 0)] , (5.4)

where q, is easily found to be

( 1)' ['(n + )

qon(x, 0, 0)0, qo_i(X, 0,

q_z(x,0, 0)=_21 and q_,(x, 0, O)=oo for n3.

(5.5)

These functions correspond to Havelock's Q function which are defined as'5

IIHO(X) Yo(x)]dx,

(5.6) Qo(x)

Q1(x)dx.

After fairly long calculation

in the formulas (2.3) to (2.5), we can obtain the

following relations.

(5.2)

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On the Fundamental Function in the Theory of the Wave-Making Resistance of Ships 10

_Co+t)E (p+t 2p 2

2

(5.9) 4'

Here Fig. 4. u-plane

e_z

''

2 / 3'\

e du=P)

V(n+3/2)

In the same way but after long calculations, we have also

1 rp+t

Qo(0, y, t)= --i- log [2r(p-f-t)] +,TEo(z)dz.

Finally, if y vanishes too, we have directly from the above

(5.10)22)28) (5.11> Qo(x)=log (2rx)+2Oo°'(x, 0, 0) Q1(x)=z log (2rx)+1x+2Oi°(x, 0, 0), (5.7) Q2(x)= 2

(x2-1)log (2yx)+--± x __x2+2Ozd1)(x, 0, 0),

Here r means Euler's constant 1.78108

-One of the conveniences of using our Q1) function is that each of it has a simple

asymptotic nature and tends to zero when x tends to infinity. Secondly, when x vanishes, we have by its symmetrical nature

O(0, y, t)=Q2(0, y, t)

O' (0 y, t)=P21(0,2n+i x, t)

Since the latter case is already discussed, we will consider the former only. By the integral (2.2), we may write it as

Q_2(0, y, t)= O(0, y,t)___._e_h/2 exp. [--- cosh (2u_i)] sinh u du,

2 L1+L2 2

Here L and L2 are the paths shown in

Fig. 2 but here S=0.

Now, let us deform the paths L1 and

L2 as shown in Fig. 4.

Then, the integrations along the lines parallel to the real axis cancel out each

other and after some calculations we have

Q_2(0, y, t)

cos(_a)e_ttzÇ" e_pcos2vcos y dv

}

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Qo(0, 0, t)=--- log (4yt)±ÇEo(z)dz

Q_z(0, 0, t)=E0(t)

and by partial integration of the definition formula

nQ2(0, 0, t)±(n) Q2.z(0, 0, t)

=t[Q22(0, 0, t)+Q2_4(0, 0, t)+q2,_2(0, 0, t)+qz_4(0, 0, t)} where

q(0, 0, t)=O for n3, q_2(0, 0,

t)=--qo(0, 0, t)=-, etc.

6. Neighbourhood of the Origin

Our functions have - three arguments so that we may lose their general character even near the origin.

Then, let us consider the character near the origin for a moment.

Firstly, considering the function the simplest case n 2, we have from (5.3)

and (5.9)

ix

O(x, 0,0) --- --+-- log (yx/2)+...

O'(0, y, t);-

1(p±t)

+

Namely, it may be finite near the origin because it does along three axises as we

see and this is confirmed in fact by R. Guilloton's table° (See § 9 also).

If so, it is smaller and negligible compared with q_z=-1/2r near the origin so that we may conclude from (4.8) and the definition that

O(x, y, t)= O, y, t)

+Ç {O_2(x, y, t)+q_2(x, y, t)]dx

Jo

log(4/yp)_logÇr+x

i

r(.+X)1

)

--lo[

j,

and then, differentiating it,

t tx t

O(x, y, t)=-O(x, y, t)-q_a(x, y, t)_2r(r+x)

2rp2

2p2 Secondly, let us consider P_1 of (4.2) near the origin.

} (5.12) (5.13) } (5.14) (6.1) (6.2) (6.3) (6.4)

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On the Fundamental Function in the Theory of the Wave-Making Resistance of Ships 111 Since we have28

Ko(--P)lo(4/rP)

K1(--- ) n_1)!1'4_y2, and J2( (x/2)2Th

2 2

/

(2n)!

its series equals nearly

P_i(x, y, t) log (4/rp)+

(-1)

(n-1) !

=-- log (4/rp) - 4p Eo(z)dz. (6.5)

Here its real part is to be taken.

In this formula, if (x2/4p) is small, the integral in the right hand side is also small, but this is not the general case.

Name'y, when x and p are small but (x2/4p) is very large, this integral increases

logarithmically as

Eo(z)dz-- log (4yz), so that we may obtain

P_i(x, y, t) log (2/tx). (6.6)

Xp-.O,

(x2/4p>l

This is coincident with the predominant term of (4.3) which may be valid for such

range.

Lastly, we can obtain in the same way as the above

P_2(x, y, t)z--- log

)

4 p 4p (6.7)

where the real part only is to be taken.

7. Asymptotic Property of O'

Although the function O1) is complicated in nature as we have seen, but it is

fairly smaller compared with the P function, and that it decreases monotonically and has an asymptotic expansion when its arguments tend towards infinity.

Now, let us consider that r=1/x2Hy2±t2 is very much larger than the unity. Returning to the integral (1.1) and expanding its dominator of the integrand as

1

kcos2u-1= (1 i kcos2u-Fk2cos4u-F , ...)

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O1(x, y, t) _[(x y, t) +qn2(x, y, t)+q+4(x, y,

t)+...]

(7.1)

For example, making use of (1.13) we see that

Oo"(x, y, t)E rt+p2 p=Vtz+y2 , (7.2)

2r(r+t)2

O(x, y, t) x +xE3Pz(1+t)+2txz]+ , (7.3)

2r(r+t) 2r3(r+t)3

O(x, y, t)--

2r+

p2(r+t)rx2

2r3(r+t)2 + , (7.4)

which are coincident with the asymptotic characters in the degenerate cases of §5.

The first term of the right hand side of (7.4) is a well-known mirror image term

and was given by R. Guilloton from the observation of his table9 (see §9 also).

8. Asymptotic Property of P_i

In contrast to O,,1 function, P, function takes a comparatively larger value even at the point far from the origin, and it shows a well-known Kelvin's wave pattern which

has been studied by many authors142627.

Here we consider the asymptotic expansion of our simplest function P_1 following

their methods.

Let us rewrite (3.1) as follows,

P_1(x, y, t)=--

eixfdu

, (8.1)

that is,

f(u)=cosh sinh 2u+- cosh2 , (8.2)

and apply the saddle point method2.

To obtain the saddle point from (8.2), the equation

f'(u)=sinh cosh 2u+-- sinh 2uzzzO, (8.3)

must be solved, but it

is easily found that this goes to the equation of the fourth

degree and its solution is very complicated.

Hence, keeping out of confusion we will take up much simpler cases. A) The case y=O

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On the Fundamental Function in the Theory of the WaveMaking Resistance of Ships 113

so that there may be three cases if x is smaller, larger than or nearly equal to

1/gy.

f(u)=cosh u+- cosh2 u (8.4)

2it

(8.5)

f'(u)=sinh u+

smb 2u0,

X

so that the saddle point to be used is the origin. Now, putting where we may write (8.1) as

pf(u)f(0)=f' '(0)f4)(0)...,

(8.6) (8.7) ,

f(0)=l+--P_i(x, 0, t)=eix»0) Ç eixP2/ du \ldp. (8.8)

jo

\dpj

Then, if we expand as du 2 dp f"(0). a2p° namely, (8.9) 27ri J p21

the integration of (8.8) can be carried out term by term and we have

P_i(x, 0, t)= where

exp. (t+ix+ri)

F(n+)

(8.10) (8.11) T/2x+4it ,

(it)

f(0)

a0=1 , az= 2[f' (0)12'

as we get from (8.6) and (8.9).

B) The case t=0

The equation (8.2) and (8.3) go to

f(u)=cosh u--u- sinh 2u,

2x (8.12)

f'(u)=sinh u-cosh 2u,

(8.13)

Then, the saddle points are two, that is,

(8.14)

(18)

where 4

Expanding as

we have finally

o

Fig. 5a. u-plane

this point as follows,

2

f(u2)= tan 6'),

f"(u2)=72 tan 6'e, f' "(uz)= _Te'

(8.16)

f4(uz) ,(1

7e2')

i/ 8 cos 6

Now, let us take the path of integration as shown in Fig. 5a and integrate as

P_i(x, y,

\:e_XPZ ()d

(8.17)

ip2 =f(u) f(uz)

2!1 (U2)+ f' ' V = U - Uz

(i_/ 2i

\dpl

V f (us) =o that is especially

a=

i)

If x is smaller than /j 6' in (8.14)

is purely imaginary and the saddle points

are as shown in Fig. 5a and the point U2

may be used after some consideration.

Putting 6=i6' and

i/f' '(uz)/2i

§

dv 27ri pfl+l i[f3(uz)J2

if'(uz)

a0=1 , a2= 12{f' '(u)] 4[f' '(uz)Iz

i

cosh u2=i/--i/2+e2'

namely,

p=i/5+4cos2E',

sin 26' tan 2o = 2±cos 26' P_1(x, y, O)

e'f"z/2

,

I

i±-+

2 ypsinE L 2x (8.15)

we may write f(u) and its derivatives at

(8.19)

(19)

On the Fundamental Function in the Theory of the Wave-Making Resistance of Ships 115

ii)

When x>Vy,

is real and the

-saddle points lie on the real axis as

shown in Fig. 5b.

This is a well known case in which

there are the diverging and transverse

wave-systems inner the Kelvin angle26.

In this case, f(u) and

its derivatives

-are all real and they are

where

expanding as

e/4/f,,(5)

j a,,p'

i[f' '(u1)12 if4)(u) aoz=1, a2=12{f

'(u)]3 4[f"(ui)]2

then we have

P_1(x, y, O)z

ifj)+lr/4 /

i

(i+-±...).

2xf (ui) 2x

iii) When x is nearly equal to the preceding two formulas give wrong

approximation.

In this case, the most reasonable formula owes to F. Ursell2 and we will follow

his analysis. P_i(x, y, O)=

e2

dz dp dp

v=uui

Fig. 5h. 1 \3J2

cosh3u5( 1+_e21) e/(2cosh E)

cosh2u3 \

(8.21)

f'(u1)=e

,

),

)

3 ±

where j=1 or 2 and the double sign is taken as the upper one for 5=1 and the lower for j=2.

In a usual way, let us integrate along the paths shown in Fig. 5b, namely,

u-plane '1 r (8.22)

}

(8.23) (8.24)

(20)

Let us consider as real and put (8.12) hs

f(u)= +v(8)+),

namely,

[3

2/3 1

__{f(ui) -f(uz)}] , (E)=-

[f(ui)f(uz)]

then the saddle points in v-plane are

where

Fig. 5c. u-plane

that is the Airy's integral28.

The coefficients a0 and b0 are found to be

Now, if we may expand as

ii3

)ldu

P_i(x, y, O)=--r exp.

[x(_-+

vp -H I dv

j dv ire

-

[aoAi(_px212)+.2_Ao(_x213)+ x13

A(z)='cos (-+z)d

7JO du a,(v2p)"+v dv ,.-O (8.26> (8.27)

and take the path of integration as shown

in Fig. 5c, we can integrate as follows,

(8.28)

F

i

Finally, it is easily seen that the formula (8.28) is applicable for the imaginary 8,.

that is, outer region of Kelvin angle in which case p of (8.25) is

negative and the

argument of Airy's integral changes to positive27.

9. On the Numerical Tables

There are many tables of such functions prepared for the object to compute the

wave-making resistance or the wave profile and pattern.

The followings are their list available for us in our notation.

b0J

LVf"(uo)VIf"(u)I j/

2 (8

29)27)

F

(21)

On the Fundamental Function in the Theory of the Wave-Making Resistance of Ships 117

I) a) P(x, 0, 0) for n=0(1)9 and xz=0(0.4)4.4,5(1)40

where the number in the parenthese means the interval of the parameter. with 4 significant figures by T. Jinnaka9.

2 2

Po(x, 0, 0) and {Pi(x, 0, 0)-11

7T 7r

for x=0(0.1)1.0(0.2)10.0(0.4)50.0 with 4 figs., by T. Inui'.

P(x, 0, 0) for n=-7(1)1 , x=0(0.5)2(1)16 with 7 figs., by M. Besshoo).

P%(0, 0, t) for n=-7(2)5, t=0-i0, with 6 figs. and U,(t)-(-1)'P_2,,_1(0, 0, t) for n=0(1)31, t=0-.6 with 10 figs. by M. Bessho6.

---Q0(x) and--Q1(x), for x=0(Q.1)1.0(0.2)10.0(0.4)50 with 4 figs. by T. Inui16.

27r

O,'(x, 0, 0) for n=-2,-1, 0,

x=0(0.5)2(1)16 and O(x, 0, 0)dx and

0, 0) for n=1(1)4, x=0(0.5)2(1)16 with 7 figs. by M. Bessho6.

\dxj

Qz(0, 0, t) for 2n=-6(2)4, t=0(0.1)1.0(0.2)10 with 6 figs.

Eo(t) E0(t)dt for t=0(0.1)1(1)10 with 10 figs. and E(t)=(-1)Q_2_2(0, 0, t) for

n=0(1)9, t=0'-3.2 with 8 figs. by M. Bessho.

JI) a) P3(x, 0, t)

for x=0'-.60, i/ t =0-4.0 with 4 figs.

by National Physical Laboratory, Ma/16/150225.

P_1(x, 0, t) and O' (x, 0, t) for x=0'-49.5, t=z04.0, with 4 figs. by T. Takahei30>.

O(x, 0, t) and the wave elevation by a point doublet along its path, with 4

figs. by Tokyo University31.

P_1(x, 0, t) and O°(x, 0, t) for x=0'-27, t=0-5.2 with 8 figs. by T. Iwata32.

These four tables are prepared for the computation of the wave-making resist-ance and the wave profile.

P(x, 0, t)

for n=-7(1)2, x=O-16, t=0-6 and O,'(x, 0, t)

for n=-3(1)-1,

x=0'-.46, t=0-.6 are prepared for the computation of the submerged body

problem by M. Bessho6.

III)a)

, and

a-i-z

_

g [gJxI gy -g(a+z)

2LV2 v2'

setting g/V2=0.4, for gx/V=0'-20, gy/V2=0-.'4 and -g(a+z)/V2=0--0.8 with 3

(22)

b) ---Zz(q, O)Z(q, O) log (2q)+Oo°'(x, y, O) , for q=2y2O -2O, O=tan'(y/x>

2ir 2

=O-48O° given almost by figures by T. Jinnaka20.

10. Conclusion

The preceding analysis shows that

the function considered is represented by single integral instead of double integral so that the computation may become simpler.

the various limits of the function are considered and related to the known functions as far as possible so that the general feature may be elucidated.

We have the similar work by R. Guilloton in which he showed heuristically and

numerically its property and the extraordinary way of computing the various quantities

of the velocity field around the ship with the aid of his tables, but, mathematically

speaking, his method has some difficulties which we hesitate to proceed with.

Our final outcome must be the same as his and this work might be the second step in the attack on this problem.

References

Bateman, H., "Partial differential equation of mathematical physics ", Cambridge, (1932)

Bessho, M., Report of the graduate course, Tokyo University (1955)

'î , Journal of Zosen Kyokai, Vol. 98 (1956) if Vol. 106 (1960)

, VoI. 106 (1961) , Doctoral thesis (1961)

, Memoirs of Defense Academy, Yokosuka, Vol. 2 No. 4 (1963)

Guilloton, R., Bulletin de la Association Technique Maritime et Aeronautique, Paris, (1956)

, Transaction of Institute of Naval Architects, Vol. 102 (1960)

Goodwin, E. T., Mathematical tables and other aids to computation, Vol. 10, p. 96 (1956)

Hanaoka, T., Journal of Zosen Kyokai, Vol. 90 (1956)

Havelock, T. H., Proceeding of Royal Society, A. Vol. 103 (1923)

if if 'î 'î t, Vol. 108 (1925) if 't 't Vol. 121 (1928)

î, 'î 'î Vol. 135 (1932) 'î î' 'î Vol. 138 (1932)

Hogner, E., 'î 'î i' 'î Vol. 155 (1936)

Inui, T., Journal of Zosen Kyokai, Vol. 100 (1957)

Jinnaka, T., 'î ti 'î Vol. 84 (1952)

, Journal of Seibu Zosen Kyokal, Vol. 11 (1961) or Doctoral thesis (1960)

Lowan, A. N. & Abramovitz, M., N.B. S. Applied Mathematics Series No. 37, Washington,

(1954)

Lohmander, B. & Rittsen, S., Department of numerical analysis Table No. 4, Lund University, Sweden. (1958)

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On the Fundamental Function in the Theory of the Wave-Making Resistance of Ships 119

Magnus, W. & Oberhettinger, F., "Formeln und Sätze für die speciellen Funktionen der mathematishen Physik" Spriger, Berlin, (1948)

Maruo, H., Journal of Zosen Kyokai, Vol. 81 (1949)

Shearer, J. R., Transaction of north-east coast Instituion of Engineers and Shipbuilders,

Vol. 67 (1951)

Stoker, J. J., "Water Waves" New York. (1957) lJrsell, F., Journal of Fluid Mechanics, Vol. 8 (1960)

Watson, G. N., "Theory of Bessel Functions ", 2nd edition, Cambridge, (1952)

Wigley, W. C. S., Proceeding of Royal Society, A. Vol. 144 (1934)

Wave-Resistance Subcommittee of Japan Towing Tank Committee, Pamphlet W2-1 (1960)

i' W3-4 (1961)

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