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OCR for page 133
Numerical Evaluation of the Complete WaveResistance
Green's Function Using Bessho's Approach
H. T. Wang
Naval Research Laboratory
Washington, USA
J. C. W. Rogers
Polytechnic Institute of New York
New York, USA
Abstract
This paper presents a numerical method for calculating
the total Green's function for the wave resistance case of a
source in steady translation below the free surface. Starting
with a representation of this function in the complex plane
given by Bessho a series of transformations of variables are
used to reduce it to three real single integrals. The
integrands are regular and are entirely in testes of elemen
tary functions. Two of the integrals are even in the direc
tion of flow while the remaining integral is odd. The even
and odd integrals may also be conveniently expressed in
terms of the near and far field components. While the
method is applicable nearly everywhere (except for the well
known difficulties at the limiting cases when the source is
located near the free surface or the field point is close to the
source), computer time varies in the computation domain.
A computer code has been written to implement the
above method. Sample calculated results are given in
several forms to show the accuracy and computer time
requirements of the code. A series of line and contour plots
are given to show typical shapes of the integrands at dif
ferent locations in the computation domain as well as to
exhibit the relative behavior of the various component
integrals.
1. Introduction
For a number of decades, interest in the important
Green's function for a submerged nonoscillating source mov
ing at constant forward speed in fluid of infinite depth has
largely centered on its far field characteristics. This was
both due to the greater simplicity of the far field evaluation
as well as its direct applicability to finding the hull resis
tance component due to wavemaking. The far field wave
pattern required only the evaluation of a single integral with
a regular integrand over a onedimensional wavenumber
space while the complete Green's function involves the addi
tional calculation of a double integral with a singular
integrand over a twodimensional wavenumber space. The
applicability of using only the far field analysis to obtain the
wave resistance was aided by the pioneering work of
Michell [1] who showed that reasonable estimates of the
source strengths modeling the hull surface for the case of
thin ships could be obtained directly from ship geometry,
without need to ascertain the near field mutual influence of
the sources. The landmark paper by Eggers, Sharma, and
Ward [2] presents a comprehensive survey of different
methods of using the singleintegral far field Green's func
tion to obtain the wave resistance.
In recent years, with the availability of ever larger and
faster computers, interest has been enlarged to include the
near field terms of the Green's function as well as a more
accurate calculation of its far field behavior. An evaluation
of the near field terms would give a more accurate determi
nation of the source strengths for hull forms which do not
conform to thin ship theory as well as a more detailed defin
ition of the flow field and pressure forces on or near the
hull. In the far field case, the use of modern day remote
sensing technology makes it of interest to assess the ship
wake for wavelengths which are significantly shorter than
those applicable to the wave resistance problem.
Efforts at rendering the initial double integral represen
tation of the Green's function amenable to numerical calcula
tion usually involve expressing it as a series of single
integrals. Noblesse [3] gives several alternate single integral
representations. A popular form is to express the Green's
function as two single integrals consisting of a near field part
N which is even in the flow direction x, and a wave distur
bance part W which is defined only downstream of the
source. The near field part N has an integrand which is in
terms of the higher order derived exponential integral func
tion. Noblesse [4] and Euvrard [5] have conducted detailed
studies of the behavior of N and W. especially at limiting
regions of the computation domain.
In terms of actual numerical implementation, Newman
[6] has developed a procedure for calculating N in terms of
extensive sets of tabulated coefficients of Chebyshev or ordi
nary polynomials. The coefficients take on four different
sets of values, depending on the radial distance R from the
source. In the case of the wave disturbance part W. Nob
lesse [7] and Newman [8] have developed procedures for the
specialized case of the vertical xz centerplane while Baar and
Price [9] implement a more general calculation for the entire
domain with the exception of the region near the free sur
face. In these studies, the computation region is again
divided into several regions depending on the values of x/z
or x/v y~ + z ~ . In many cases, the solution is in terms of
133
OCR for page 133
Image Sink
_ (X,Y,  Z)
Free Surfaced 0 Free Surface
(x,y,z) ~ /—
Field Pod
Y ~
~ x 
.(X,Y,Z) Current U
Source Point
Figure 1. Definition of Coordinate System and Flow Configuration
Go=—km dd dk
or ~—o —~ O
exp [—k(z +Z) + ik (x—X) cos ~ + ik(y—Y) sin 8] (2)
k cos2 ~—1—incise
In the above equation, all length variables have been made
dimensionless by multiplication by the factor g /U2 where g
is the gravity constant. Go represents an inverse double
Fourier transform over the twodimensional k—
wavenumber space. The parameter ~( > 0) is added to
ensure that the radiation condition is satisfied at infinity.
Also, following Ursell, the terms (x—X), (y—Y), and
(z + Z) appearing in Eq. (2) are simply replaced by x,y,
and z for the sake of convenience.
2.2 Concise Statement of Bessho's Approach
The usual way of simplifying the double integral
expression for Go is to perform an initial integration over k
or 8, thus reducing the double integral to a single integral.
The integrand, however, is not entirely in terms of elemen
tary functions but contains the higher order derived exponen
tial integral function E~(u) defined by
~ e&
E~(u)=i d'A (3)
u ~
2.
a series of functions, and there is often a delicate balance
between the regions of convergence of near field and asymp
totic expansions.
Our paper presents a numerical implementation of a
representation of the Green's function developed by Bessho
[10]. By means of a series of ingenious transformations, he
succeeds in reducing the entire Green's function to a single
integral along a curved path in the complex plane. The
integrand is regular and is entirely in terms of elementary
functions. This remarkable representation has been noted by
previous investigators [4,6] and has prompted Ursell [11] to
give a more complete derivation, including the justification
of an important intermediate step. Our work differs from
previous numerical implementations in several aspects. Our
work is entirely in terms of integrals as opposed to the use
of expansions in series. Our implementation is for nearly
the entire computation domain and for the complete Green's
function as opposed to specialized domains or particular
parts of the Green's function.
The paper starts with a concise statement of the critical
points of Bessho's contribution. Then, a more complete
description of the derivation, as given by Ursell, is given in
outline form. One reason for giving this derivation is to
point out the starting point for our numerical work, which
occurs before the final single integral representation is
reached. A detailed description is then given of the transfor
mations required to convert the original complex integrals to
three real integrals, two of which are even in x (Ge) and one
is odd in x (Go). The simple relationship between Ge and
Go and the commonly used N and W is pointed out. Pecu
liar features of each of the three integrands are discussed,
and the procedures used for their integration are pointed out.
A special limiting process is used to obtain Ge and Go on
the axis directly downstream of a source at the free surface.
The variation of computer time requirements and accuracy
. . .
Of the calculated results in the computation domain are dis
cussed. Numerical results are presented to check on the
accuracy of our procedure and to illustrate in graphical form
the behavior of the integrands and their integrated values.
The paper concludes with a summary of the principal find
ings.
2. Derivation of Bessho's Single Integral Representation
1 Initial Representation in Double Integral Form
In this work, we will consistently follow the notation
used by Ursell [11]. Figure 1 shows the coordinate system
used in our work, where x corresponds to the direction of
the current flow U. y is the horizontal direction perpendicu
lar to x, and z is the vertical direction positive downwards.
For a stationary nonoscillating source located at (X, Y,Z) the
Green's function G(x,y,z; X, Y,Z) at field point (x,y,z) is
given by
G(x,y,z) = Rid — R + Go (1)
where Ri = [(x—X)2 + (y_ y)2 + (z_Z)2]~/2
R = [(x—X)2 + (y _y)2 + (z+z)2]~/2
In Bessho's approach, the initial integration to reduce
the double integral to a single integral is not performed at
the outset. Instead, after a series of transformations of the
double integral, including changes of variables, deformation
of the paths of integration in the complex plane, expressing
Go as derivatives of more convergent integrals, and inter
changing order of integration, the double integral is finally
converted to single integral form by means of the following
crucial equality
I ~ exp (imP) dm (4)
_= m —Q
Pi exp (i P Q) (P > 0, Im Q > 0)
—Sari exp (i PQ) (P < 0, Im Q < 0)
0 (P Im Q < 0)
134
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where P is real and Q is complex. Thus, the order of
integration is reduced entirely in terms of elementary func
tions. The following section gives a summary of Ursell's
detailed derivation and justification of Bessho's approach.
2.3 Summary of Ursell's Derivation
Using the equality
1 cos ~ d ~ = 1 0 —cos ~ d
rewrite Go in Eq. (2) as the following sum of A and B
Go (x,y,z) = A(x,y,z) + B(x,y,z) (6)
where A and B are now integrated from —~r/2 to +'r/2
over 8.
By introducing the following two sets of new variables
m = k cos 8, tanh v—sin ~ (7)
y = p sin cY, z = p cos cY, x = p sinh,B, R = p cash ~ (8)
and writing A and B as
A = Ao + i at Al, B = Bo + i aX Be (9)
the following expressions are obtained for Ao, Bo, A i, and
Be,
Ao =— dm  dv exp[—m p cash (v—ice) +imx] (lea)
Bo =— dm  dv exp[—m p cash (v—icY)—imx] (lob)
A 1 = 1 km I dm I ~ do exp[—m p cash (v—ion) +imx]
7r `o O = m—coshv—is
(lOc)
Be= slims dml" do exp[—mpcosh(v—in)—imp]
7r To 0  ~ m—cash v +ie
(led)
The reason for introducing the new integrals Al and Be is
that they are more strongly convergent than the original
integrals A and B.
The double integrals Ao and Bo are simplified by
Ursell by making the change of variables v—ice—v, and
using equality (9.6.24) of Ref. [12] to convert the integrand
in terms of the Bessel function Ko
Ao + Bo =—lo dm KO(mp) cos me
2 2
= , =
Vp2 + x2 R
Actually, the integral Ao + Bo can be more directly
simplified (without the need to use Bessel functions) by not
ing that the order of integration can be interchanged since it
is absolutely convergent, carrying out the simple integral
with respect to m, and making the change of variables
w = ev in the resulting single integral to arrive at the same
result given above.
By making similar shifts in the paths of integration of
Al (v = for + 2 Hi + w, w real) and Be (v = ion—
2 Pi + w, w real), the sum A ~ + B ~ takes the form
Al + B.` = —km  dm  dw
7r ~—o —~ —co
exp (—imp sinh w + imp sinh A)
+ (single integrals
m—~ s~nh(w + fix)—ze
due to residues at poles crossed by the shifts) (12)
By interchanging the order of integration, Bessho notes
that the integral with respect to m is of the form given in
Eq. (4) and thus the double integral is converted to a single
integral entirely in terms of elementary functions.
Ursell points out that the double integral in Eq. (12)
does not satisfy the sufficient (but not necessary) condition
of absolute convergence in order to justify the interchange of
order of integration. His proof of the validity of the inter
change basically consists of extending Bessho's idea of gen
erating derivative functions, shown in Eq. (9), to generate
the following still more strongly convergent integrals A2 and
B2
A2 = 1 lime dm5 ~ dv exp[—mpcosh(y—ict) +imx]
To 0 = (1—i)(m—coshv—ie)
(13a)
B2= 11iml dam do exp[—mpcosh(v—in)—imp]
7r To O = (l+im)(m—coshv +ie)
(13b)
which are related to A ~ and B ~ as follows
Al = (1 — at) A2, BY = (1 — at) B2 (14)
Ursell proceeds to operate on A2 and B2 in a manner similar
to Bessho's transformations of Al and Be. That is, once
again the paths of integration are shifted for
A2 (v = for + 2 Hi + w, w real) and B2(v = ice—
2 pi + w, w real), and the resultant residue evaluated at
the pole of the integrand crossed by this shift (for some but
not all m) where the pole V is defined by
135
OCR for page 133
m + it for B2 In either case, Eq. (17a) is expressed as the following sum
cash V(m) = m —it forA2 (15) of two single integrals
By making the further change of variables from m to V, as
given in Eq. (15), in the single integral resulting from the
residue evaluation, the following expressions are obtained
for A2 and B2, where each is given in terms of a single
integral and a double integral
A2 = —km dm I°° dw
7r~o 0 1 —im =
exp (—imp sinh w + imp sinh I) 16
m—i sinh(w + ICY)—it ( a)
—2i 1 1= dV exp(—pcoshVcosh(V—icY)+ixcoshV)
2 ~` 1—i cash V
BY = ~ km I dm I dw
~ so 0 1 + im =
exp (imp sinh w—imp sinh A) (16c)
m + i sinh (w + in) + it
2i 11(~ Or ) dV P( P cash Vcosh (VZCY)—ix cash V)
(16d)
where cY is taken to be—0 in Eqs. (16b) and (led).
Ursell shows that the double integrals (16a) and (16c)
are absolutely convergent, and thus it is permissible to inter
change the order of integration. By writing —m for m in
Eq. (16c) the two double integrals can be combined into a
single form, with order of integration interchanged
1 1= dWI" dm exp(—impsinhw+impsinh~g
7r  ~  ~ (1—im) [m—i sinh (w + ice)]
= ~ I dw l(w,O (17a)
The presence of the extra factor (1—im) in the denomina
tor makes the equalities given by Eq. (4) not directly appli
cable for evaluating I(w,,8). Instead, Ursell evaluates it by
means of an elaborate contour integration, accounting for the
residues due to the poles which are enclosed for O
:0' 2 i]
['25
(is, in)
O
—\
L0~ 2 iJ 1 ~
Figure 2. Bessho's Integration Paths
at 1 Hi co+ 1 Hi
Go (X,y,z) = 2 50 + Ib+i
In the following, we express the above complex
integrals as the sum of real and imaginary parts, and con
sider only the real part. By means of a series of transforma
tions, we reduce the resulting integrals to compact form.
3.2 Conversion to Real Integrals
Consider first G1 given by Eq. (24a). By first expand
ing sinh (w + ice) and then making the change of variables
cosh w dw = dw', Go becomes the following
G ~ = 2 5 [cos ~ ~ cos(x sin ~ ~
—yw A) + sin ~ sin (x sin ~ ~
—yw ~w2 + 1)] e xcosa w + zw2 dw (25)
Let us now consider G2 and G3 given by Eqs. (24b)
and (24c). By using the well known relationship
e+'XC°st = cos (x cos 0+ i sin (x cos 0, the real part of
these two integrals can be conveniently written out. By
making the change of variables ~ = —if' in G3, G2 + G3
can be written as the following single integral
_x
G2 + G3 = 2 1 2~ +a cos ~ cos (x cos Me PCostcos(~a)d; (26)
By making the change of variables ~ = if' + cx/2, using
various trigonometric identities for the cosine terms, keeping
only the even functions of ~ (since the odd functions give a
zero integral), and (very similar to the final transformation
indicated for Eq. (25)) making the change of variables
cos (d; = dw, the integrand of G2 + G3 takes on a form
resembling that of Go given in Eq. (25)
a
G2 + G3 = 41 2 [COS (X COS (X COS a! it)
. CX
sm
cos(xsin~2 w)—w :2 sin(xcos 2—A)
0` P (cosa+1  2w2)
sin(x sin 2 w)]e 2 dw (27)
Finally, let us consider G4 given by Eq. (24d). By
making use of the identity cosh (V—ice) = cosh V cos
ax  i sinh V sin cz the real part of G4 is written as
00
G4 = 4 1 cosh V sin (x cosh V)
cos (p cosh V sinh V sin cx) e P cosh2 vcos a dV (28)
By making the change of variables cosh V dV = dw and
recalling Eq. (8), G4 becomes
G4= 4lo sin(x~)cos(yw~)ez(w +~)dw (2
By making the further change of variables dw = sec28 d8,
we can transform G4 to the single integral form given in Eq.
(13.3b) of Wehausen and Laitone [14]
G4 = 4 1 sin(xsec28cos8)cos(ysec28 sin8)e z see dsec249db
(30)
In our numerical computations we find it somewhat more
convenient to make the change of variables sec28 sin ~ = u
and use the following alternate form proposed in [2]
G4 =410 sin(xs)cos(yu)es Z Z~ du (31)
where s(u) = [ 2 ~ = see 8.
Adding the component integrals given by Eqs. (25),
(27), and (31), the resulting form for Go in real form is
given by
Go = GX + Go + Go = Ge + Go
= 2  [cos cx ~~ cos(x sin ax
—yw A) + sin cx sin (x sin cx ~
—yw ~)]e xcos`Yw+pcos
The integrals Go and Ge are also simply related to the
commonly used near field and wave disturbance components
N and W [4,6,9]. Comparison of the expressions for Go and
W shows that
W(x, y, z) = Go(X, y, z) + Go(~X ~ Eye Z) (33)
Since W + N = Go + Ge, N is related to Ge and Go as fol
lows
N(x, y, z) = Ge(X' y, z)—Go( AX I, y, Z) (34)
3.3 Behavior of the Integrands
It is of interest to investigate the behavior of the three
Integrands in different regions of the x, y, z computation
domain in order to derive effective integration schemes.
While the behavior of the integrand for the odd integral Go
is relatively well known, we do not believe that the
Integrands of the even integrals OX and GC have been previ
ously considered.
Figures 3 to 5 respectively show the Integrands for Gx,
Gc, and Go at various (x,y,z) locations. In these figures,
the horizontal and vertical coordinates have been normalized
so that the independent variable lies between 0 and 1, and
the integrand lies between 1 and 1.
To ~ _
Z . 6
¢
llJ
Z . 2
L
o
111  . 2
J
OCR for page 133
The error incurred by using a finite upper limit UM in
the integral for Go, Eq. (32c), may be estimated as follows.
By neglecting the trigonometric factors, which have magni
tudes less or equal to 1, and changing the variable of
integration from u to s, defined in Eq. (31), the error ~ is
bounded by
~ S 4 1s es Z s ds (37)
where SM = 2 + I. By making the change of
variables t = s2, the above may be integrated by parts to
give
z `~t + z Issue Z ds (—) ds (38)
where d ( s ) = d (52 _ 1)~/2 c 0. Therefore, it is
ds u ds
necessary to consider only the first term in deriving esti
mates of the maximum error incurred by using finite values
of UM. By considering ~ and z to be given parameters, Eq.
(38) may be solved iteratively for the required upper limit
UM. Table 1 gives the calculated values of UM for
~ = (103, 104, 106) and z = (10., 1., 0.1, 0.01).
Table 1 shows the manner in which UM increases with
decreasing values of ~ and z.
Table 1—Variation of the Upper Limit of
Integration UM in Go with Error Criterion
and Source Submergence z
~ 1 Z=10 1 z=10 1 z0~1 1 z=0.01
103 0.01 6.2 77. 880.
104 0.085 8.4 99. 1100.
106 0.60 13.0 140. 1500.
3.4 Limiting Behavior as p—O
Use of the above error estimate appears to indicate that
UM ~ °° as z—0. Yet it is well known [4,8] that
Go(x, y = 0, z = 0) is given by
Go(x' 0, 0) = 2 W(X) = 2~7rYI(x), x > 0 (39)
where Ye is the Bessel function of the second kind of order
one [12]. The existence of the limit shows that a higher
order error analysis, accounting for the oscillatory tri
gonometric functions in Eq. (32c), would lead to a more
efficient procedure for the integration of Go.
In order to obtain the complete function Go for
(x, 0, 0) it is necessary to calculate Gx and Gc defined in
Eqs. (32a) and (32b) as y—0 and z—0. Since Go exhi
bits singular behavior in the neighborhood of this axis, it is
necessary to use care in taking this limit. By starting with
Eq. (29) for Go, and neglecting terms which are well
behaved near the axis p = 0 and which are of the order of
140
1/ fix ~ as fix ~ — no, we have derived in [13] the following
expression giving the first order far field behavior in the
neighborhood of the axis (x, 0, 0)
GO(x,y,z)~sgn(x)2:exZ/4(Y +Z)sin( ~ Y 2
+ ~ + ~)+2~(sinx+sgn(x)cosx)eZ (40)
This equation generalizes, for the case of large x, the
expression given in [5] for the wave disturbance W for the
specialized case z = 0. The above equation confirms the
findings in [5] that the limit z = 0, y—0 is singular, and
that (provided z ~ O) the limit y = an, z—O is indepen
dent of a. Since the near field component N is well
behaved, Eq. (34) shows that Ge must show a similar
behavior.
We find it convenient to compute the values of Gx and
Gc for y = 0, z—0. In this case, ~ = tan~(y/z) = 0.
The integral Gc may be calculated in a straightforward
manner by setting z = 0 and then evaluating the well
behaved integrand between the finite limits O and 1. In the
case of Gx, the limiting process must be applied withsome
care. In particular, taking care to preserve the previously
noted symmetry of the exponential factor about the midpoint
of the integration interval x/2z, the integral Gx has the fol
lowing limit
km Gx (x, 0, z) = km 2 5 w e xw+zw2 dw
z—o z—O 0 ~/~
= 1im25x/2z ~ w + x/z—w ~ exwdw
Z° ° L~ ;(x/z w)2 + 1 ~
50 ~~e_Xwdw+2 (41)
Thus, Go may be conveniently calculated on the axis
(x, 0, 0) by adding Go, Go, and Gx.
3.5 Computer Time Requirements
Computer time depends on various factors, of which
the values of x, y, and z, the error criterion I, the type of
integral, and the integration rule are the most important.
The simple trapezoidal rule and the higher order Simpson's
rule were used to integrate each of the three integrals over a
range of x, y, z, and c. The integration starts with an initial
partition of the given range into two intervals, and then suc
cessively doubling the number of intervals until the
integrated values from two successive iterations agree to
within the specified error e.
We have found that on average, the use of Simpson's
rule for Ge(=GX + Gc) results in a computer time which is
0.3 of the time using the trapezoidal rule, while the
corresponding ratio for Go is 1.5 [13]. These trends may be
due to the fact that the (usually) smoother integrands of Ge
benefit from the parabolic fit through the points used in
Simpson's rule, while the (often) oscillatory integrands of
OCR for page 133
Go seem to be better approximated by the straight line fit of
the trapezoidal rule. We realize that, based on the diverse
behavior of the integrands shown in Figures 3 to 5, the use
of different integration rules (for the same integral) based on
x, y, and z would result in even greater savings of computer
time. We have not, however, pursued such detailed refine
ments.
Figure 6 shows the average CPU time per field point
for Gx, Gc, and Go, respectively, using the Hewlett Packard
(HP) 9000, Model 550 minicomputer for ~ = 104 for vari
ous intervals in the region 0.1 c x, y, z c 40. It is well
known that for values of x, y, z near the origin (0, 0, 0) and
the axis (x, 0, 0) the behavior of Go is singular. For a given
xinterval, the figure shows two sets of numbers. The upper
set gives the average CPU time for a fixed value of y (indi
cated on the vertical axis) and a uniform xz grid of field
points over the interval indicated on the horizontal axis.
The lower set of numbers gives corresponding results for a
fixed value of z and a uniform xy grid. Several runs were
repeated using c= 106 and these suggest that average
CPU time is approximately tripled compared to the
~ = 104 results. Also, some runs have been made on the
Cray XMP/24 mainframe computer and the corresponding
CPU times are typically 60 times smaller.
O..
1.0 _
N
>I
10 _
40
z: (.02, .02; 1.0)
(.01, .0s;  )
( 01, .10; .10)
(.02, .10; )
(01, .25; .02)
(.01, .25;  )
x
o 1 y (.02, .05; _) 1 0 (.05,.30; ) 10(.20,1.3; )40
1 .
(.65, .20; 2.3
_ (.20,1.0; )
(.35,.20;.20'
(.20, .40;  )
(.20, .45; .02)
_ (.10, .20;  )
(.10,.10; 1.5)
(.05, .25;  )
(.10, .10; .15)
(.02, .20;  )
(.02, .25; .02)
_ (.02, .20;  )
(01, .25; .01) (.01, .30; .01) (.10, .45; .01)
Figure 6. Approximate Computer Times
per Field Point for ~ = 104
Most of the computer time trends can be directly
inferred from the expressions contained in Eq. (32) or
shown in Figs. 35. One trend is that the overall sum of the
CPU times for all three integrals tends to be a minimum
around (1,1,1), which may be inferred from the smooth
behavior of all the integrands at this point. Another trend is
the increase in CPU time with increasing x since this leads
to increasing oscillatory behavior of the integrand. With
increasing z, computer times decrease for Go (larger effect
of the exponential decay factor) and GX (smaller values of
x/p) but tend to increase somewhat for Go (more confine
ment of the exponential factor to a narrow region near
w = cos(cx/2)). It may be noted that in the analysis given
in [9] for W. which is closely related to Go' the calculation
is limited to values of y /z less than tan (86.4°) = 15.9.
4. Numerical Results
We first present various checks which have been made
on the accuracy of our formulation. We then present con
tour plots of the various integrals to show their overall
behavior. Finally, we give line plots of these integrals to
show the finer details of their individual and relative
behavior.
4.1 Numerical Checks
Newman [6] gives accurately calculated benchmark
values of N(x,y,z) for the field points (R. 0,0), (O,R, O),
and (O,O,R) where R = 1, 4, 10. The last two sets are
most convenient for us to calculate and we shall discuss
these first. For both of these cases x = 0, where Go = 0,
and Eq. (34) then shows that N _ Ge. In addition, since x
= 0, then x /p = 0, the integral GX —O. reducing the cal
culation to N= Gc. For an ~ = 0.0001, our calculated
values [13] agree with those given in [6] to at least four
decimal places.
The set (R. 0,0) provides a more thorough check on
the accuracy of our formulation since all three of the
integrals Gx' Go, and Go would be involved. Unfortunately,
our upper limit of integration for Go— ~ as z—0. We
perform this check in two ways. One way is for us to calcu
late the three integrals and form N according to Eq. (34) for
small values of z. For an ~ = 0.001 and z = 0.01, we
obtain agreement to at least two decimal places with the
benchmark results [13]. While this method is computation
ally expensive and inefficient, it does show the stability of
our formulation as z—0. The second and more convenient
method is to calculate N(x, O. O) with the aid of Eqs. (39)
and (41) for Go (x, O. O) and GX (x, O. O), respectively.
Using this approach, our results agree with those in [6] to
six decimal places for ~ = 106.
4.2 Contour Plots
Figures 7a to d respectively show contour plots for the
component integrals Gx, Gc, Go' and N for the case of z =
0.1 over a 101 x 41 rectangular grid with 0 s x s 50,
and 0 s y s 20. To illustrate the effect of z, Figures 8a
and b respectively show Ge form = 0.1, and 1.0 for the
same rectangular grid.
Figure 7a shows that the largest waves Of Ox are con
fined in a triangular region which does not extend to the x
axis. Figure 7b shows that the wave pattern for Gc is rela
tively simple, consisting of waves whose crests are nearly
parallel to the y axis. Figure 7c shows the familiar far field
wave patterns confined to the Kelvin sector. Figure 7d
shows that the near field integral N has contour lines which
are nearly circular and asymptotically decay as 2/R.
Figures 8a and b show the well known disappearance
of the clutter corresponding to the short waves as z
Increases.
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OCR for page 133
/
~# ~^ a =
Figure ah. Conlour Plol of at, z = a.
^ Z = ~
Figure ad. Contour Plot of ad, z = 0.1
Figure 8a. Conlour Plot of at, z = 0.1
~ ~^ z =
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OCR for page 133
4.3 Detailed Line Plots
Figures 9a and b respectively show line plots of the
component integrals Gx, Gc' and Go at y = 0 and 4 for z
= 0.1 and 0 s x c 20. To obtain a view of the behavior
of the integrals over a wider range of x, including upstream
values, Figures 10a and b respectively show plots of Go at
y = 0 and 20 for z = 0.1 and—100 c x c 100.
3 .
~ . O
; . O
i.0 N\/';
3.0
G x
G c
 Go

\ / \\ /
\ /
0.0 4.0
8 . 0
A X I A L D I S T A N C E X
Figure 9a. Line Plots of Go, Go, Go; y = 0
5.0 _
Z 3.0 _
o
_
Z 1 . 0 _
11
U) _
Z : .0 _
3: _
G x
G c
G o
~_~_~\ ~ ~ ~ yN
16.0 20.0
Figure 9b. Line Plots of Go, Gc' Go; y = 4
5. 0 _
Z 3.0 _
o
L)
z
11
u)
z
I1J
TIC
1.0 _
I 1 .0 _
3.0 _
AXIAL DISTANCE X
Figure lea. Upstream and Downstream Line Plots of Go, y = 0
z
z
u)
1 . C
LL
c:
3.0
1.0
3.0 _
5 .0
60 .0 20 .0 20 .0
AXIAL DISTANCE X
60.0 100 .0
Figure lob. Upstream and Downstream Line Plots of Go, y = 20
Figure 9 shows that the relative behavior between the
component integrals varies with the location y. Figure 9a
shows that on the x axis, y = 0, the even integral com
ponent GC resembles the odd integral Go, while the com
ponent GX serves as a correction factor indicating the differ
ences between Ge and Go. At the off axis location y = 4,
Figure 9b shows that the reverse is true, i.e., now GX
resemble Go' while GC tends to serve as the correction fac
tor.
Figure 10 shows the expected trend that Go — 0 far
upstream of the source. This is a verification of our
approach in that the separate calculations of Ge and Go do
combine to give the required annulment far upstream. This
figure shows that the local disturbance is nonoscillatory,
decays rapidly near x = 0, and then shows a slow but per
ceptible decay far upstream.
5. Conclusions
A numerical procedure has been developed to calculate
the complete wave resistance Green's function Go for the
case of a nonoscillating source translating below the free
surface. The numerical implementation is based on the
unique work of Bessho, who succeeds in representing Go as
a single integral in the complex plane. We have recast this
integral as three real single integrals Ox' Gc' and Go, where
GX + GC corresponds to the double or even integral Ge and
the third integral to the single or odd integral in the usual
representation of Go. By simple rearrangement, we can also
express our results in terms of the more physical near field
and wavelike components N and W. Computer time depends
on a number of factors, principally the submergence of the
source z, the required accuracy a, and the horizontal dis
tances x and y from the source.
A number of checks have been performed on the accu
racy of our numerical analysis and computer code develop
ment. For field points located on each of the three coordi
nate axes, our calculated values for N agree well with accu
rately calculated benchmark values. Our comparison for the
points on the x axis is facilitated by using specialized limit
expressions for y = 0, z—O for two of our three integrals.
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OCR for page 133
Also, the calculated results show the required mutual annul
ment of Ge and Go far upstream. A number of line plots
are given to shown the often dramatic differences in the
behavior of the integrands in different regions of the compu
tation domain. Contour and line plots show the relative and
detailed behavior of the various integrals.
6. Acknowledgments
This work was conducted as part of a research program
in free surface and marine hydrodynamics supported by the
Naval Research Laboratory and by Code 12 of the Office of
Naval Research. The second author performed this work
under the sponsorship of the U.S. NavyASEE Summer
Faculty Research Program.
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144