Theoretical study of the wave resistance of sidewall hovercrafts

(1)

CHALMERS UNIVERSITY OF TECHNOLOGY

DEPARTMENT OF NAVAL ARCHITECTURE

AND MARINE ENGINEERING

GOTHENBURG

-

SWEDEN

TEHORETICAL STUDY OF THE WAVE

RESISTANCE OF SIDEWALL HOVERCRAFTS

by

NIKOLAI PLISSOV

DIVISION OF SHIP HYDROMECHANICS REPORT NO 51

(2)

1 . S ULIT.L: 11 Y

. INTRO DITCT TON 2

3.

FO Ri.11. LA S FOR T.,--TAVE RE S I S T ANC E _{- 4}

4-, RESULTS 2,..i.:11) Ti-'IR. DISCUSSIO 13

5,

CONC LUDING RELIARK 15

,

Q. A C ICTOVIL ED GE1.. .1T _{' 15}

70 REFERENCES .16

8 . liUlER.ICA L C 01UTAT ION OF

INTEGRALS Ii AND

12 17

C 01TUT ER PROGRAM 20

-FIGURES

(3)

This report is concerned, withthe wave

resistance of simplified

sidewall hovercrafts

traveling over water of infinite depth with

steady speed. An approximate approach is used

to derive

suit-able exPressions.for the wave. .drag. Unlike prOriOs theoretical

woks in this field his report- contains -vast calculations in

which the separation of wavemaking

coMponents'cfrom walls,

air :

cushion and interaction between them) of.sidewail hovercraft

is shown. Comparisons. are also dene with available theoretical

results. Some details with 'respect to practical computingand

(4)

2. INTRODUCTION

The wave resistance of a eijeawall hovercraft

depends on

the

type and design of the creft because of

the

sidewalls.it

more important or this typ

of .air-cushion vehicle

than for

the.amphibieus oneec.. The .alide'6oards can probably change the

two humps of the. resistance and corresponding Froude

num-bers.

There are many unanswered questions in this field. A

few papers concerning the

,problem

are available

and

are short,

ly considered below.

In the work

by

Chaplin [1] it is recomended either'to neglect

the wavetaking

of

the sidewalls or to regard it as the

addi-tional wave resistance of two separate thin hulls. HowevIr

this approach seems to be too primitive. There are two szecial.

theoretical. investigations on the subject by.Hurthy and Yin [2,5J.

Murthy [2] assumes that the sidevalls can be replaced by two e

uniform pressure distributions of rectangular form travelilime over the water.surface.together with the main pressure

distri,-bution from the air cushion. The value of the pressure in these

. side distributions is

determinated .by the draught of_the

side-walls.

The wavemaking'of

the forward and, aft seals

is

neglec-ted. These assumptions made it possible to

derive.at,a formula

for wave resistance. Hurthy computed several examples bud didn't

separate the interference part of the wave resistance.

Yim

[3]

used Eretensky formula for wave resistance of

a

:Pressure

distribution in a channel of finite width. The sidewalls were

regarded as two linear source distributions along their

center--planes. This assumption corresponds to the

well

known slender

ship theory. Since the same source distribUtions was taken as

for isolated thin hull without air Cushion effect

and this

dis-tributions were supposed not to

affect

the wavemaking of the air

cushion. it means that only Wave sUperposition without reflection

was considered. A formula was derived and some calculationS

for

infinite rater were done. As a result theaUthor could,

show the influence of the Froude number and of

some

geometri-cal factors on the wave resistance. He also made the first

(5)

ef-feet by increas'ini,; the rushion_pressure in the local areas near

the seals. In thi;?; aph the Sizes of the areas. are not de,..

fined. however. The infi of sidewalls on the total wave

resistance hasnot been shown.

Experimental invecidge-Lion,s, cif the wave drag of sidewall

hover-crafts are no known.

The aim of present investigation is:

to split otal wave resistance in components

to carry out more complete calculations than before

3° to show how the sidewalls wavemaking and the

inter-ference between them and air cushion depends on

(6)

-FORMULAS FOR WAVE IIES=ANCE

An appreximate apbrea-J11. oimA.ar to reference

DI

is used to

derive to a.forml).1a fo± the ave resistance. The drag of the

seals is nelecT,ed and .aal vfivemaking is regarded to

be.g.ene-rated,by the aj.1- cushion and .the sidewalls. The former is

presented as a ctang area of uniform pressure distri- .

bution.

The sideboards are assumed to be

thin and

symmetri-cal with respect to their centerplanes. Figure 1 shows the

main geometrical characteristics of craft and of the moving

coordinate system oxyz. The x-axis lies on the water

sur-face and coincides with the speed vector The

2;.;.akis is

directed' upwards

Fig 1 Vain geometrical characteristics

(7)

/

c.

where 12

, density of

the.water

and

= a1 + al

tr b b b 1 1 ' /69:

ces

3 0Q'9

Acco.,.ding to [4] (see also 11Costyukav [5 ] ) the wave profile

at a great distance behind a moving body can be' fOund in the

Mtn: r

Z(zpp=

a(69-cosEcLW069

(-2* cose?

&e:/2.

-/

fa2

Le)--,-,,Lvo(-cvs62

:Ye ./2. 6))

jj

c/62

( 2 )

where

and the wave amplitude is

az(62) gro)

(4)

The probleft is

find a()) and. b(e)

we write a(8) and b(8) as

a

1 2

b =

bl

Havelock's formuda

41 tu:ed for the wave resistance

speed of the craft

(8)

In [5]

a1 1 .aerminate.the contribution from the sideWalls

15

derived at by. afftion..of w%ves from two Walls (a1 bl

and

al , bi). b, determinae the contribution from the air

cushion pressure

distbu4-io1.

We thus accept the same

assumP-tion as in reference [3],

ize

we consider only superposition

of waves_ fro :2 differe:-It sources. Wavemaking at great distance

from an origine

is

regarded to be the same as if the origine'

is

'isolated.

This

approach is fairly

approximate and the

fol-lowing numerical results can be interpreted- mainly qualitati,'

vely0 However they must be closer to reality the greater the

distance between the hulls and the smaller their thickness and draught may be

As a starting point two known expressions for waveprofile

ordi-nates at great distance behind a thin ship

[5]

and behind a

pressure distribution according to [6] are taken. They

are respectively: 2

(16)

#5/. J

ir

Z4

(x?:iyi)=1 ---7.---.7

rdS

e

(

_COL52/

_CM36)

(7) where

to,

cog

624-

(041r--are coordinates of a point of the hull surface S

q is the

density of sources distributed, over surface S

and

-yz

,hitib

s6;7- (1,

-);;j6

)

where

()Co1.6)

L °

p is the pressure distributed over the arrea

a

7 a)

(8)

(9)

Then

Formula (7) the coordinate system

01 xl yl

zi

fixed to a 11;A1., In t1L:: sl(:nder ship case

[51

the

sourcedi-stribution over

the

center-olane of the ship.

Considering cylindrical hulls with

cons-tant form of the watea-iiy. we can obtain. according to [5]:

K) r7J77

,.,e

d

(9)

where

7-1/(1)

is the equation of the waterline. In the

fol-lowing calculations we put

Qpa

)

"Ja

/

I 2r

_r

-

Z2

To derive final exp-.L.essns

for

al; bit al; 131 and

hence for

If

and

we must next take some successive steps.

a1 u1

1. SUbstitude (11) into

(7)

and integrate the .latter

regarding dS = d

d1 and -0;5

L1ZCi Z

- Hi

7C/Z0

where Hi is the "effective" draught,

of the sidewall (see later)

(12)

1

where

0,

x,

t1

are the coordinates fixed

to

the left

board and

o1 °1:1 2,1

to the right one..

2. Transform the results to the coordinate system oxyz

using the relations:

= x

xi

X1

=x

y = y -. y + 0.5B

(10)

8

-3.

0oi.o.a.2e the e:KSsions found in step 2 with formula

and separateHt.hus a(8). and b(8).

Ommiting int,F:rnediate e..(7mutt.iftions we give the.final.expre6sions

.0

The .integration with respect to C. in formula (7) is D rformed

from 0 to

HI.

where H. is the

mean

value between outside and

in-' I

-sidedip.ghts at zero speedj

I-4

a

6-p

"

(14)

In (14) -H - outside drawjht of sidewall

p - air cushion pressure

(11,- a.)- inside draught.

specific

weight

of the water

This approximate approach has been ..used. to take into account the

difference between the, waterievels outside and inside the

ide-walls.

Similarly one can reduce foildula (8) to (2) and obtain

_{a22 b2.}

o

#4szr

#.0,543

CO59t9

f

fl

-6:64

-0.6-8

And after. integration

a2

= 0

8/5

_)18.S01

S6'2(

)/Zi

)(7 l'S6'n

20 2 Cola

7-~

" t Ce75

co5 6

4.714;1,0 dg.ebi.

(13)

(15)

(11)

Now.froa (i3) and (i5) the expression for the wave

resi-stance is easily obtaindo

F/2 . q where /.2

=

Tri9-2fi

42(6)0053,62,o/6i

--Lt 0

=

- 61-/*P2

r. cop& sz)zz/ 1-48

Se.44.0) .16,f22(

)I.L

ild& (17)

y. y_

i'-' 1 307,220

(

2! ciasz

(

2.005(i

0

5,

t)" viA.

.-R

_V

_R14,1

S

,z

...)0.2f2/itz(0.067.-5362ci

-/o2qp-vrvz

4,ei

)/. 8,..911?

0Ocog

_st.:1,71

vz,

Ze

_;

z

* CO"32(

r

2

co5z 6 e -

2.e6456

zy

cog 640

26-6-(p /cof3

r

7)1' ₀ z

s

(:;?

2c0$621

"Z

(19)

-vH4

e

Y---1-3-1-L--f.161. ).

--CCth

,Y 16i

)

LAI cou32

y

L) c,/2.

_{2Cai Y}

-

2))

Rwc is

the wave resistance of the air cushion, ioe of. a

rectanu-lar'pressure distribution.

Rws is the wave resistance of the sidewalle

ARw is the wave resistance due to interection between the

cushion and the sidewalls.

(16)

(le)

We shall not coliipute :awe since the extended calculations of this

part made by Newnan. and Poole are 'available [7]. The wave resi

4°. eff4.4.

cos.'

(12)

GW4

stance of

gations of catamarans.- r.nere, are however

two

reasons that have .

made us to c67-sin-te this i)art in the present

work.,

.First the

draught of the sidewalls than available data from

catamarans. Second it ii7.batter to compare the tart of

eachcom-ponent of total wave resistance taking the. same sidewallform

both (18) and

(19).

Nov attention

will

be paid to calculations

of avi and cIRW. Since dimensional values - are unctinvinient for practical employment the following non-dimensional characteris-ticsare introduced.

IU

cwu: in principle be taken from investi-,

The coefficient

a,

gives the relation between the buoyncy of the

sidewalls and-the air cushion lift. .

Instead

_{of RWs-and}

ARW non-dimensional wave resistance

coeffi-cients

Cw1 and Cw2 related to air

cushion lift

are introdUced.

CW- -I

P-

L B _ApiTm F L 1' S (20) ( 21 ) (22)

Taking (20) - (22) into account the e.:cpreSsions

for

CW/,

Cw2 can

easily be obtained from (18) and (19)

"24/

2

ir(e-'7'/Fez;coczo.iz

Ke2.8.-)3

c6:S2(V-V

- cos

.E

ars 6 Ecrirl-(

_2k

_{z cci}

_tsi)

v

₂

ki

_l`ft.

r2_

17 .a

----n

'

CO:3( ,---j- 2 7 ejiciy3-iodo3 (23).

(13)

92'4.1

C't

(x2 #,)2

2

two

of[ie

''PL

2 N

For computation we reduce the integrals

(23) and

(24) by the

substitution

1 cos

_e

_ 1

Then

= G 1 1 w2 G2

where

2048

_t2

_Kc: n,

-

2 B ° P 11

2 26-6

e'

'C2

c

6.3'61)

cos362

)6-.

Re

ad t.

1-

/

Y&Ago?

lee

zn(2-R

CO30) ./-

11-M

.Sti>27

)

--

R

/

) d

2 C°-) (

'C'e2C46 0 !J

in formula (23) is

11

.17

Rea

3 /3. (L

A

2

G

256

Ks 2 -n

5

_Ke

(z4-1)_7- C3. coil

Re

2k0

.

66:

6,

24:.1.), 1r

0,2(

(24)

(30)

G2z

9:3+1).

VC

(23)

(26)

(14)

es°

C,i/Xa.7%

2;

=7

e

Nia LIZ 1-11

Pe

, )17

214 2 14.1)2' (Z2-1-2)

I

ke

y)7

jj

2k0

Q

The

final

exprssions 32) are used for computation.

Numerical computation of (30) and (31) faces some special

fea-tures, Both integrals- have

infinite upper limits

though they

are convergent. The integrand in (31) is undertermined at x=0.

The intergrands osscilate and this is the reason for the large

number of steps necessary for numerical integration. The cal7

culations have been done on 'Hewlett - Packard digital computer

In the

end

of the report details on practical computations of

integrals (30)7.(31) and also the computer program, are given. Relative error in numerical integration is about 1%.

In formulas (30) and (31) following designations have been in-troduced

=7. 0

(15)

40 RESULTS AND THEIR

,72SION-The results of computation are given in figures 2 - 14. The

total wave resi.2tnce'coffcient C is a sum of wave-resistance

coefficients cbmItedc-ding to formulas (28)

'(31)

and wave

resistance coefficient Ow, of the moving pressure field from

[7]

thus

Ct C11 Cw2 Ow,

:awe

where

C=

T M

C .t4 +.3

Fig. 2 and 3 show the variation of each component as 7211 as

to-tal wave resistance with Froude number. In the figures s61.1e

theoretical results from [2] and [3]. are also given for

cOM:-parison. One observes two humps at Froude numbers Fr

0935

and

Fr 096. The latter is biggest. The wave resitance of the

sidewalls (Cwi) i8 negligibly small but the drag due to interac-tion of the air cushiori :iLnd the sidewalls is about 20-30 percents

of air Cushion wave drag at Fr 095 - 0.96. At a low TroUde

number (Fr F4

0,35)

the interaction resistance is negative and

at Jr 095 - 096 it is largest. Comparizon.With calculations

carried out by Tim

E3]

shows good qualitative and quantitative

agreement. Murthyis results [2] givea bit higher wave

resi-stance coefficients at Fr 0,4 and move the first hump to

lo-.

wer Froude number. The qualitative agreement is satisfactory

however.

Figures 5

5

illustrate variation. of "additional" wave

resi-stance coefficient K with Fr Ke,

Coefficient K is Cw K c.,we

(35)

(34)

(16)

Thus K show.: hOw the ai..(:%.m.;hion-Wave dra increases due to the sidewalls.

It appears f-for: fiz 6 doesnit'practically depend on

but

essential; 'i.. E..-on-the

air

cushion pressure under

gi-ven sizes of P., Than the greater a (i.e less

pressure) the greater is relative sidewalls effect. When the

buoyncy of the sidewa:Ils is about 50% of the air cushion lift the additional wave resistance due to the sidewalla is about.'

50 - 60% of air

cushion Wave d,-a. It is interesting to notice

that the relative width of cushion

2

and draught R of the

boards affect this coefficien fairly little. Values of the to-,

(17)

- 15

ON'aLUDI-1\111 REMARK

The calculaion:

bere-Iik-previoustwo papers

on this subject

are based on a.7, approach. Further theoretical Work

in this field oan be cono-rnc,d with formulas which take into

account the-reotion

and

distortion

of

waveSfrom different

parts of a hovercraft, However, this theoretical troo-.

blem

is

fairly complex.

At

any rate experitents with

measureent of wave resistance of sidevall hovercraft sees. now to be more desirable to check the simplified.thecries and to give the answer about practical expedience of the developing of more exact theories.

6. ACKNMILEDGF2,IENT

This Work was carried out under Sponsorship of Division of Ship Hydromechanies of Chalmer's UniVersity of Technology asa ipart

(18)

REFERENCES

Chaplin, E.7i.

"Some J.-J Priniles of Ground Effect Machines",

Section D Drag D;:i.v.J.d Taylor HodelBasin

Report 1.966

MUrthy,

"The Wave Resistance of a Sidewall Hovercraft

Cata-marans and Trimarane, .

University of Southamvton-,,A,A.SO.

Report NQ

296 1969

Yim, B.B.

"On The Wave Resistance of Surface .Effect Ships",

J.S.R. .15

₍₁₉₇₁₎

1

4.

Havelock,

"The calculation of wave resistance",

Ibid vol

144, lig 85, 1934

Kostyukov, A.A.

"Theory of Ship Waves and Wave Resistance",

Leningrad

1968

6. Lunde, J.K.

"A Pressure Distribution 1.Aoving at Constant Speed

of Advance on the Surface of

Deep

or Shallow Water'",

Skipsmodelltanken, Norges Tekniske

Hogskole,

Trondheim, 1951

t.

Newman, j0J,, Poole, F.A.P.

"'Wave Resistance of a .Moving Pressure Distribution. In a Canal"

(19)

8. NME4ICAL 00YPUTATION OF INTEGRA1Z 11 AND 712 Then (z).(i% 0 grals A If

and

So far. as (31) are convergent one can replace.

infinite.

up-_91er.itLit 7:y,c=e number N. It is possible :to

esti-mate the accuracy' d'ue to thi3 replacement. Let the integrand in

(30)

be F1(,:). ancI. in (51) - :F2(x)c

A =

10-

1

f(ZYZ +if 617*/z

2

To find the error of A

1

and make its estimate

2

K 4 K 2

.

C, C_

m ₊ _-r

2 5 8

truncation in (30) we consider

inte-zjz

L

ikjo2

C3 z

i`c

3

3 /2

4/4

r

/11(

A le

19 A

67/W9

AV

/

01/Ke

(20)

-at given r:Lc.f:.,

L;

1061

N1 ;

ey:

M

/

/ 1CI

The same app:cac iitI reL,;pect to

z

ko

,

a/.2

._

P-2

(7)

-

C/2 Z.

(

)

2-

/

+ C

3 _(a-2-#.-/)_ai(rZ a 1`

2)

1 2.7 A72 /V2

3/

)

dz

- /ife

Z.

_C3

)0/2 _Alo2 (.3

.2'

a

7 _/12

(

7.

9 7 /

1//442

IL 7476 /

0/ z

Rp

z opy26

Thus where 2 C,

T°

2 T = 2 8 + and A2 =

1052

One-Usefull property of the integrands was found at some

preli-minary computations. They are with great Accuracy constant at

x 00.01 at any value of XC1' C C. . This property.has been.

o' 2' 5

used to eliminate the difficulty of numeridal integration of

22(x)' at x = 0 and to simplify

comraltation program.

12

was thus

computed as

6

r

--2

(ezi

e2

.F2(x)dx gives

(36)

(37)

(38).

(21)

Where c

2 sHane

0,01 and greater than P6

301!

uni-formity of 1)talli th

sayar.: method has been used for

al-though F.1 io:detemine at 0. An other-uSeft111 point came

from PreliMinry oalcIdaons

Because of integrands

oscil-lations and eMe..".!,1 meanings it

is better to de.Vide the

integra,-.tion interVal. in 2evera1 parts.

The 'number of

steps and

cot-puter.time cai. e red7loed in

this ways All these observations

have been regarded in the following computer program.. The

pro-gram

is

written in the computerianguage "Basic" which is used

(22)

9. COMIJUTER PROGRAM

1. Di7,F.F::: (X) ;....FNA(X)-(o7 DIP FNP (x) FNB(X)

Statent 1:is vc7Lripus, If

Cr1 is computeithen

FNF(X) . i(x), if Cw, - FHF(X) FNB(X)

5. PRINT"-ET:GINTTM"

70 PRINT "INPUT FR, EL, Hp 39 ES, A IS REQUIRED"

100 INPUT Al, A24 A3, A4, A5, A6

Because of Basic features we :inust use another desigpa7

tions in practical program* Statement 10 demands of

the input of main initial data:

Al = Fr, A2 = Ke, A3 = A4 = S, A5 = b, AG =

200 LET i'd0 = 1,333 A*- A3 * .A21- 2/A6/ A4 %i A5

In

the:2..ca,,Lri

MO p

30. LET HO = A- 110/2

HO

n

in the present program

400 LET KO = Al 4 2

50. LET Cl HO/K0

60. tET C2 . A4/K0

70* LET C3 = A20K0/2

100. LET

TI = (1o4/12)

4-

031.2/8 +KOt2gC3/5

mn

1100 LET T2 + C3/6

115.

PINT "INPUm Zl Z2

IS REWIRED"

120. INPUT Z11 Z2

°Statement 120 requiresZ1 = 81 and Z2 =52 to be input*

81 and&2

determinate the accuracy of calculations (See

formulas (56) and (37))

(23)

'..

140. LET

N2.

((LO_;- ;T2 *10 t Z2) )/6)

145. PRINT t2 WILL PRINTED"

150. PRINT 111 N2

Ni, N2 ar.the up.-ber limits for the integrals computed

160. DEF

= xi

+ 1

170. DEF _(X:= 1 -

P.Ci

PNZ(X) 2)

180. DEF FNP(X) = A2/2/1C0*FNZ(X)

190. DEF FNC(X) = t 2/FNZ(X)* SIN(FNP(X)) - 03*COS(FNP

200. DEF F11(X) = SQ,R(X t 2 4. 2)

210. DEF FND (X) = C2* FAX) * FR(x) X

220,, DEF FITE(X) = SIN(FND(X))

230, DEF FNG(X) = COS(FND(X)/2)

240. DEP F.NK(X) = SIN (FNZ(X)/2/K0)

250. DEF FNA(X) = (FN2,:(X) t 2 * FNG(X) t. 2* FNC(X) 2)/(FNZ(X)1%

14*-PNQ,(X)

260.

DEF 1M(X)

F3(X)/(FITZ(7)*X3i-PNg(X))

270, DEF FNB(X) F11(X)*FNL(X)*FNIC(X)* FTIC(X)/FNZ(X)/FNQ,(X)

Statements 160-270 after all define two integrands

(x) = (X) and_ 7E02(x) . FNB(X)

2800 DIM T ( 6 )

All integration interval 0-N (or 0-N2) is devided on six

parts. 6 numbers in matrix T. are the bounds of these

sub-intervals

2900 PRINT "INPUT LIAT T"

300. laT INPUT T

Statement 300 requires o above-mentioned numbers to be

in--out. The first tuber must be less or equal 0.01, the last

equal

Ni rhen C. or rhon Cw20

3100 PRINT "INRIT El"

320, INPUT El

Statement

320 requires El.- to be input. El determinates

(24)

subinteri,

taken equal 10751 (or 10-62) then

the absoluto srror over interval

0-N1 (or

0-N2

is ,

6 0

10 (or 6 '0 10 `-`-)

330..

LEI' VU

TN

340.

LET FO 217(7/0)

350.

LET- YO VU 1.1;!0 360 PRINT "YO"

370.

PRINT YO

Statement 370 prints the first part of integrals computed

according to (38) 375. LET Y4 YO

380.

FOR K = 1 TO 5

390.

LET X1 = T(X.)

4000

LET X2 = T(K + 1) 410. PRINT "INPUT N" 4206 INPUT IT

Statement 420 requires N to be ihput, The .number 2N

equal to to number of steps -in numerical integration for

everSi subinterval.

4300

GOSUB 800

Statement 430 is the address to the subroutine for

numeri-cal

'integration

440. PRINT "Y, E9"

450.

PRINT Y, E9

Statement .450 .prints.

5 pairs

Y2 E9corresponding to

5 in.-.

tegration subintervals begining- from the second.. 1. is the

meaning

of

integraliand E9 Shows ho r the integral has been

computed. E9 can be equal. 0; 1; 2,,

.IF E9 0 it means that required accuracy has been reached

IF E9. 2 it means the opposite situation because of. small N and IT must be increased.. The case E9 = 1 means that the-required accuracy cann't be obtained because of rounding

(25)

460. LET 14 14 + I

470.

NEXT K

545,

PRINT "THE VHOLE ijEGHAL

YI"

550.

PRINT V,

Statement 550 Prints the value of the whole integral from

0 to N.. )

555.

PRINT "INTPLIT W = 1 FOR R1 OR VT = 2 FORR2"

560. INPUT W

Statement 560 requires W to be input

W

1 when

Cti

computing and W = 2 when Cw2

570. IF W > = 115 TEEN 605

575.

LET 01 (65202 ftKO*A5 )/(42 t 2 itA4 LIO)

580

LET R1 = G1 4-Y4

585.

PRINT "R.1"

590.

PRINT R1

Statement 590 irints siciewalls wave resistance coefficient

Cw1 = R1 600. .G0 TO 1250

605.

LET G2 = (81.6*K0*,15)/(A2*A4) 610. LET R2 . G2 *14 615. PRINT "R2" 620. PRINT R2

Statement 620 prints "interaction" wave resistance coeffi-cient

Cw2 =

6303 GO TO 1250:

The following statements are the Subroutine for...numerical

integration of a given fuction, it uses trapezoidal rule

together with Rotberg's extrapolatien method. This

inte-gration method gives the smallest truncation error in com-paring with other numerical methods at the same number of

steps.

The .trogram_has been

developed from the similar:

Subroutine

for

-Fortran. -(see IBM.Apulication Program.

(26)

Q

24.

800,

DIM A(3o)

810. LET A(1) ,=70e:5

820. LET H = X2

X1

830.

LET TIEN -1170

840. IF H

0 THEN 1200

850,

LET

H1 = H

860.

LET F El /ABS (H)

870,

LET

D2 = 0

880,

LET

P =1

890.

LIT2

Ji = 1

900.

FOR

I

= 2

TO N

910.

LET Y

= A(1)

920,

LET

D1 = D2

930.

LET

H2 =

940.

LET

H1 = 0.5* H1

950.

LET

P

= 0.5P

960,

LET X X1 +111

970,

LET S 0

980. FOR J

1, TO J 1

990,

LET S

S + FNF(X)

1000,

LET X

= X + 112

1010.

NFAT J

1020.

LET

A(I) = 0.5 if A(I

F S

1030.

_{LET Q=}

1

1040,

LET

33 =. I

1

1050,

FOR

J

= 1

TO 33

1060.

LET

15 =

J

(27)

1090.

LET A(I5)-.. '

WI5+1

A(I5))/(5..1)

11000 NEXT

1110.

LET

D2:. ABS(Y

A(i))

1120.

IF (I

5) LO

HE

1150

11300

IF (2)

= 0

HEN 1200

11400

IF (D2- D1) s,>= 0

THEN 1220

11506

LET

31 = 31 +-31

1160.

NEXT 1

1170,

LET

E9 = 2

1180.

LET H si-A(1)

1190.

GO TO 1240

1200.

LET

E9 =

1210.

GO TO

1i0

1220.

LET E9 . 1

1230.

LET Y

= H*Y

1240.

RETURN

1250.

END

Input data for this numerical integration subroutine are:

X1 - the lower integration

unit

X2 the upper integration limit

2N is the number of integration steps (N- DIM A in

statement 800 so if it is required N 30 one must changc the statement 800)

B1 - the upper bound of the reauired absolute error

FNF(X) the definition of the external function subprogram

used

Output data

Y - the resulting approximation for the integral value

so-called error taratetr

E9 0 or E9 =-1 or E9 = 2. .See the notes to

state-' ment 450.

(28)

r

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,...,

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irr

'''

- '''' T IT T TT tT 11-1-11-111 :11111, IT, 114 ilit ip,p1 1_,_c. 111,1. - 1 ,11rIti r," II, .-11111. .. 1 .. -IL. 111,111. I . ' 1 ul -I 1, 1 11 1 1111 1,1 i$1,11 11.,1,;:i41:11:;: 1 :,, -11...1.1 . 11 .'4044' .1-1.'1' S i ;1111hr ;--, -I) ,J11-;h:1 ::-;::-:.:- 1 .:::,/:::: :

(29)

1:1112:!

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(30)

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'''

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