5 e sbouwkunde sctìc Hocechoo, DOCUMENIATiE
4'
-DATUM ¿U U t. UM k N I A'-it!
Kuniharu Nakatake
THE DEPARTMEN OFMìiy,
IiIrEcr
D D MIIRINE ENGINEERiNGTHE UNIVERSITY OF MICHIGAN
COLLEGE OF ENGINEERING
7
6t'I
ARCHEF
No. 025
June 1969tab. y. Scheepsbouwkunde
Technische Hogeschool
DellL
Translation
ON THE WAVE PATTERN
CREATED BY SINGULAR POINTS
Marine Engineering
Translation
ON THE WAVE PATTERN CREATED BY SINGULAR POINTS by Kuniharu Nakatake
Translated from Japanese by Young T. Shen Edited by A. Reed and S. D. Sharma
ON THE WAVE PATTERN CREATED BY SINGULAR POINTS1
by Kuniharu NAKATAKE2
Abstract
In this paper the author proposes a method to calculate numerically the wave patterns created by a point source or a point doublet in uniform
streams with free surface. Numerical examples are
included.
Paper presented to Seibu Zosen Kai at the 31st regular meeting in October 1965, and published in the Journal of Seibu Zosen Kai, No. 31, February 1966, pp. 1-18.
2
Kyushu University, Engineering College, Department of Naval Architecture.
When moving in steady motion on a free surface, a ship generates waves; it is these waves which result in the wave
making resistance of the ship. By studying the wave patterns
on the free surface, one can gain insight into the causes of
wave making resistance. The waves which are generated by the
ship can be divided into two components, the free waves, and the local disturbance; it is the free wave component which results in the wave making resistance.
Wave patterns on a free surface were first investigated
by Lord Kelvin2, who used a point impulse traveling with a
constant speed on the free surface, along with the idea of a
"stationary phase" to explain what is called the Kelvin wave
pattern. Since Kelvin's time the wave patterns for ships have
been developed by Hogner3, Havelock4, Wigley5, Kaminaka6, and
others; by a variety of methods. Even more recently Ikehata7
and Kajitani8 have calculated these wave patterns in more detail. In his calculations Havelock used the concept of elementary waves, and showed that the wave resistance of a ship could be
calculated from its wave pattern which asymntotically approxi-mates to a free wave train behind the ship.
Pien9 and others have used this concept to obtain the
wave making resistance by measuring the free wave patterns
behind the ship. Unfortunately, the limitations of this method
-2-When we try to analyze the wave pattern on the free surface in the near field (neighborhood) of a ship-form of minimum wave making resistance, we observe the total wave;
and what happens to the free wave, which is directly related
to wave making resistance, is not quantitatively clear. In
order to solve the above two problems, it is necessary to calculate the wave pattern generated by the ship, carefully
and in detail. In this paper a simple method (different from
that of Kajitani) is described for computing the wave patterns on the water surface using singularities in the water, i.e.,
point sources and point doublets.
2. EQUATIONS FOR THE WAVE PATTERN
Let us assume that our singularity, i.e., point source or point doublet is in a uniform flow, and located a distance f below the undisturbed free surface of an inviscid and
incom-pressible fluid. The Cartesian coordinate system Oxyz is
fixed in space, with the xy-plane on the free surface, and with
the positive z-axis vertically upward. The singularity is
located at the position (O, O, -f) in this coordinate system.
The flow is in the positive x direction with speed V, and the
acceleration due to gravity is g. A field point (x, y, z) is
transformed into cylindrical coordinates (r, o, z) by the
following equations:
X = rcosc,
y = rsin,
z = zThe disturbance of the free surface, i.e., the elevation of the wave () caused by a point source of output 4c can be expressed in the following equation.
z Fig. i Va k0f3ec29 g [4k sec3ûe cos(k,sec2O)dC 2 + 2k0 Ç 2
secO dû ncosnf k0sec2ûsinnf "
I
2kcc29)2 ne dnJ (2. I)
o
where
k0=g/V2, o-_aw, w=xcosû+yiinO=rcos(O - a).
"or the doublet in the direction of the x-axis with
strength 4i, the wave elevation can be expressed in the
following way: k0fsec2O sec4ûe sin(k0sec2û.t)dO 2k0! 2k Ç2 sec2ûdO !1sinnf+kosec2Ocos0f J (2+f2532 7, o n2-- (ksecO) ne dn (2.2)
For convenience, we will define the following non-dimensional quantities.
csvcrl(gf), C=Vp/(gJ), F=kJ=gJ/V2,
Rorr/f, s s/(fC3), D D/(fCD). and p=Fsec2&, q=R cos(Oa) ç2.3) (2.4)-4--from (2.1) and (2.2), we can write
.s=f+
where ¿ 2 /2 -p 1F p e cospqdo, (2.5) (2-6)We shall first consider the problem of finding the
free-wave pattern. The wave elevations and
Df given in equations
(2.6) and (2.9) can be evaluated by dividing the range of the integral into between 20 and 80 sub-intervals and applying
Simpson's first rule with O as the variable of integration. In
this case, the integrand becomes zero at O = ±ïr/2. It will be
2F'12 p'2dO Ç Ucos-1sinpu ue_PQUdU (2.7) and DDf+Lfl, (2-8) where flf= 4F p2esinpqdo, (29) DI = (1 +R2)3/2
2F p2dO cosPu+usinpu ueVdu.
(2-10)
In this case, p1"2 is equal to / secO, if - < O < or - sece
if O > R is the non-dimensional horizontal distance from
the origin; F is the reciprocal of the Froude number squared. In the above expressions for non-dimensional wave elevation and
,
and represent the non-dimensional free wave
elevation, while and represent the non-dimensional
elevation due to the local disturbance.
that is to say:
noted that as the values of F and P. are increased the number of intervals into which the range of the integral must be
divided will also increase. However, this comnutation is
relatively easy.
The rest of this section will be devoted to evaluating
the more complex local disturbance. Let's define the functions
K1(p,q) and K2(p,q) as integrals with respect to U of the double integrals of equations (2.7) and (2.10)
K1(p q) ucosPu-T-sinpu UC"dU
K2(p, cospu+usnu
These two functions can then be combined in the complex function K(p,q)
K(p,q)zS c-(pqp)u
=K1(p, q) iK2(p, q),
where equation (2.4) defines the ranges of p and q as follows: p>O, q>0.
We shall now consider the properties of the new function K(p,q) in three cases.
(1) the case where both p and q are small:
By contour integration the integral of equation (3.2) is
found to have the expansion:
these equations can now he evaluated by use of Simpson's first rule.
It now becomes necessary, by careful studies of the in-tegrations obtained above for K1(p,q) and K2(p,q), to determine
the applicable ranges of equations (3.3) , (3.4) , and (3.5)
With this in mind the values of K1(p,q) and K2(p,q) were
com-puted for p = 1, 2, ..., 20; and for q = 0, 0.2, 0.4, .
10.0; by use of equations (3.3) , (3.4) , and (3.5) . The results
of these calculations for p = 1, 4, 17; and for q = 0, 0.2, .
1.0, 2.0, . . ., 10.0; are shown in Table 1.
For the case where q is removed from the zero point, the integrand of equation (3.5) would tend to indicate that it will
yield the most accurate results. The calculations using equation
(3.5) were made with y varying between O and 20 with an increment
of 0.2. When q = O the integrals K1(p,q) and K2(p,q) as given
by equation (3.5) diverge; and it is found that equation (3.3) which takes into account the finite part of the divergent
integral, is the most accurate. After a careful study of the
above arguments, and the calculated results it is found that the dividing point between the ranges of validity of equations
(3.3) and (3.5) is a value for q of about 0.4.
Taking N of equation (3.4) to be 10 a study of effects of
a variation of p is made. From a study of the right hand side
of P and Q for equation (3.3) it is learned that any value of
n such that
p/(nn!)>104
gives valid results.From a further study of K1(p,q) and K2(p,q) it was con-cluded that equation (3.3) should be used for p<l7.0 and q<l.0, equation (3.4) should be used for all p>l7.O for any
-8-range of q, and equation (3.5) should be used for n<17.0 with
q>l .0.
Using the values of K1(p,q) and K2(p,q) obtained above,
and become s D 1/2 2F p3'2K(p,q)dO, - = - 2F 2F Ç2 1+R2)3'2 - 17 p2K2(p, q)dB
For the integration with respect to e the range of the integral is divided into 20 sub intervals and integrated by Simpson's first rule.
At the points e = T/2, where p-, expressions for
K1(p,q) and K2(p,q) given by equation (3.4) may be used to study the behavior of the integrands in equation (3.6)
As p- we find that: j N (n-1)? p312K1(p,q)
2(lq25fl2S1°
N (n-1! cos2r, p'K2(p,q) - ,2 1+q2This would show that equation (3.6) can be used in the
conventional manner to obtain values of and
D2
as p-.
By using the above methods the amplitudes of the free
waves and the local disturbance, the values of and
generated by the point source and noint doublet of equations
(2.5) and (2.8) can be obtained. Equations (2.6) and (2.7)
make it clear that in the case of the point source different
limiting values of and are obtained as one approaches
the point R = O along different values of c. However, the sum
at the point R = O is independent of a, as it should be.
(3-6)
4. NUMERICAL RESULTS AND DISCUSSION
Using the methods described above the values of and
of equations (2.5) and (2.8) were calculated using an
OKITAC-5090H machine at the Kyushu University Central Computing
Station, with F = 0.5, 1.0, 3.0, and 5.0. For the free wave
computation of
sf and Df' a was varied from Q0 to 40° with
an increment of 2°, and from 45° to 160° with an increment of 5°; with R varying from O to 10 with an increment of either
0.2 or 0.4. For the local disturbance, the and
D9. were
calculated with a between 0° and 90° with an increment of 10°; and for R varying from O to 10 with an increment of either 0.2
or 0.4. Values of were calculated with F = 1.0 and a = 0°
s2
for R between () and 10; and can be found, along with those of
Wigley10, in Table 2a. Table 2b contains a comparison of the
results obtained by Kajitani8, through direct double integra-tion, with those obtained by the above method for F = 0.5,
and 1.0; and a = 0°, 30°, and 60°. The results of these two
comparisons were very satisfying.
Since and
D change in a simple manner with respect
to a, intermediate values may be obtained by interpolation
between a's. Adding the interpolated values to the values
obtained for the free wave pattern, the total waves and
may be obtained. The results obtained by applying this method
to the point source are shown in figure 2 as wave profiles taken along the lines a = 0°, 10°, 20°, 40°, 60°, and 90°. In these figures, the solid lines represent the total wave,
and the dotted lines represent the free wave. Similar results
10
-figures 2 and 3, the effects of the local disturbance become negligible after a distance on the surface of about 5 times the depth of submergence of the singularity.
Furthermore, the free wave pattern, the local disturbance pattern, and the total wave pattern are plotted in figures 4
through 11, with respect to the parameter F. In these figures,
a water elevation higher than that of the undisturbed free surface is taken as positive and plotted with a solid line, while an elevation below the undisturbed free surface is taken as negative and plotted with dashed lines.
[Editor's note--It should be noted, that the wave patterns depicted in figures 8 through 11 are opposite in sign to
4
those computed by Havelock . The explanation of this is, that
in the present coordinate system (figure 1) , a dipole of
negative intensity (i.e. one with its axis in the minus x direction) would be required to generate a sphere of positive
volume.]
After studying these figures it is possible to make the
following remarks. The pattern of is that of a slender
body for small F (F = 1.0 or 0.5) , and it becomes the pattern
of a blunter body as F is increased, where the limit is of
circular shape. The pattern of for small F is shown in
figures 8b and 9b; the curve is expanded in the OE = 900
direction with the shape of the sector of a circle as F
increases. The sign of the disturbance changes in the
neighborhood of R = O as F approaches 3.0, and there is a
peak at R = 0. In the total wave patterns and for
stream and in the neighborhood of = 200 the divergent wave
appears.
5. CONCLUSIONS
In order to obtain a clear picture of the propulsive
properties of a ship it is necessary to study the flow pattern about the ship along with the wave patterns and wave making
resistance. In this paper, as a first step, the wave patterns
caused by singularities in a uniform homogeneous flow have
been computed. In the future, we should be able to apply the
technique of streamline tracing to the wave pattern and wave flow and thereby determine the singularity distribution
representing a ship-like body.
From the computed results we have found that the effects of the local disturbance are negligible after a distance of approximately 5 times the depth of submergence of the
singu-larity. Therefore, if one wants to measure the wave making
resistance for a ship from an experimentally determined wave pattern the measurements should be made at a distance behind the ship of at least 5 times the draft of the ship.
Finally, I would like to express my deep appreciation to Professor Keizo Ueno for his guidance, to professor Masahiro
moue for his continuing encouragement, and to professor
Ryusuke Yamazaki for his detailed discussion on wave resistance
theory. Thanks also to Messrs. Kohei Den and Kiyoshi moue
for their assistance in the writing of computer programs and in the drawing of the figures.
12
-RE FE -REN CE S
T. Inui, "Wave Making Resistance of Ships," Trans. SNAME Vol. 70 (1962) PP. 282-326.
Lord Kelvin, "On Ship Waves," Proc. Inst. Mech. Eng. (1887).
3 E. Hogner, "A Contribution to the Theory of Ship Waves,"
Arkiv for Math. Astronomi och Fysik, Vol. 17 (1922-23)
4. T. H. Havelock, "The Wave Pattern of a Doublet in a Stream
Proc. Roy. Soc. London Series A, Vol. 121 (1928) pp. 515-523.
5 W.C.S. Wigley, "A Comparison of Experiment and Calculated
Wave Profile and Wave Resistance for a Form having Parabolic Waterlines," Proc. Roy. Soc. London Series A., Vol. 144
(1934) pp. 144-159.
Tetsuo Kaminaka, "Investigation of the Relation between Ship Form and its Waves," Kyushu University, Doctoral Thesis (1960)
Mitsunao Ikehata, "Second Order Approximation to the Wave Resistance Theory of a Singular Point, and its Applications," Tokyo University, Doctoral Thesis (1964).
Tadashi Kajitani, "Second Order Approximation to the Exact Relation between Source Distribution and Hull Form in Wave Resistance Theory, and its Applications," Tokyo University, Doctoral Thesis (1965)
P.C. Pien and W.L. Moore, "Theoretical and Experimental Study of Wave-Making Resistance of Ships," Proc. Intern. Sem. on Theor. Wave Resistance, Ann Arbor, Michigan (1963) pp. 133-182.
W.C.S. Wigley, "A Note on the Use of a Certain Integral in connection with Wavemaking Calculations," Trans. I.N.A. Vol. 91 (1949) p. 378.
Table I K1 (p,q) and K2(p, q) obtained from Threc Methods
0.0 .1839 O -.5754.. I .0000 O -.1641 0 -.109610 0
0.2 -. 741710. L -. 7823_ 1 -.588010 0 -. 6454o_ 1 -.627310.. 1
0.4 -. 7433o- 1 -. 739o- i 5369io- 3 -. 2548. 1 -. 2544o_ 1
0.6 -. 6070_ 1 -. 6O69_ 1 . 5096o_ -. 2829o_ 2 -. 2816o_ 2
0.8 - . 4620_ 1 -. 4624o_ 1 -. 4l07. 1 . 8436,_ 2 . 8470_ 2 1.0 -. 3427io_ 1 -. 3449_ 1 . 4535.. 1 . 1298o_ 1 . 129310.. 1
2.0 . 8520o_ 2 -. 6107a.. I
-.
2 . 1O08_ 1 -. 1818io O3.0 - . 2989_ 2 -. 2987_ 2 . 567510_ 2 4.0 -. 134510_ 2 -. 1344.. 2 .3480 2
5.0 -.71O1_ 3 -. 7O94o_ 3 2
60 -4183io. 3 -.4175o.. 3 .i645o.. 2
7.0 -. 266510_ 3 -. 2656o 3 . l223_ 2 8.0 -. 180lo_ 3 -. 179l.. 3 .9426o_ 9.0 -. 1274io_ -. 1264_ 3 74ó8.. 3 10.0 -. 935Oo. 4 -. 9244_ 4 . 6048_ 3 p = 17.0 q (3.5) (3.3) (3.4) (3.5) (3.3)
0.0
3lo- 1
-. 130110_ 6 °300io O .9815o_ 2 -.39ó4 20.2 -. 7358 3 -. 1551. 2 -. 1563_ 2 -. 33S4 2 -. 3447 2
0.4 -.2406_ 2 -.2421o_ 2 -.2241_ 2 .2292_ 2 -.229i_ 2
0.6 -. 2536_ 2 -. 2536o_ 2 .2536_ 2 -. 1 153o 2 -. 1i53_ 2
0.8 -. 2242o_ 2 - . 2242o_ 2 -.224210.. 2 -. 3487_ 3
-.
31.0 -.182910.. 2 -. 1833o 2 . 1829 2 . 1217o_ 3 3465iu-2.0 -.5442.. 3 -.3i58 6 -.5L4210.. 3 .4516_ 3 -.2265 6 3.0 -. 19961o_ 3 -. 199S.. 3 .2871o_ 3 4.0 -. 9132 4 . 9l28j_ 4 l833.. 3 5.0 .4862o. .48571o_ 4 .124310.. 3 6.0 -. 2877io.. - .2871o_ . 8909_ 4 7.0 -. I8380. 4 -. 1831o_ 4 .6663 4 8.0 -. 1244a_ -. l237o_ 4 5154. 4 9.0 -. 8812_ 5 -. 874Oo_ 5 4 10.0 -.64721o.. 5 -. 5398.. 5 . 3322a =1.0 q (3.5) K2(p,q) (3.3) (3.4) (3.5) K (p,q) (3.3)
0.0 -.2224o O -. 1156o 1 .0000w -.5972o_ I 3O28 O
0.2 -. 8690go O . 884l O -. 3125o 6 . 458810 0 . 4685 O
0.4 -. 631Oo O -. 6312 O . 11l6 6 . 5l97 O . 5199io O
0.6 -. 4322o O -. 432210 0 7l56 5 4984 O . 4984io O
0.8 -.291O O -.2939 O . l21Oo 5 4443o O .4443 O
1.0 -. 1961 O -. 196010 0 -. l254 5 3826o O 382ô 0 2.0 -. 3433. 1 -. 3340_ I . 1238 3 . 172210 0 . 1753o 0 3.0 -.8975_ 2 .1564 I .9063_ 1 4.0 -.3082_ 2 -.6335.. 1 .5486_ 1 5.0 -. 1269_ 2 -. 1816.. i . 365O.. i 6.0 -. 594610_ 3 -.4161 o- 2 . 2593o_ 1 7.0 -.3O65_ 3 -. 1l16o.. 2 .193210.. 1 8.0 -. 170110.. 3 -.396810_ 3 .1491o.. 9.0 -. 1002o_ 3 -. 1696_ 3 . 1i83_ I 10.0 - . 6194 4 -. 8563_ 4 9596. 2 p =4.0 q (3.5) (3.3) (3.4) (3.5) (3. 3)
)
(3.4) -.409110 6 . 1233 6 . 1592io 6-.
5 -.3202o 5 -.161510 4 .216310 2 -.333010 1 -. 19O1 O . 1350_ 1 . 2304_ 1 .1891 0-149I.. 1 . ll92i_ 1 9727io- 2 (3.4) -.790610 0 . 145810.. 1 . 2383 O -. 2489_ 1 -. 3239_ 1 . 4242o.. 2 . 1016o_ . 5675_ 2 .348210_ 2 .2321_ 2 .1649_ 2 . 122310.. 2 .948510_ 3 . 7540io- 3 . 6134_ 3 (3.4) .39641o_ 2 -. 3447.. 2 -.229120_ 2 -. 11331o_ 2 -. 348710_ 3 . i2l7_ 3 .4316_ 3 .2371_ 3 . 1833_ 3 . l245_ 3 . 8928_ . 6689.. 4 .5186o_ 4 .4l34,_ 4 336o.Table 2a. Comparison of ata =0° F=1.0, a=0° R A B A : obtained by Kajitani B : obtained by author R Table 2b. F =0.5 a=0° A B Comparison of at various a a=30° A B A B 1.0 -0. 3663 -0. 3759 -0.2555 -0.2588 0.0170 0.0137 2.0 -0.1627 -0.1626 -0. 0646 -0.0662 0. 0598 0. 0572 4.0 -0.0613 -0.0628 -0.0011 0.0005 0.0348 0.0335 6.0 -0.0337 -0.0350 0.0049 0.0081 0.0202 0.0200 10.0 -0.0150 -0.0137 0.0040 0.0062 0.0087 0.0087 F= 1.0
a=0° a=30° a=60°
R A B A B A B 0.51.0 -0.9860-0.6028 -1.0300 -1.0228 -1.0200 -0.4628 -0.4600 -0.6100 -0.5964 -0.5963 -0. 1796 -0. 1800 2.0 -0. 2544 -0. 2539 -0. 2352 -0. 2345 -0. 0080 -0. 0078 3.0 -0.1372 -0.1367 -0.1184 -0.1166 0.0160 0.0160 5.0 -0.0588 -0.0605 -0.0452 --0.0400 0.0152 0.0155 7.0 -0.0328 -0.0338 -0.0224 -0.0141 0.0108 0.0115 10.0 -0.0168 -0.0165 -0.0096 0.000/ 0.0064 0.0069 0 -1.5656 -1.5631 -0.6008 -0. 6100 2 -0.2520 -0.2539 3 -0.1368 -0.1367 4 -0. 0824 -0. 0867 5 -0.0552 -0.0605 6 -0.0384 -0. 0445 7 -0.0312 -0.0338 10 -0.0160 -0.0165 -sl obtained by Wigey o1)taifled by author
-5
-o a 2V
e -Le
-tvtal cave nrnflle
Io a .0' -5 0.,C' r
..
-' -- It - '-.1 $ - -, a-toT.
r e-tvFig. 2a Wave Profile of Point Fig. 2h Wave Profile of Point
Source at FO.5 Source at F1.O - totalcave profile
tree cave profile
Fig. 2c Wave Profile of Point Fig. 2d Wave Profile of Point
-- total aove profile
-- - free oaoe profila
-
16
a
-ot*l wave profil. fra. aove profi)..
Fig. 3c Wave Profile of Point
Doublet
t F=Ø
-e--Fig. 3d Wave Profile of Point
Doublet F=-5.Q
R
Fig. 3a Wave Profile of Point
Fg. 3b
Wave Profile of Pointo
Fig. 4a The Free Wave Pattern of Point Source at F0 .5
-o.,
---.o2
03 03 --..-o t. I I -.3 -4 r /Fig. 4b The Local I)isturhance Pattern of Point Source at F-0.5
-Cs -2 -, O O-o .10
j'
--- ---./
/j
.,, -- I O, .00 'oR
¡ 2 4Fig. 4c The Total Wave Pattern of Point Source at F- 0.5
o 0.3 os Il t ç 'o R 7 .5
t:
.3 4.5,
so
18
--j
o
Fig. Sc The Total W ave Pattern of Point Source at F 1 .0
4
- R
F g. 5h The I ocal 1)1 sturhance Pattern of Point Source at F 1 .0
- oJ.
- R
-j 2
j
t-
q /R
-.
Ç/>2)/
/2,1i
¡/' 1' f7" -3 -4 -3 .J -1Fig. 6b The Local Disturbance Pattern of Point Source at F3.O
Fig. Bc The Total Wave Pattern of Point Source at F3 .0 4
- R
o
7 2 4 5 6 7 q
20
-o 2
Fig. 7a The Free Wave Pattern of Point Source at F 5.0
o ' I' 3 4 S 7 / / \ ¡I ,_ .. - -
\
\ ¡L / S ' L'1 L L - L L'L'/'/o
L \ L o ¡ 2 oFig. 7 b The Local Disturbance Pattern of Point Source at F= 5.0 4
¡;j
L'LLL/'(
'! o R o .3 'O!'ig. 7c l'he lIfta! Wave Pattern of Point Source at F5 .0
as
'o
I L I
=03
Fig. 8c The Total Wave Pattern of Point Douhiet at F-i- 0.5
I I
7 Io
R
-J -2 -I 0 f 2 3 4 fc
Fig. 8a The Free Wave Pattern of Point 1)oublet at F=-O .5
R -0.2 / I, -o
I-
---t ...-' I / / s I _----\ I -s I / \ f ' 07 I \(
'
f ,// ,i/'II
i ¡ It t I - - i I i I k¡I,
I iIt
i -.5 4 -J -2 -, o 1 2 3 4- R
Fig. 8b The Local Disturbance Pattern of Point Doublet at F=0.5
-3 -2
22
-Fig. 9a The Free Wave Pattern of Point Doublet at F= 1 .0
--Fig. 9c The Total Wave Pattern of Point Doublet
at F 1.0
- R
Fig. 9h The Local Di sturhance I'attern of Point I)ouhlet at V 1 .0
-- R
-3
.1
Fig. ¡ Oa The Free Wave Pattern of Point Doublet at F3.O 0,3
Fig. lOb The Local Disturbance Pattern of Point Doublet at F 3.0
/
f f,'5-, 1 /
(U"
Fig. i Oc The Total Wave Pattern of Point Doublet at F=3 .0
o
-p R
'o R 8 7 4-* R
2 J -J -2 -/ 2 7 JII qJ
-2 -3 / 1 /r
-
2 -z o -3 - -3 -2 -/ 0.2 ¡.0 4Fig.
lia
The Free Wave Pattern or Point Doubletat F5 .0
e-:.
S02,',._ __..-o. '
// '
/j_ ...,ò. I
/ 2
Fig. 1 lb The Local Disturbance Pattern of Point Doublet at F=- 5.0