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5 e sbouwkunde sctìc Hocechoo, DOCUMENIATiE

4'

-DATUM ¿U U t. UM k N I A

'-it!

Kuniharu Nakatake

THE DEPARTMEN OF

Mìiy,

IiIrEcr

D D MIIRINE ENGINEERiNG

THE UNIVERSITY OF MICHIGAN

COLLEGE OF ENGINEERING

7

6

t'I

ARCHEF

No. 025

June 1969

tab. y. Scheepsbouwkunde

Technische Hogeschool

DellL

Translation

ON THE WAVE PATTERN

CREATED BY SINGULAR POINTS

(2)

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

(3)

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.

(4)

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

(5)

-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 = z

The 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.

(6)

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)

(7)

-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.

(8)

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:

(9)
(10)

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

(11)

-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+q2

This 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)

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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

(13)

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

(14)

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.

(15)

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.

(16)

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 O

3.0 - . 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 2

0.2 -. 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

-.

3

1.0 -.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.

(17)

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

(18)

-5

-o a 2V

e -Le

-tvtal cave nrnflle

Io a .0' -5 0.,C' r

..

-' -- It

- '-.1 $ - -, a-to

T.

r e-tv

Fig. 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

(19)

-- 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 Point

(20)

o

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 'o

R

¡ 2 4

Fig. 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,

s

(21)

o

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

(22)

-.

Ç/>2)/

/2,1

i

¡/' 1' f7" -3 -4 -3 .J -1

Fig. 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

(23)

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 o

Fig. 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

(24)

=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 i

It

i -.5 4 -J -2 -, o 1 2 3 4

- R

Fig. 8b The Local Disturbance Pattern of Point Doublet at F=0.5

(25)

-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

(26)

-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 q

(27)

J

-2 -3 / 1 /

r

-

2

-z o -3 - -3 -2 -/ 0.2 ¡.0 4

Fig.

lia

The Free Wave Pattern or Point Doublet

at F5 .0

e-:.

S02

,',._ __..-o. '

// '

/j_ ...,ò. I

/ 2

Fig. 1 lb The Local Disturbance Pattern of Point Doublet at F=- 5.0

-p R

4

o R

Q /0 2 3 4 5 6 7 q

Cytaty

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