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
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
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+picosuO(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
}
P2(x, y, t)=(l)"e
sin (x sec u) cos (y sec2 u sin u) cos2 uduf/2 2
P2,5i(x, y, t)=(l)'1
e sec cos (x sec u) cos (y sec2 u sin u) cos2''1 uduThese 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 thenO,'(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)
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 dkdu47r -r o
and we learn from this definition that
a a
qn= --qn--i
Ox Ota a a
qn+tqn
at+ xqn + yqn 0,
ax ayand 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)2Here 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 OtO)
Q, j O°) Qn j= O4'L1)Q')
Qz
-+qi
+qU-2(02 02 0
o,»'
-+ 0t2+
)
Q J = ox (02 \0y2 01' Fig. 1. v-plane / 0 02 02 1O'
02 0 00x4+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 (viafl4rJ-
jom+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)
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 coshvcosh(my sinh v)[+ih
y
47rj_coshvjo
]dm
i
°siflhVdÇ0_,,,tcosh,(ih)[th
same as the above]dm=O,4j_coshv
joas easily from the symmetric character.
Hence, adding the latter integral to the former and integrating in the same way as the above, we have:
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)
. duJo sinu
1 -'-
(icosh u)
+-
exp.[{psinh(uia)x}sinhu]
. du2.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.
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 Çui--)jdv
4p 2 2which 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.
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
21\
+ 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)
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)
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)
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
}
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)]dxJo
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)
2rp22p2 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)
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)2Th2 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 , ...)
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)rx22r3(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
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 p21the 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)
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 6Now, 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 especiallya=
i)If x is smaller than /j 6' in (8.14)
is purely imaginary and the saddle pointsare 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)J2if'(uz)
a0=1 , a2= 12{f' '(u)] 4[f' '(uz)Izi
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)
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 dpv=uui
Fig. 5h. 1 \3J2cosh3u5( 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)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 dvj dv ire
-
[aoAi(_px212)+.2_Ao(_x213)+ x13A(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 (829)27)
F
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 and0, 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)
, anda-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
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.
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