Experimental procedure for the determination of wave resistance in a towing tank

EXPERIMENTAL PROCEDURE FOR THE

DETERMINATION OF WAVE RESISTANCE

IN A TOWING TANK

by

David D. Moran

Sponsored by

Bureau of Ships Fundamental Hydromechanics Research Program Naval Ship Research and Development Center

Contract N00014-68-A-0196-0005

IIHR Report No. 133

Iowa Institute of Hydraulic Research

The University of Iowa

Iowa City, Iowa

August 1971

This document has been approved for public release and sale; its distribution is unlimited.

TABLE OF CONTENTS INTRODUCTION EXPERIMENTAL EQUIPMENT EXPERIMENTAL PROCEDURE COMPUTER OPERATION WAVE RESISTANCE

RECOMMENDED EXPERIMENTAL PARAMETERS APPENDICES

POWER-UP POWER-DOWN PROCEDURES FOR 1801

LISTING OF LONGITUDINAL-CUT WAVE-PROFILE DATA COLLECTION AND CALIBRATION PROGRAM "WAVES"

III LISTING OF DATA PUNCHING PROGRAM "MORAN"

A LONGITUDINAL-CUT METHOD FOR COMPUTING THE WAVE RESISTANCE OF A SHIP MODEL IN A TOWING TANK

LISTING OF WAVE RESISTANCE PROGRAM LISTINGS OF EXTRA SUBROUTINES

VII LISTINGS FOR ITERATION METHOD

ACKNOWLEDGMENTS

This report is based upon research conducted under the Bureau of Ships Fundamental Hydromechanics Research Program administered by the Naval Ship Research and Development Center, Contract N00014-68-A-0196-0005.

The author would like to gratefully acknowledge Chii-el Tsai for his valuable aid in computer programming.

ABSTRACT

The technique, equipment, and experimental procedure presently used at IIHR to obtain a record of the water-surface profile at a fixed

point as a ship model is towed past it, is described in detail. The

equations and computer programs used to obtain the wave resistance of the ship model from the acquired data are also presented.

EXPERIMENTAL PROCEDURE FOR THE DETERMINATION OF WAVE RESISTANCE IN .A TOWING TANK

, INTRODUCTION

The drag associated with the waves generated by a ship moving

on

the surface of a towing tank may be determined from a longitudinal-cut'

wave profile. This report presents the technique, equipment, and

procedure used at IIHR to determine the wavemaking resistance of ship model. This report iS intended to be used as a guide to all aspects of

data collection and reduction. The procedures and computer _programs

described herein are complete and may be used as presented, This is not

to imply, however, that use of the longitudinal-cut technique is routine.

The experimenter should use the recommended values of the _parameters

record length, transverse position, etc. only as guide lines and should

verify the applicability of' these values for the specific testing

conditions,

II. EXPERIMENTAL EQUIPMENT

The following is a list of the various pieces of equipment which

are used in various phases of data collection Or wave-study _towing-tank

preparation,

1- _{Aluminum Channel to mount wave gauge on}

Check to make sure that the carriage will clear the

channel and capacitance gauge before making an runs.

Four wing nuts, are supplied with channel _{to screw it to}

one of the towing tank rail braCkets (starboard side _at

foot. of stairs.

0

4, A A. s. A A A 4- C. *4 1. IL. %A.

Figure 1, Aluminum Channel

-2-Mount wave gauge bracket on the wooden block which slides

in the aluminum channel. Hold to channel with C clamps.

Figure 2. Wave Gauge Mounting

Capacitance Wire Black Box

Figure 3. Capacitance Wire Black Box

The electronic circuit of the capacitance-wire wave gauge is

shown in Figure

4.

Photocell. Plug into multiplexer terminal 03

Figure 5. Photocell

3.

SO K4-17.

_XTAL

s+

8),

Figure 14.

/fi

.01

COARSE

A D7UST

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N 6 7

10K

-E

ADSU ST

10K

tOOK

10K

,\A"tv

CAPACrTANCE

TRAMDUCE

.01

1001k

_VWVi

10K

SA-2-OUT

-4-Light source. The light source is mounted on the carriage.

Use a light bulb with a wattage rating of 100 or larger.

Figure 6. Light Source

Board to skim water surface

Figure 7. Skim Board

-5-III. EXPERIMENTAL PROCEDURE

The complete experimental procedure for collecting longitudinal-cut data is as follows:

Lower water level in tank to at least 6 inches below wave

dissipating boards.

Skim the surface of the water in the tank by towing the

skimming board from West to East in the tank. Tie up board

and leave in tank at East end. Flush out surface scumwhen

the water level is raised again.

Mount boat model. Make sure that it is rigidly fixed so

that there is no force on the dynamometer. Check movement

and stability of model.

Connect tachometer to overhead cable.

Connect all electronic equipment. Plug- output of wave gauge

in Multiplexer Terminal 00. Plug photocell output in

Multiplexer Terminal 03. Connect counter near carriage

operator to record speed of carriage.

IMPORTANT Check to IiLke sure that there is adequate

clearance between carriage and ship model and the wave

gauge probe and support mechanism. Run the carriage slowly

past all equipment to check clearance.

Check operation of photocell by observing flash of red bulb on the photocell unit as the light source passes the photocell.

Position carriage and calibrate probe. Be sure that the probe

will clear the carriage and model before moving carriage. Measure the transverse position of the wire, y, relative to the ship centerline.

The origin of the x-coordinate should have been measured

already by measuring the x distance between Ship CL and

Light Source, and the x distance between photocell and wires.

Return carriage to end of channel to start run. _{Wait for}

the waves to subside to an acceptable level. Make run and punch data.

Return carriage to end of channel and wait for waves to subside

(20 - 30 minutes). _{Check wire for floating bubbles before the}

next run. 10,

-

-6-IV. COMPUTER OPERATION

The procedure for starting (powering) the computer is given in Appendix I.

The wave profile along a longitudinal-cut i,: recorded by the 1801 computer and punched on cards in a two pass process.

The first program (Appendix II) which is used to collect data and store it on disk is a "process" program and is loaded into the

computer by pressing the process interrupt switch 02 (PSIW 02) or by

loading the name "WAVES" into the queue table of the computer process monitor.

When properly loaded, the computer will type the line labeled

A in Figure

8.

In order to type the set of messages indicated as B, Data Entry Switch 15 should be set to the on position and the button labeled START

should be pushed on the console of the 1800. As the messages indicate,

three program choices are possible. The first, a calibration program allows

individual wave elevations to be read each time you push START. The third

is an exit routine which allows you to end the computer operation at any

time during the calibration process. (Set DESW 13 and push START.) The

second option is the data collection program described below.

The numbered WAIT's refer to the contents of the A register, i.e,., if the hexidecimal number /0004 is shown in the A register and the computer is in the wait mode then the program is currently stopped at point WAIT 4.

If after line A is typed, the messages are not desired, set DESW 15

off and push START. In either case, the program will next halt at WAIT 1.

The desired option should be chosen by setting the appropriate Data Entry Switch and pushing START.

In the example, the calibration part of the program has been

loaded by setting DESW 15 on and pushing START. The message labeled C

is immediately typed and the program stops at WAIT 2. Each time you push

START the water surface elevation is recorded and typed. The data are

numbered for future reference. A set of sample data is labeled as D in the

figure. The actual numbers have no physical meaning except that +32,767

corresponds to +5.000 volts at the multiplexer terminal 00. The numbers do,

-7-longitudinal-cut. It is important therefore that the calibration be performed

using these values with the associated actual wave height.

When the calibration has been completed, (the computer is still at WAIT 2) either set DESW 13 and push start to exit and clear the 1801 for

the next user, or set DESW 14 and push START to obtain the data collection

program.

In the example, the data collection program has been loaded.

The message labeled E is typed by the program and then halts at WAIT

4.

The starting-loop time should now be entered in 1/8's of milliseconds

(e.g. 3.625 m sec requires a DESW setting of /001D). The starting-loop

time allows a time delay between the time when the ship model passes the

photocell and the initiation of data collection. If no such delay is

desired, set the DESW to /0000 and press START. In either case, the program

will pause at WAIT 5. The program looping time (Delta T = the time in

1/8's of milliseconds between consecutive samples recorded)should be loaded

into the DESW in 1/8's of milliseconds. This time delay, delta T, must be

greater than 2 milliseconds (2 msec. corresponds to a setting of /0016 on

the DESW). If a time delay of less than 2 msec. is entered, the program

will type the message labeled F (***Delta T < 2 msec....) and the program

will pause at WAIT 5. Reload delta T and push START. When this parameter

has been correctly specified, the program will type the starting loop time

and delta T as indicated by line G in the example. Both values typed are

in eights of milliseconds.

The program is now looping and reading the voltage at multiplexer terminal 03, waiting for the light source mounted on the carriage with

the ship model to pass the photocell. When this occurs, the program

collects data at the specified time delay, stores the data on disk, and automatically exits leaving the computer free for the next user.

The data must now be punched on cards as desc ribed below. If

more data collection is attempted by reloading the data co-lection program H before punching the data from a previous run, the message labeled I is typed.

The program used to punch the longitudinal-cut data on cards is

a "non-process" program named "MORAN". This program is executed by loading

the two cards indicated in Figure 9 into the cardreader hopper followed by

a deck of blank cards. The data is punched in a hexidecimal format, 20

-8-PIPR5 * WAVE GAUGE PROCPAM *

RE'lt 17 For

MEF!7Arrr

WAIT 1 * CHOOSE

PROGRAM

*

SET DESV 17

For rALI9R"\TIor

SrT REf'W 14

To COLLECT

nATt

SET RE711

13 TnEXIT

WAIT 7

* CALIBRATION

+nnnnn

+nnnn2

*** RE [TA T

) 2 M!;:EC

*

ENTER AnpIr ***

+00000

+onolr

PIPR5 * WAVE

GAUGE pRoonm * SET

ncsu 15 FOR

MESStGES

*** DISK FULL * PUNCH DATA

* * *

Figure

8.

Example

1801

Output

PROGRAM

*

Fu!,r F,TrRT

WAIT 4 * DATA COLLECTION

* ENTER START Loop TIME

Ir

ElrrTs oF rf7rc

WAIT 5 * DATA COLLECTION

* ENTER DELTA T 1r

EIGHTS Or Mr:2EC

* PELT >

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TEC

CALIBRATION PROGRAM # 001 +00350 -# 902

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# 903

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005

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# 006

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# 007

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# 009

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# 010

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RATA COLLECTION PROGRAM

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-10-V. WAVE RESISTANCE

The analytic procedure for obtaining the wavemaking resistance

of a ship model in a towing tank is presented in Appendix IV.

The solution of the set of simultaneous equations required by

the longitudinal-cut technique is accomplished by inverting a matrix of

coefficients and matrix multiplication.

The FORTRAN computer program which may be used to reduce the longitudinal-cut data and compute wave resistance is given in Appendix V. The program consists of a main program and eight subroutine subprograms which must be loaded and run simultaneously.

The input data to the wave-resistance program requires (1) a leader card, (2) a series of calibration information cards, and (3) the output deck of the IBM 1801 program MORAN.

(1) The header card contains the constants with the associated

FORMAT configuration noted in parenthesis.

NRUN Run number or alphameric designation (A4)

TRNGTH Waterline length of model in feet (F7.3)

SURFA Wetted surface area of model in square feet (F7.3)

Towing-tank width in feet (F7.3)

Distance from ship center plane to the measuring

device in feet, normal to the direction of motion (F6.3)

XBEGIN Distance between ship center plane and measuring

device, parallel to the direction of motion, at initiation of data collection (when the light source

is opposite the photocell. Sign is negative if the

wave gauge is ahead of the model center plane and positive when aft of the center plane (F7.3)

SPEED Speed of model in feet per second (F6.3)

DELTT Delay time between successive data points in seconds (F9.6)

NDIM Number of data points read into memory to be used in

the analysis (I5)

Truncation constant or order of the analysis (I3)

XZERO Lower limit for finite integrals (F7.3)

XT Approximate upper limit for finite integrals (F8.3)

NCAL Number of points on calibration curve read into the

computer (I4)

All constants are punched on one card beginning in card column 1

according to the overall FORMAT (A4, 3 F7.3, F6.3, F7.3,

F6.3, F9.6, 15, 13, F7.3, F8.3, 14).

The calibration curve is read in on from one to a maximum of four cards with the water surface elevation in feet and the associated calibration constant (proportional to voltage)

second, 5 pairs per card, according to the FORMAT (5[F7.3,F9.1]). A linear interpolation is used between the specified points

on the calibration curve.

The data deck (output of IBM 1801 program "MORAN" is punched

according to the hexidecimal FORMAT (20Z4). Normally,

256 cards are included for each run.

VI. RECOMMENDED EXPERIMENTAL PARAMETERS

The program in present form averages the first twenty wave elevations, so the wave profile must include points for the undisturbed water surface.

The best choice for the transverse position of the probe appears

to be y/b = 0.30. This position is outside of the wake region for most

(2)

-12-models, but several positions should be used and the results compared if the width of the wake region is unknown.

The time delay DELTT must be chosen with care. A physical

spacing of 0.1 foot is recommended for the wave resistance program. This requires that DELTT be given by

DELTT = 0.1 / SPEED (f.p.s.)

The constant NDIM should be equal to 1024. For NDIM > 1040,

the dimension statements of the FORTRAN computer program must be charged.

For Froude numbers greater than 0.15 use M = 25. For smaller

Froude numbers, a larger value of M should be considered (i.e. M = 30). Increasing M dramatically increases the computation time and memory requirements.

The integral limits XZERO and XT should be chosen such that

xo/b 21 1 and 6 xT/b

5 _{10. The values XZERO = 10' and XT = 70' are}

recommended for the Iowa towing tank. Several values of XT should be

tested to assure that the wave resistance is approximately constant with variations in the downstream truncation point.

-13= Appendix I.

Power-Up Power-,Down Procedures for 1801

I. Power-Up Procedure

Turn on the power supply located

in

the

1826.

The switch is

located directly below the oscilloscope.

B. Press and release the POWER ON button located at the top of

the console on the 1801.

C, _{Press and release the 11PRO button on the 1442 card reader.,}

punch.

Press and release the two START buttons

on the 1810.

Wait until both READY lights, on the 1810 light before proceeding,.

F, Press and release the IMMED, STOP button located at the bottom

of the console on the 1801.

Press and release the START button. located at the bottom of the

console on the 1801. If the following message

COLD START PROGRAM.

EXITS VIA CALL VIAQ AFTER UNMASKING INTERRUPTS.

is

printed, the system is ready to execute either process, or

nonprocess programs, If this tetsage is not printed,., a

cold-start operation must be executed,

II. Cold-Start Procedure

Set the WRITE. STOR PROT BITS switch to the YES position and, all other console switches down.

Press and hold the CLEAR STOR button located at the top of the console, on the 1801, and then press and release the START

button located at the' bottom of the console on the 1801.

Release the CLEAR STOR button, and press and release the EMMED STOP button. Turn on Program Switch T.

Load the

1442

_{with cold-start cards followed by a blank and}

ready the

_1442.

Press and release the PROG LOAD' button located at, the top of

the console on the 1801. After the message

-TURN OFF WRITE STORAGE PROTECT SWITCH

is printed, press and release the START 'button. 'Do not turn _off'

.1(

the WRITE ,STOR PROT BITS switch. After the message,

ENTER TIME THROUGH DATA SWITCHES

is. printed, enter the time in hexadecimal,: hours

in

switches

0-7, and. minutes in switches 8-15. After' the message

COLD START PROGRAM.

EXITS VIA CALL VIA Q AFTER UNMASKING INTERRUPTS.

is printed, the system is ready to execute either process or nonprocess programs

III. Power-Down Procedure

'Terminate all process and nonprodess

jobs,

Press and release the DIMMED STOP button, and press and release the START button,

After the message

COLD START PROGRAM.

EXITS VIA CALL VIAQ AFTER UNMASKING INTERRUPTS,.

is printed, press and release the IMMED STOP button:.

Press and release the two STOP buttons on the 1810, and the

NPRO button on the

1442.

'Press, and release the POWER OFF button oh. the 1801, and then

tura off the power supply located in the

1826.

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2:00120 2 226'00E142 2 a 2 2 200 2'282'2 111 1 I I, II I 1 I I I, 33331:3 333133:3 3130130:0 30313 3.30330333/0:3 31 444414:4446444:144 4 4 4004 44140404 4444 444 044 4' I I I I I I I II I I 1 555515550 5115:5 585:0 5000 5150/50 50 5 55:50.5.5 50,0 1 II 1 I 616 6 6 6166 6, 6 16 6 6 616 616, & 6,6 6 6 6,6 6 6:0 43 6 i4 0 I 1 1 1 7 7 7 717 7 700p 73 7 717 7 7'7117 771 717 7 7 7 7 7 7 7 7 747 7771 1 I 1 1 a 82222 8 8 8:2 81 8448 0;88810313 8388 a SOC 3;8 3 83,8 4 ₁ II I 9,9999899900 9 99 919 9 9 9 9 9 9 9 9949 S 59 6559 9 '39 9 SIS 41 47 413 44 4%41 41 44146 $051 12 53 14 5456 5/ 51 51 1 61 62 63 64 15175 61 66 49 70 71 72 13 74 7543 /7 III 76 63 OD Z1411110 00'

01

0 0212 20

ED:

11 222 4, 11211120220; la all

:4 0 0 60 0 0 0I3 CCO 010 0 OCI: OD130 0 DO 0 000102 000 ea 0 0i0 0 0 CI 0

k-,1 1 2 7 4 I: I11 10 11 17 13 14 73410 I/ HI /9 20 71 22 2374 25176 17 21 21 30 31 31 23 31 ssiso 3/ 14 31 40 41 42 43 44 45146 47 411 4115d an 6 6111.6 a 66 6 I ' 7771711111 7 I ./ 9 1161 818 8 it 8 8

az

.10 u,o ra 013 1 i 1 II 1 02000n22o2ooloac:262 0110 0 0 0 010 0 0 0 1; 51 51 13 54 ssisi 57 54 SS 60 6142 63 64 4.fr,6 67 61 6140 n 71 73 /4 75171 7171 7160

1 Ill 11110 1 8 1 1 1.10:01HIL111.1 11,1 8 8 Ili

II I 1 2 2 2 2 242 2 2 22 2 21 2 210 2 2 2 2 2221217121 2., I , I 1 12 3, 3 30 30.3.0 3 3001 303 3 3 3 13 313 3 3 3,3 I 1 4 4 4 4 414 4 44 4 1 1 1 g 5.0 515 5:0 5 50 5 55 sa 5 55 55 5 55 545 5 55 5 v I 1 I Ci. 6 6 6 6 616 6,6 6 6 6 ,6 6 6 6143 6 66, 6 6 6 6, 6' EI5 G G 6 6 1 I 1 7 7000 7 7 7.7 77 7 7 11 177 7 7 7 71747 7 7 7 7 . I II 1 6 8 8 SO18 8 68 3 8 181 84 8 5 8 811 0 0 0 418 8 8 &III 1 111 1 111 1 111 2 22 2 212 2 2 22 1 3 3 3 3 313 3 13 3 1 6 6 6,6,616 6 616 G G. 66 613 61 6 6 6 411, 1 .1 1010 0 0 0/10 0 0' 0 0 0 0 01040 0 0 0 0 61 62 6.3 64 15460 61 62 69 74 il 12 12 74 71176 71 71 71 13 1 1 1 111 1 1 11 2 22 2 2/2 2 222' 3 313 313 3 3 32 4444 414 444 4 4 4 4 4 4144 4 4 1 5 5 5 5 515 515 5 55555 5155 5 5 5 G 66' 6166,0 66 7 7 7 7 717 7 7 7 1 7 7 7 7 717 7 7 7 7 7 7 7 11,11 7'? 1? 1 8 8 30 siasIssausslasasa ; 818813814s 1 1 9999929999999950195605 11 II 12 61 65166 17 66 61 hi st ra 74 ;sin n 16 11 60 0 11 1 4 3 I I I I I 6 I 1 1 0

Appendix II

Listing of Longitudinal-Cut Wave-Profile Data Collection and Calibration Program "WAVES"

Assembler for IBM 1801

-16-START

CAI.l BRA WOW Pito 6 'WAIT IReAlb D.E. S. Te

PiCK

PRO &RAM

ReA

I'IPXR

00

TYPE

I:47A

MaSSA6-6 01Sic. PULL

TIMER

I MITE RRuP T RoOT/Ase" [ REL.° /11: T/ME.It

A

I

RE A b AND

S To 12. ceArA RES E

Mire

U P r

*WAveS *

Wilve

PP° F/ LE - PRO & PAP/

RETURN Ness AGES RE/ID D.E.$.

To

Pi<

PRO GRAM"

0*TA

c.0 i.Lecr/o/ft PRo 65SA6.ES. ve/ 1' 'i Rs"re

D.(

F-h_os-5720 [wcfs =

.16 SecToRS STOP TiemElt. MACK b*J7ERRuPT$ REA 0, F/IEST hit).

$7O20

ONitf st<

Gam r4A RE To co itee wear) Pea

l="Sk

FWLL.

WAIT

Desk Empry

j

14eS$A6.6

Z N Mot 6 TA RT LooP Dei.A le mu 01 hes OF Agsex DELA

READ Da. s.

s Top -rilvieR A

IYes

67 o

tooRoj

i*Z4E/ID ? /MC Itemem7 we at. Coo N T

'WA

Fo R /NT& RR,0 P T

f

ISTAar -nrreft

Cr.o ReAD _PlPYk

03 Fog THE

PworeD cE 1_ YO,L.74

-- "TYPE

[Mop 4 DELAY

WAIT

ti*

R,eAt

I '067,A k;,#6 mseq

6-o

C AIP

To

REF Voi-TS

AT< Z1

coif,

_{DELA y}

Te

2 ivy sec

-17--M

00

A/0

N04 READY READER

If JOB 0000011111

"4)

//

ASH WAVES *LIST

*PRINT SYMBOL TABLE

***** **************** ***** ** ******** ***************

-18-*WAVES*

WAVE GAUGE PROCESS PROGRAM. PISW BIT 02. INTER- *

RUPT LEVEL 08. THIS IS A PROCESS PROGRAM FOR

CALIBRATION OF WAVE GAUGES AND COLLECTION OF DATA AT A GIVEN DELTA T.

CALIBRATION * MPLXR 00 IS READ AND TYPED.

DATA COLLECTION * TIME VALUES ARE ENTERED VIA *

DESW. PROGRAM LOOPS READING MPLXR 03 UNTIL CAR-*

RIAGE PASSES PHOTOCELL. DATA IS RECORDED EVERY *

DELTAT VIA TIMER INTERRUPT ROUTINE. 5120 (16 *

SECTORS) WORDS ARE STORED ON DISK BEGINNING AT *

FIL05. NON PROCESS PROGRAM 'MORAN' IS RE*

QUIRED TO PUNCH DATA TO CARDS IN A 20 HEX WORDS

PER CARD FORMAT. THIS PROGRAM WILL NOT RUN UN- *

LESS /7FFF IS STORED IN THE FIRST WORD OF DISK *

FIL05 BY THE NON PROCESS PROGRAM 'MORAN'.*

MORAN, 710201

**** ***** ******************************************

0000 20 23A17155 START LIBF TV PEN

0001 0 2002 DC /2002

0002 1 1529 DC DMESE-1

0003 0 0000 DC 0

0004 20 23A17155 LIBF TYPEN

0005 0 0002 DC /0002

0006 0 70FD MDX *-3

0007 20 17064885 LIBF PAUSE *** WAIT ***

0008 1 OODD DC MASKB

0009 01 000000D2 X10 IOCCA READ DESW, SET DES 15 FOR

000B 01 B40000EE CMP WAIT1 MESSAGES.

0000 0 7009 MDX ADD RA

000E 0 7008 MDX ADDRA

000F 0 1000 NOP

0010 20 23A17155 LIBF TYPEN

0011 0 2002 DC /2002

0012 1 1553 DC DMESA-1

0013 0 0000 DC 0

0014 20 23A17155 LIBF TYPEN

0015 0 0002 DC /0002

0016 0 70FD MDX *-3

0017 20 17064885 ADDRA LIPF PAUSE *** WAIT ***

0018 1 00EE DC WAIT1

0019 00 C4000008 LD L /000B

001B 0 0016 STO SAVES

001C 00 2C400008 STS

L 10008,64

CLEAR SPB

001E 01 00000002 XIO L fOCCA READ DESW, CHOOSE PROGRAM

0020 01 B40000EF CMP L WAIT2

0022 0 7002 MDX QUITS CALL EXIT IF DESW 13 SET

0023 0 700F MDX CALIB CALL CALIB IF DESW 15 SET

0024 0 704B MDX WAVES CALL WAVES IF DESW 14 SET

*********** ********************* ***** ** *************

0025 0 COOC QUITS LD SAVES

0026 00 04000008 STO L /000B

0028 00 2C410008 STS L /000B,65 SET SPB

002A 30 24554480 CALL UNMK

002C 1 OODE DC MASKC 002D 1 OODF DC MASKD * * * * * * * * * * * * * * * * * * * L L

rind )(SIG 0I NSIO GV313

wisla

A811 56119110 OZ 8/00 idino 1 OIS 911000110 TO 9100 IIIVM G1 8100 0 SLOO 84SVW 00 0000 I 1100 VASVW 00 3000 1 £100 NSVW 11V3 081190411 0£ ILOO (3 U3W11) StI3111 dOIS 03001 OIX S3AVM 5980 0 0100

141.441**0.44.440411.4144********44****4.11.4iii***************

Pd001 Xaw 0301 0 3900 £-* XOW 03OL 0 3900 Z000/ 30 Z000 0 0900 1S3I N3dAl 3811 SSILIV£Z OZ 3900 0 30 0000 0 8900 Z-V3bV 30 9110 I VS00 ZOOZ/ 30 ZOOZ 0 6900 INIOd VIVO 3dAl N3dAI 1811 SSILIViZ OZ 8900 tI 30 1000 0 1900 V3bV 30 OTTO I 9900 V3tIV 30 8110 I 5900 0000/ 30 0000 0 1900 tidlOH 3811 605£6S80 OZ £900 8+V3bV 00 OZIO I 1900 30NI8 3811 £0ISSZZO OZ 1900 SVIVO 1 01 63000010 IO 1500 L+V32JV 1 OIS --AII00010 TO 0500 94.V3/111 1 OIS 31100010 TO 8500 Z+V3tiV 1 OIS V1100010 TO 6500 I+VRIV 1 OIS 61100010 IO LSOO V3bV 1 OIS 8II00010 TO 5500 91 VlS OIOI 0 1500 V321V 30 8110 I £500 30N18 1811 cOISSZZO 01 1500 NdWVS 1 01

0000013

TO 0500 I'NdWVS 1 XOW £300101/ TO 3100 i-* XOW 030L 0 0100

OHO/

00 0010 0 0100 IS31 NidIV 3811 508151I0 OZ 8100 0 00 0000 0 V100 bX1dW 00 0300 I 6100 SVIVO 00 6300 I 8400 00II/ 00 OOTT 0 1100 t:IXdW OV31 NIdIV 3811 X1:100V 50815110 OZ 9400 53AVM XOW

WO/

0 51100 XbOOV xaw TOOL 0 1100

SUM)

xaw 1301 0 £100 ZIIVM 1 dWO 13000018 TO 1100 WVHDObd 3S00H3 'MS30 OV321 V0301 1 OIX 10000000 TO 1£00 ZIIVM 30 1300 I 3£00

ilym

3SIIVd 1811 Pd001 588190/1 OZ 0£00 NdWVS 1 OIS £3000010 TO 8£00 91

VlS

0101 0 V£00 XOW

010/

0 6£00

1000/

00 1000 0 8£00 N3dAl 3811 SSILIViZ OZ 1£00 0 00 0000 0 9£00 I-IS340 30 LIST I Si00 ZOOZ/ 00 ZOOZ

01O0

N3dAI 1811 811V3 SSILIViZ OZ £i00

4444,44...4....41,044471.411441411M4Mi4,4404i44*441.44

00 S3AVS 0000 0 Z£00 11V0 0091WIZ-11£

okuu--

DVIA

b3WII IbVIS 03301 1 OIX 80000000 TO 3100 4 -

_ LI 11.11

fr

?-k --H .1 _ 0079 0 1000 DC /1000 007A 1 0176 DC --- 01ITPT 007B 0 0000 DC 0

007C 20 04262495 LIBF DISKN TEST DISK

0071 G 0100 DC /0100 007E 1 0126 DC OUT PT 007F 0 70FC MOX *-4 0080 01 C4000128 LO OUTPT+2 0082 0 BO5F CMP H7FFF 0083 0 7002 MD' ADDRG 0084 0 7001 MDX ADDRG 0085 0 7008 MDX ADORE

0086 20 23A17155 ADDRG LIBF TYPEN DISX FULL

0087 0 2002 DC /2002

0088 1 1611 DC DMESF-1

0089 0 0000 DC 0

008A 20 23A17155 Li BF TV PEN

0088 0 0002' DC /0002

008C 70FD MDX *-3

0080 0 7097 MDX QUITS

008E 20 23A17155 ADORE LIBF TYPEN

008F 0 2002 Ot /2002

0090 1 1603 DC DMESC-1

0091 0 0100 DC 6

0092 20 23A17155 LIBF TYPEN

0093 0 0002 DC /0002

0094 0 70FD MDX

0095 20 17064885 ADDRD LIBF PAUSE *** WAIT ***

0096 1 00F0 DC WAIT4

0097 0 083A X10 IOCCA READ DES, START LOOP' TIME

0098 0 0048 STO STLOP

0099 20 17064885 ADDRK LIBF PAUSE *** WAIT *** ft:

009A 1 00F1 DC WAIT5

0098 0 0836 XIO IOCCA READ DES, DATA LOOP TIME

009C 0 D048 STO DELAY

009D 20 23A17172 LIBF TYPE2

009E 1 00E4 DC STLOP

009F 0 0002 DC 2

00A0 0 C044 LD DELAY

00 Al 0 8050 CMP SIXTN

00A2 0 700A MDX ADDRJ

00A3 0' 7001 MDX ADDRH

00A4 0 7008 MDX ADDRJ

00A5 20 23A17155 ADDRH LIBF TYPEN DELAY TOO SMALL

00A6 0 2002 DC /2002

00A7 1 1622 DC DMESG-1

00A8 0 0000 DC 0

00A9 20 23A17155 LIBF TYPE NI

00AA 0 0002 DC - /0002

00AB 0 70FD MDX *-3

00AC 0 70EC MDX ADDRK

00AD 0 1010 ADDRLY SLA 16

00AE 0 9036 S DELAY

0 AF 0 D035 STO DELAY

0080 0 9033 S STLOP

0081 00 D4000004 STO L /0004 LOAD TIMER A

0083 0 CO39 LD MAXNO

0084 0 D032 STO COUNT

0085 0 CO34 LD ATIMT

LOAD ADM Of TIMER MNTERUPT

0086 00 D4000008 STO L /0008

0088 G CO32 LD ADOUT

0089 0 102C STO POINT DATA SET POINTER

00BA 20-012578D5 LOOP? LIBF AIPTN LOOP READING PHOTOCELL

0088 0 1000 DC j1000

00BC 1 00E8 DC PHOTO

0080_1_ 00E1 DC MPLX3

L

00 BE 00BF 0000 00C1

-e)

00C2 00C3 00C4 0005 0006 0007 0008 00 CA 00CB 00CC 00CD 00CF 00 DO 0 0000 20 0125781)5_ 0 0000 70FD CO25 8028 0 70F5 0. 7001 0 70F3 0 080C 30 14062480 1 CODE 1 OODF 0 -3000 01 74FF00E7 0 70FC 0 7038. 00D2 0000 00D2 0 0000 00D3 0, 0740 00D4 0 8000 00D5 0 0420 0006 0 0000 0007 0 0420 _ 00D8 0 2000 00D9 0 0420 OODA 0 0000 OODB 0 0721 OODC 0 1DEO OODD 0 0000 °ODE 0 3FFC OODF FFC0 00E0 0 1000 00E1 0' 1003

__DDE2A7FFF

00E3 0 0000' 00E4 0' 0000 00E5 0 0000 00E6 0 0000 00E7 G 00E8 0 00E9 0 00EA L 00EB 1 DOEC 0 DOED 0 00EE 0 00EF 00F0 00F1 0 00F2 0 0000 0000 0000 00F3 0128 E69C 1400 0001 0002 0004 0005 0010 00F3 0, 0000 00F4 D COFO 00F5 00 D4000004 00F7 0 08E2 00F8 0 4878 00F9 0 1000 DC LIBF DC MDX LD CMP MDX. MDX MDX BEGIN X10 CALL DC DC TWATT WAIT MDX L MDX

MDk

********************** ********

*********M********tir

/0010 AIPTN /0000 *-3 PHOTO TESTX LOOPF BEGIN LOOPF IOCCB MASK MASKC MASKD COUNT,-1 TWA IT EXITS BSS E 0 IOCCA DC DC /0740 IOCCB DC /8000 DC /0420 TOCCC DC 10000 DC /0420 IOCCD DC /2000 DC /0420 DSWTM, DC /0000 DC /0721 MASKA DC /10E0_ MASKB DC /0000 MASKC DC /3FFC MASKD DC /FFCO MPLXR DC /1000 MPLX3 DC /1003 H7FFF DC /7FFF SAMPN DC ,*-* STLOP DC *-* DELAY DC *-* POINT DC *-* COUNT DC *-* PHOTO DC *-* DATAS DC *-* ATIMT DC TMINT ADOUT DC OUTPT+2 TESTX DC -6500 MAXNO DC -5120 WAIT1 DC /0001 WAIT2, DC /0002 WAIT4 DC /000A WAITS DC /0005 SIXTN DC 16 *********************** TMI NT DC ID . STO L X10 BOSC NOP *-* DELAY /0004 DSWTM -2+

START LOOP DELAY IN MSEC DATA LOOP DELAY IN MSEC DATA SET ADDR POINTER_ COUNTER

**Itif*************************

TIMER INTERUPT ROUTINE RELOAD TIMER A

LOAD TIMER

05W-7-.--CLEAR INTRPT AND SKIP NEXTOP

01FA 20 01257805 LtBF AI PTN READ WAVE ELEVATION

00FB 01 1100 . DC /1100

LOOP' TILL MPLXR 3 IS POS

START TIMER A

WAIT FOR I NTERUPT

READ DES START TIMER A STOP TTMERS START TIMER C TIMER OSW' MASK INTS 4 5, 7 II 9 10 I

MASK INTS NONE MASK INTS 2-13

MASK (NTS 14 - 23

MULTIPLEXER TERMINAL 00 MPLXR 3 , READ AT START Loop

ADDR Of TIMER tNTERUPT R0UTK1 -ADDR Of OUTPT AREA

MAX ALLOW WORDS -0 0 0 0 3 0 -0 0 *

*

* *

0109 0 08CC EXITS X10 IOCCC STOP TIMERS

010A 0 COE2 LD MAXNO

010B 0 DO1A STO OUTPT

010C 20 04262495 LIBF DISKN WRITE DISK

010D 0 3000 DC /3000

010E 1 0126 DC OUT PT

010F 0 0000 DC 0

0110 20 04262495 LIBF DISKN TEST DISK

0

0111 0 0100 DC /0100 0112 1 0126 DC OUTPT 0113 0 70FC MDX *-4 D 0114 01 4C000025 BSC L QUITS *

************ ************ ***** ********* ***** *********

* * 1529 0 0011) DC DMESY-DMESE

152A 0026 DMESE DMES '4RPIPR5 * WAVE GAUGE PROGRAM * SET

153D 0014 DMES DESW 15 FOR MESSAGES'E

1547 0000 DMESY BES 0

1547 0 000B DC DMESZ-DMES1

1548 0016 DMES1 DMEs 'R'RCALIBRATION PROGRAMIE

1553 0000 DMESZ BES 0

1553 0 00AF DC DMESB-DMESA

1554 001B DMESA DMES '2RWA1T 1 * CHOOSE PROGRAM

1561 001E DMES * SET DESW 15 FOR CALIBRATION'

1570 0039 DMES 'R'25X* SET DESW 14 TO COLLECT DATA'

158D 0030 DMES 'R'25X* SET DESW 13 TO EXIT'

15A5 0021 DMES '2RWAIT 2 * CALIBRATION PROGRAM *

15B5 000A DMES PUSH START'

15BA 001E DMES I2RWAIT 4 * DATA COLLECTION *

15C9 0023 DMES ENTER START LOOP TIME IN EIGHTS OF

15DB 0004 DMES MSEC'

15DD 0010 DMES 'RWAIT 5 * DATA COLLECTION *

15EB 0022 DMES ENTER DELTA T IN EIGHTS OF MSEC *

15FC 000D DMES DELT > 2 MSEC'E

1603 0000 DMESB BES 0

1603 0 0000 DC DMESD-DMESC

1604 001A DMESC DMES '2R DATA COLLECTION PROGRAWE

1611 0000 DMESD BES 0

1611 0 0010 DC DMESX-DMESF

1612 0020 OMESF DMES '2R*** DISK FULL * PUNCH DATA ***'E

1622 0000 DMESX BES 0

1622 0 0014 DC DMESW-DMESG

1623 0018 DMESG DMES 'R*** DELTA T ) 2 MSEC *

162F 0010 DMES ENTER AGAIN ***1E

-, _*

0116 0 0008 DC 8

0117 0 81C0 DC /81C0

0118 000E AREA BSS 14

0126 31 06253C35 OUTPT DSA FIL05

0129 1400 BSS 5120 OUTPT AREA

0102 0 COE6 LD DATAS

0103 01 Dk8000E6 STO I POINT

0105 01 740100E6 MDX L POINT,1

0107 01 4C8000F3 BSC I TMINT

-OOFC 1---00E9 DC DATAS

00ED 1 00E0 DC MPLXR

00FE 0 0000 DC 0

OOFF 20 01257805 LIBF AIPTN TEST

,

0100 0 0100 DC /0100 0101 0 70FD MDX *-3 . -'

-1637 0000

1638 _ 0000

ADDRA 0017 ADDRD 0095

ADDRJ 00AD ADDRK 0099

ATIMT 00EA. BEGIN 0007

DELAY 00E5 UMESA 1554

DMESE 152A DMESF 1612

DMESY 1547 DMESZ 1553

H7FFF 00E2 IOCCA 00D2

LOOPF 00BA LOOPJ 0030

MASKD GOOF MAXN0 00ED

PHOTO 00E8 POINT 00E6

SIXTH 00F2 START 0000

TWAIT 00CC WAIT1 00EE

WAVU_0070

NO ERRORS IN ABOVE ASSEMBLY. WAVFS

OUP FUNCTION COMPLETED ___a_XESIVIAVES L

N04 READY READER *CCEND

CtB, BUILD WAVES

R05 WAVES UNMK 0028 LEV.2

RO5 WAVES VIA° 0031 LEV.2

RO5 WAVES MASK 0072 LEV.2

R05 WAVES MASK 00C9 LEV.2

ROC ANINT 0022 LEV.0

ROC ANINT 00101 LEV,0

ROC

ANINT 0023

LEV.0

CORE LOAD MAP

TYPE NAME ARG1 ARG2

*CDW TABLE *IBT TABLE *F10 TABLE *ETV TABLE *VTV TABLE *PNT TABLE -MAIN WAVES PNT WAVES LIBF PAUSE CALL UNMK CALL VIAQ LIBF AIPTN

L1BF INDC

LIBF HOLPR CALL MASK LIBF TYPE2 CALL GAGED CALL UNGAG 2002 200E 2010 202D 2036 2046 204A 2048 3682 3696 36E0 3742 37C4 3812 38EE 3914 3968 3978 000C 000F 0010 0009 000F 0004 2036 2039 203C 203F 2042 DMESW BES END 0

START LAST CARD

-SYMBOL TABLE

ADORE 008E ADDRG 0086

ADDRX 0046 ADOUT 00EB

CALIB 0033 COUNT 00E7

DMESB 1603 DMESC 1604

_DMESG 1623 DMESW 1637

DMES1 1548 DSWTM OODA

IOCCB 0004 IOCCC 0006

MASKA 000C MASKB &ODD

MPLXR 00E0 MPLX3 00E1

QUITS 0025 SAMPN 00E3

__MOP 00E4

TESTX 00EC

WAIT2 ODEF WAIT 4 00F0

** ***** ****** ***** AD0RH 00A5 AREA 0118 DATAS 00E9 DMESD 1611 DMESX 1622 EXITS 0109 IOCCD 00D8 MASKC OODE OUTPT 0126 SAVES-0032 TMiNT 00F3 WAITS 00F1 it 111

-CALL ANINT 3986

CALL _PELT_ 3AEO

CORE 3B2C 0404

CLB, WAVES NI_ --NX NO4 READY READER

-24-

-25-Appendix III

Listing of Data Punching Program "MORAN" Assembler for IBM 1801

NEW CAR°

READ A

sEc

ToR

FROM DISK

_

3o won*

V WRITE /8000 IR FIRST viD

OF SEC ToQ

CoNVER7 20 WoRDS

To HEX.

PUNCH

ONE

CAI 0

-26-V

110W MucH

DATA HAS

Geriv PuAr-i-P

* MoRAM *

PuNco/

WAVE

Pito

FILE

FROM

DI SK

NEoA/ SecTo R

ADD ONE

To SEc TOR

ADDRESS LI

N04 READY READER

// JOB 0000011111

// ASM MORAN *LIST

*PRINT SYMBOL TABLE

************************************** *************

-27-*MORAN*

THIS NON PROCESS PROGRAM 'MORAN' PUNCHES 5120 WD* (16 SECTORS) FROM DISK BEGINNING AT FIL05

FORMAT IS 20 HEX NUMBERS PER CARD. THE PROGRAM * STORES /7FFF IN THE FIRST STORAGE LOCATION OF *

FIL05 MORAN, 710201. ****************************** ***** **************** 0000 0 1010 START SLA 16 0001 0 0043 STO COUNT 0002 0 C046 LO CN320 0003 0 D04B STO ELEVS 0004 0 D047 STO DUMMY

0005 20 04262495 LOOPC LIBF DISKN READ DISK

0006 0 1000 DC /1000

0007 1 004F DC ELEVS

0008 0 0000 DC /0000

0009 20 04262495 LIBF DISKN TEST

000A 0 0100 DC /0100 0008 1 004F DC ELEVS 000C 0 70FC MDX *-4 0000 0 1010 SLA 16 000E 0 8036 CMP COUNT 000F 0 700B MDX ADDRA

0010 0 700A MOX ADDRA

0011 0 CO35 LD H7FFF

0012 0 DO3B STO DUMMY+2

0013 20 04262495 LIBF DISKN WRITE DISK

0014 0 3000 DC /3000

0015 1 004C DC DUMMY

0016 0 0000 DC 0

0017 20 04262495 LIBF DISKN TEST

0018 0 0100 DC /0100

0019 1 004C DC DUMMY

001A 0 70FC MDX *-4

0018 01 74010050 ADDRA MDX L ELEVS+1,1 ADD ONE TO SECTOR ADDRESS

001D g 62F0 LOX 2 -16 16 20 WD CARDS PER SECTOR

001E 0 CO2C LD ADELV

001F 0 D026 STO POINT

0020 0 61EC LOOPD LOX 1 -20 20 WORDS PER CARD

0021 0 CO28 LO ADPUN

0022 0 D003 STO ADDRB

0023 01 C4800046 LOOPB LD I POINT

0025 20 02255227 LIBF BINHX

0026 0 0000 ADDRB DC *-*

0027 01 74040026 MDX L ADDR8,4 4 LOCS PER DATA WORD

0029 01 74010046 MDX L POINT,1

002B 0 7101 MDX 1 1

002C 0 70F6 MDX LOOPB

0020 20 03059115 LIBF CARDN FEED

002E 0 3000 DC /3000

002F 0 0000 DC /0000

0030 20 03059115 LIBF CARDN TEST

0031 0 0000 DC 0

0032 0 70F0 MDX *-3

0033 20 03059115 LIBF CARDN PUNCH

0034 0 2000 DC /2000 0035 1 0192 DC PUNCH-1 0036 .0_ 0000 DC

/0000_

- __

-,,

MORAN

OUP FUNCTION COMPLETED // DUP

r

PNT MORAN 2030 LIBF BINHX 2222 2036 CORE 2258 1048 1 CLB, MORAN LD XQ 1

DUP FUNCTION COMPLETED

104 10.335 02 1442 NOT READY

Tt---R

ADDRA 001B ADDRB 0026 ADELV 004B AD PUN 004A, CN320 0049 _

COUNT 0045 DUMMY 004C _ELEVS 004F H7FFF 004T LOOPB 0023 1

LOOPC 0005 LOOPD 0020 MAXNO 0048 POINT 0046 .PUNCH 0193. 1'

UITS 0043 START 0000

i"

NO ERRORS IN ABOVE ASSEMPLY.

0037

MBA

20 03059115 n nnno Lior DC CARDN TEST 0 0039 0 70FD MDX *-3 003A 0 7201 MDX 2 1 003B 0 70E4 MDX LOOPD 003C 01 74010045 MDX L COUNT,1 003E 0 C006 LD COUNT -003F 0 B008 CMP MAXNO 0040 0 7002 MOX QUITS 0041 a 70C3 MDX LOOPC 0042 0 7000 MDX QUITS

0043 30 059C98C0 QUITS CALL EXIT

-00145 0 -0000 COUNT DC *-* NO OF SECTORS * 11334.

0046 a 0000 POINT DC *-*

0047 0 7FFF H7FFF DC _ /7FFF

0048 0 0010 MAX NO DC 16 MAX NO OF SECTORS READ

0049 a 0140 CN320 DC 320 _

004A 1 0193 AD PUN DC PUNCH

0048 1 0051 ADELV DC ELEVS+2

004C 31-06253C35 DUMMY DSA Fl L05 004F 31 06253C35 ELEVS DSA FIL05

0052 0140 BSS 320

0192 0 0050 DC 80

0193 0050 PUNCH BSS 80

01E4 0000 END START

SYMBOL TABLE'

CLB, BUILD MORAN

CORE LOAD MAP

TYPE NAME ARG1 ARG2

11

CON___IA_B_LE__2002 000C_

*IBT TABLE 200E 000F

*F10 TABLE 201D 0010 1

*ETV TABLE 2020 0009

*VTV TABLE 2036 0003

*PNT TABLE 203A 0004,

MAIN MORAN 203E.

*STORECIX 0 MORAN MORAN COLDS

-29-Appendix IV.

A Longitudinal-Cut Method for Computing the Wave Resistance of a Ship Model in a Towing Tank

by

D.D. Moran and L. Landweber Institute of Hydraulic Research

The University of Iowa Iowa City, Iowa

for

16th American Towing Tank Conference Sao Paulo, Brazil

-30-A Longitudinal-Cut Method for Computing the Wave Resistance of a Ship Model in a Towing Tank

by

D.D. Moran and L. Landweber Institute of Hydraulic Research

The University of Iowa Iowa City, Iowa

II.

Introduction

The wave resistance of a ship model can be computed from

measure-ments of the surface disturbance produced by a moving body. The current

trend in wave-resistance analysis is to use Eggers (1]. expressions for the

surface disturbance derived for a laterally unbounded fluid and to employ only the data taken upstream of the first wall reflection in the towing tank. The remainder of the record, which is required to be of infinite length by the method of analysis used, is approximated by an asymptotic expression

valid far astern of the model. The asymptotic expression, first introduced

by Newman [2], contains several unknown coefficients which are evaluated by

a least-squares fit to the measured data. This approach neglects wall effects

and the near-field terms in the expressions for surface disturbances. The

first assumption, that only the far-field expressions are significant, results in major error in experiments where the ship-length to tank-width

ratio is of the order of unity. In this case, all of the data must be

collected near the model,thereby violating the far-field assumption.

The present study develops a method of predicting wave resistance, using a significant but finite amount of data measured along a line parallel

to the direction of motion of the model. This single longitudinal cut is

considerably longer than is used in the aforementioned procedure, including

waves over a range of several wall reflections. The upstream truncation

point may be taken far enough downstream that the near-field effect may be neglected without error.

-31-II. The Single-Longitudinal-Cut Method.

Figure 1. Sketch of Waves in Towing Tank.

The expression for the wave pattern produced by a body moving on or near the free surface may be given by

C(x,ko) = y [c (k ) cos wx

s

(k ) sin wx] cos _27T127

m

m o in o in

m=o

provided that we are sufficiently far enough downstream to justify neglecting

the near-field portion of the wave equation. This assumption is easily

satisfied for x >

The other variables in this expression are

ko = g/U2

km =o2 + vm2

= (ko + km)

The wave resistance produced by a body is given by

kb

0o _k Cw = o [Co2 + So2 + in (Cm m2

+ S2)]

22,2 m=1 k + k (2) 0 in (1) m b

-32-where the coefficients are the same as those appearing in the expression for wave elevation.

This set of unknown coefficients may be determined by the following

where p(n) is a set of integers chosen such that for a given downstream truncation point xT, the difference

2rp(n)

);-T

-n

is a minimum. We further define

x + 2TIP(n) o wn

Icos

wnx dx m

cos wx

sin unx 2rp(n) x + U) f 0 =

}

n Lp(n) xo sin unx ' cos w x sin u n dx

(4)

(5)

procedure. First define

FP(n)}

Gnp(n) 2rp(n) (3) xo + cos wnx I = ,(x,y) n sin w x d x L

n

o mx inn Tm .= cos 2221

(6)

After integrating Equation 1 and truncating the series at in = M,we obtain

the set of 2(M + 1) linear equations

FP = [C IP T + S J T ] n=0, 1, M (7) n in inn in in inn in m=0 Inn wn ,

C'S

_min

-33-G=

[C KP

T +S LP

TI

n=0, M (7)

n in inn m m inn m

m=0

involving the set of 2(M + 1) unknowns

m=0, 1, ..., M.

Solving the set of equations for the unknown coefficients and substituting

this set into the truncated series of Equation 2 yields directly the

wave-resistance coefficient Cw

III. Numerical Study.

The accuracy of the method was verified both experimentally and

and numerically. The numerical study used a body generated by a known source

distribution. For simplicity, a linear source distribution was chosen, with

the source strength given by

M(x) = constant x < ft, - < z < 0

'

The exact wave resistance computed for this source distribution may then be compared with the wave resistance coefficient computed by the present method.

The results of the numerical study indicate that the method may be used to obtain wave-resistance coefficients which are consistently within

1% of their exact values. _{In most cases this error is due solely to the}

truncation of the series expression for the wave-resistance coefficient (as opposed to the error due to poor conditioning of the set of linear equations for the coefficients).

The accuracy of the method is independent of the transverse position

of the cut y/b,except that the analysis requires _{that the following condition}

must hold:

IV. Experimental Results

Physical experiments were performed in the towing tank at the Iowa

Institute of Hydraulic Research. The towing tank is 10 feet wide, 10 feet deep,

and 300 feet long. The longitudinal-cut wave profiles were measured with a

-34-Variation of the model-length to tank-width ratio i/b yielded consistently good agreement between the exact and computed values.

Numerical experiments at various Froude numbers indicated that the number of terms required in the series expression for Cw increases with decreasing Froude

number. Twenty terms are required in order to obtain one percent accuracy for

Froude numbers greater than or equal to 0.25. For Froude numbers in the range

0.25 > E > 0.15, 25 terms are required for comparable results.

The most significant feature of the analysis was the identification xrj-xo

of a critical length of wave record ( for various values of

crit.

the parameters kok, y/b, and M. For record lengths shorter than the

critical value, the error increased without bound with decreasing length of

record. Above the critical limit, the error was essentially constant and

less than one percent of the exact value. A summary of the critical

wave-record lengths is given below.

Summary of Critical Wave Record Lengths

ko 1 2./b y/b )CT x0 0 b )crit. 4.20 0.345 .30 .32 20 5.1 4.20 0.345 .30 .32 25 2.8 5.55 0.300 .30 .32 20 5.0 5.55 0.300 .30 .32 25 2.4 4.20 0.345 .60 .32 20 7.2 4.20 0.345 .60 .32 25 3.5 ) k/b

-35-capacitance-type surface piercing wave gauge which was located 150 feet from

the end of the towing tank. The wave gauge was attached to a horiZontal rail

which was in turn fixed to the channel wall. A ship model was attached to

the towing tank carriage and towed the length of the channel at a constant speed. The wave profile was recorded by sampling the output of the

wave-gauge circuit at constant time increments. This was done automatically by

the Institute's IBM 1800 computer. The sampling process was initiated by an

electrical pulse, generated when a light source fixed to the carriage passed

a photocell attached to the channel wall.

The ship model employed in this study was a 10-foot Series-60

model of 0.60-block coefficient. The model has the following characteristics:

Displacement = 275 lb., Waterline Length = 10.17 ft., Wetted Surface Area = 17.64 sq. ft.

Longitudinal-cut data were collected using this model for several different speeds and transverse positions of the cut.

The results of the experimental study are shown in Figure 2. The

curve of wave resistance (CT _C) was constructed using the total drag data of Wu [3] and the viscous drag data obtained from a wake survey by Tzou [4]. The wave resistance computed by the present single-cut technique is indicated

for three different values of y/b. In each case, the wave-record truncation

points were taken as xio= 10 ft. and

xT= 70 ft. The series was truncated at

M = 25. Also shown in this figure are Ward's results for the same model using

his method [5].

At higher Froude numbers the agreement between the measured wave

resistance and its expected value is quite good. The rapid variation of C

with I near the point E = 0.32 may be expected since the longitudinal-cut technique is more sensitive to the effects of reinforcement and cancellation of the bow and stern waves than methods not based directly upon the measure-ment of wave amplitudes.

The good agreement between various values of the transverse position y/b suggests that the effect of the wake on this longitudinal-cut method is

-36--

,

negligible since the waves measured at y/b 0.2 are close to the wake region

These results also imply that there is MO optimum choice of y/b for experi,

mental work. No additional restrictions on the choice of y/b were detected

in the experimental results.

References

If

I

II

K. Eggers, "Ober die Ermittlung des Wellenwiderstandes eines

Schiffs=-modells durch Analyse seines Wellensystems," Schiffstechnik, Bd.

9,

Heft

46, 1962, pp. 79-84.

J.N. Newman, "The Determination of Wave Resistance. from Wave Measure- '11

ments Along a Parallel Cut," Proceedings of the International Seminar.

on Theoretical Wave Resistance, Ann Arbor, Miohigan,,August

1963.

Jin Wu,and L. Landweber, "Variation of Viscous Drag with Froude Number,'' Proceedings of the Tenth International Towing Tank Conference, Teddington,

September

1963.

K.T.S. Tzoil, and L. Landweber, "Determination of the Viscous Drag of 0.

Ship Model," Journal of Ship Research, Volume

12,

number 2, June

1968.

Lawrence W. Ward, "Experimental Determination of Ship Wave Resistance. 1$

from the Wave Pattern," Webb Institute of Naval Architecture, May

1964.

t

Wave, Rests. Coeff. C

0.0030

0.0025

-0.0020

0.0015

-37-=MN

Ward, x-y Method,

1964

Model: Series 60, L =

10.17, CB =0).,6

Longitudinal Cut . Analysis: M = 25

xo/b = 1.0

xT/b = 7.0

I

a

0.30

Figure 2. Wave Resistance of Series £o

Model.-0

0.35

1 06

A

tpIPS 0.0010 e .0 0.0005 <> CI '0

0 0

0,

0

0

0 Co 1

0.15

0.20

0.25

IF =,

.(Tzoul 1967)

.61M

Wu, 1963)

y/b =r 0.2

o

y/b

0.3

y/b = (L4

0

= =

- C

0

0

-38-Appendix V.

Listing of Wave Resistance Program Fortran IV for IBM 360

! FORTRAN IV G LEVEL 1*9 MAIN DATE = 71222 08/04/29.

FINITE INTEGRAL METHOD NEGLECTING NEAR FIELD TERMS COMPUTE WAVE RESISTANCE FROM LONGITUDINAL CUT PROFILE IN TOW TANK COMPUTE AND INVERT THE COEFFICIENT MATRIX AND PRINT IT THIS METHOD IS A SINGLE CUT METHOD MATRIX RANK IS AMP2 = 2*(M#1)

*

P IS THE UPPER LIMIT ON THE

CMNPX INTEGRALS, Y IS THE 01ST. OF THE CUT FROM THE CENTER LINE-. THIS ROUTINE REQUIRES A P FOR EACH N SO THAT THE UPPER LIMIT ON THE INTEGRAL * X042*P*PI/OMEG4N * IS APPROXIMATELY A CONSTANT_ EQUAL TO THE CONSTANT

CUT, WHICH IS THE MAXIMUM LENGTH OF THE

WAVE RECORD

TH1] FUNCTIONS F & G MUST BE READ

CSUBW IS THE WAVE COFF BASED UPON MODEL LENGTH

ELENTH.

CSUBAS

IS BASED UPON WETTED SURFACE AREA

SURFA.

REQUIRES SUBROUTINES GETFSG FANDG CSMNPX SSUBM COEFS MINV .REQUIRES SUBROUTINES SIMPS

; 0001 IMPLICIT REAL*P(AHpOZ,$) 0002 REAL*8 KIERO/KSUBM . 0003

DIM1.:NSI0N A(52,521, AF(52952)

0004

DIMENSION FGS(52), CSS(52), LDUM(52/0 MDUM152)

0005 DIMENSION CMNPX(41, VECOFP(26) 0006 EQUIVALENCE (A(1),AF(1)) C C C s s

* * * * * * * *

*. * *

* * * * *

se ***** .*

* * C C

NOTE THAT IDIMENSION STATEMENTS my&T BE THE SAME AS AMP2 = 2(M*1)

C C * *

** I

*

****

*

** *. * *

4, * 4 * * * * * * 4. * * * * * *

C C 0007 16 FORMAT(111,5X0WAVE RESISTANCE st 'COEFFICIENT',//,4X, 1.M1,14WC(Al',22Xp'S(M)1122XOTERM',22X0CSURA'#/) 0008 17 FORMAT(I5.4F25.16) 0009 18 FORMAT(

/////

00X,'FROUDE NUMBER =11E8.5 p

3X0CSUBWILENGTHI'l

1' =',F20.16, 5X, ICSUBW(SURFACE AREA) 10E20.161

0010

22 FORMAT(' ERROR IN FACTR ROUTINE NUMBER',I3)

0011

23 FORMAT(' ERROR IN RSLMC ROUTINE NLIMBER'fI3)

0012 PI=3.1415926535897930O 0013 TWORI.PI+PI 0014 100 CALL GETFSG(M,FGSIVECOFP,KZERO,Y,XZEREWLENTHISURFA,B/ 0015 MEN0=M+1 0016 MAP2=2*MEN0 0017 CON = TWOPI*Y/B 001$ FROUDE=1.000/DSORT(KZERO*ILENTH) 0019 CCOFF=KZERO*8/(ILENTH*ZLENTH) 0020 CCOEF=CCOEF.ECCOEF 0021 DO 500 IP1=11pMEND 0022 0023 ZI=OFLOATIII 0024 TSUBI*DCOS(CON*1I4 PAGE 0001 7, 7 11 , 7.3 - :3 -] 41 - ,

j

C 3 I=IP11

3,1272r. 4C.M. i ; g " fp, -IFORTRAN 0025 0026 0027 0028 0029 0130 0031. 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0146 0047 0048 0049 0050 .0051. 0052 0053 0054 0055 0056 0057 0058 0059 0060 0061 0062 0063 0064 IV G LEVEL 19 MAIN DATE 3 71222

no 500 .01.101FND J.JP1=.1 P.VCCOFPIJP1, CALL CSMNPXII,J,P,XZEROIKEERO,CMNPXI A(IP1,JP1)=CMNPX11/*T5U111 A(IP1+MEND,01)=CMNPRI2/*T5UBI A(IP1,01+MEND)=CMNPX13/*TSU81 AIIPI+MENDIJPI+MENDI.CMNPXI41*T5U8I:

500 CONTINUE

t

DO 550 I.1,MMP2 on 550 J=1,MMP2 '550 AF(I,J)=AllpJ/ .

t

CALL MINVIAFOIMP2,DETFLOUMODUMi

t

DO 750 J.1,MMP2 CSS(J1.0.0D0 DO 740 I.1,MMP2 ,,. C55(J)=CSSIJI+AFII,A*FGSCI4 740 CONTINUE 750 CONTINUE A.

COMPUTE THE WAVE RESISTANCE COEFFICIENT WRITE 16.164 111.1 1=0 _{TER4=C5SIIPMSSAIP)+CSSIIP+MENDI*CSSIIP.MENDI} TERM.TERm*CCOFF CSUBW.TERM WRITE 16,171 1,CSSIIPI;CSSIIP.MENDIgTERMgCSUOW DO 810 IP.2,MEND CALL COEF5(10(ZERO,DUMO<SU8M,DUM) TERM=C5SIIPPItSSIIPI+CSSIIPMENDI*C5511P+MEN01 TERM=TFAM*XSUBM/IXIE80+XSOBMI

_

.t

.

TrRM=TERM*CCOFF CSU3W.CSUBWTERM WRITE (6,17) I,C55(IPI,C5S(IP+MEND),TERMgCSUBW

810 CONTINUE

CSUENS*CSUBWoZLENTH*2LENT14/15URFA.40001 WRITE 16,181 FROUDE,C5U8W,CSUBW5 GO TO 100

999 CALL EXIT END .00/04/29 E PAGE 0002 , . -IP ,--...,-,, ... ..., -s f I=IP-1

0004 0005 0006 0007 . _ . ...,,,... -FORTRAN) IV G LEVEL 19 GETF SG --DATE 71225,- _ - .. 21100/15--. .21100/15--.. ,... PAGE 0001 ....1 .- -2..-, -... ---...1 0001

5U3F OUT INE GETFSG( M,FGS,VFCGRP,K ZERO. V, X ZEROILENGTH4SURFA

. ....

1.. ....

.-_

..

--.

- __,..--.,-, . . ... (

EXECUl E SUBPCiUTINE FANDG

C

THIS PROGRAM ACCEPTS DATA ON LONGITUDINAL CUT. WATER-SURFACE

PROF.IL...., ..

...r.-,..

C

AND SCALES THL DAT A TO FFET USING A. LINEAR CALIBRATION

CURVE., THEN _ ...- -.,

...

..,__

_

C

INT EGR ATTS THE CUT TO COMPUTE

. F (No PO:II AND G I N, Pt QI

USING-E0(4-27,-... ..,...-...,,....,_.:

.,

... 44. ,

..

. . _ 1 c

IN CAR CONT PACT PROPOSAL OF 1970

c

C

THIS PROG

AVERAGES

THE FIRST .100 DATA POINT S. TO _GET-ZERO CAL111---...t.--..,...--...c,..,.

. ..,..-4. 4 .. ...=. c 1,-DISTANCE FE Om

WA.VT GAUGE TO CENTER OF .ChANNEL

...---...,

_ . --.,. . . ..., i..,-, 'C LENGTH-LENGTH OF MODEL 1 t C 0002 ono?

suekouTINEs .FANEIG SIMPS COEFS

es../14

.0

IMPLICIT REbL*81A-M0.7i.l,$) REAL*8 KZFRO.LENGTH

ital.r4.44--... -44 4. V.. I NT F-GF R*2 I Z . _ _. _ .____ - - --- . ___ . . - - . _ , ....-..--..-..--1 DIMENSION X(1040).1(1040 ,II120),JZ(20)417120) ,VT( 20 )...,, ---...,.... ..._ .,..,_. -DIMENSION F GS (52) e VECOFP 126 ) . . --,-...., -.... ...--,,,,,,---,-..--,--.----. _,,,...,... -1 COMMON X, Z .... ,. .. -,-....,...1.-- .5.4"...*. ... ..,...., 4.1. .. I -..=1. , '`.."4. .i:. 000E 1 FDRMAT (A4, 3F7.3,F6.3,F70.3,F6.3tF9.6,15,13,F7.3,F13.3,14)...---,-,-,.., ....,.... -, . COOS 2 FORMAT (4(14 tI6X) I aa. 5, .554555

,

r,,,,, ...,, 001C 3 FOR.mt.T(5(F7.3,F91 ) ) , t ..., 0011 4 FORmATI6(F1C31F127) I ..

----.=.

-0012

5 FORMAT ( THO , 40X ,' LONGITUDI NAL CUT PR OF IlLk

- * !DEPTH I N ..FEET1/6... ,.,..,.. ,,-,,,

..

1 ( 6X 9 ' X' ,I3X 'DEPTH ' 1 a ) . . . ...., .1 -... 0013 6 FDRMAT (1001 3.5) ,..F4...14 44. 444 F.. 4,4=1. , 55 -,- .i. 0014 7 FORMAT (30X t 10F103 ) ,

,..,,....

,..-- --,-- , ,,----r.,... i 001 5

8 -FOR MAT ( INT, // it' LONGITUDINAL CUT WAVE RESISTANCE

*

RUN NUME47.:R!....,...-:-...-,-.

1,

2 X tA.4, /

,

5X,' MUDEL/TANK CHt CTER IST ICS '

, 6X g ' MC:DEL LENGTH "--.1 IP -,,,,,,, .:. _ __ .... . .. -2F7.3, 4X t'W:-.:TTED ARCA, .".' ,F73, 5x, 'TANK WIDTH =',F7.3, / 1.. 5X, /4,..6...., ,I. -... _,6 S. 3' CX PLR I mrNT AL COND .7 1 IONS

7X, 'CUT POSIT ION

Y=1 ,F6.3, 3X, . ... . . -..,. -....,.. . ..._,..--,--, -... .,...

4 , X OF FIRST DATA =',E7.3, 1X,'MODEL SPEED =',F6.3, 6X ,11(ZERC

,s1t...._... .='-...-=5..

5F15...M.,

/

,3F.X, 'DELTA. T (SEC) =',F96, 1X, 'DELTA X (FT

,F10.13,---,

..,..-,

6 ix,.Nn 0 POINTS =',I5, /

.

5Xt 'ANALYTIC PARAMETERS' ,11X,ITRuNt.

... ... ... ,-7' CA TIP! °DINT Pi, ,,,'

, I3i 2X, 'X SUB ZERO

=1 ,F7.3, 6X, 'X, SUB T =I* RF 31 10X, 13, / I 0016

9 FORMAT (1H1, /// ,30X,' INTEGRAL FUNCTIONS F AND G', It

0-9X 0141.,, 113x,0F.,24x,,P0,15x,.P0, 4W2PI*P/OmFGAN,, // ) ...

--....

---.... 0117

10 FORMAT (110 ,2F2E..16 IF6.0gF103)..

-..

_ _ ..--_ . -..,..

,

-.1 .4. , ...,, , oni e

11 FoRtd6T135X , 'CALM,/ ATI ON,' t I3,' POINTS, 1ST POW IS

ELEVATION..AN0-2-,5.,,,. .-, ,.,-,,

.

. 1NO IS VGLTACE1 L... -_... ... ' :=.... 0015 14 FORMAT I, CFIRCIR IN CALIBRATION RUN ABORT-File-) ... .... -- ---'-- --,--.-.. .. 4 ,... ...:.- ,._.,. ,.4 002C 17 FOR MAT (2024 ) . _ '-C -.... --,,..,-....,.---, 0321 P1.3.141592(5358979300' -11...-4,... _,... --...5.--,,,... 0022 . TwOPIPIPI . ..,-, , ,-,... .-.,

.

002? GRAV=32.1700 11-. 0024 CAR DS=256 ..., .. _ 002 5 100 R Et.r.:

(5,1 0ENIDrr9001.ARUNiLLNGTH1S.URFAt13,LY ,XBEGINOSPEED.DELIT iNDIMa__...

...,...__..1 _ IM,X2F1417,XT 'KAI

, ...

,.... .5.. . . 0026 AZERO=GPAV/ISPEED*SPEEDI, 0027 DELTX.SPEEU*DELTT ooze NC6PC=IN0IM*19//20 4 - ' -. . ,1141. ' = REOLIPFS

FORTRAN IV G LCVOL 19 GLTFSG DATE = 71225 21/00/15 002C

WRITE (6,8) NRUN,LINGTH,SUREA,B,Y,XBEGIN,SPEFC,KURO,OCLTT,DELTXt _{INDI.,M,XZEPCOT,NCAL} CALIBRATION CATA

0030

READ

(5,3) IZT(I),V711/11=1,NCAL/

5120 INPUT [TA ON 256 CARDS, 20 PER CARD

oni

RFA0

(5,17) (17(11,1=1,20)

0032 0033 0034 0035 0C3E 0037 0031 003q 004C 0041 0342 0041 0044 0045 034E 0047 004F 0044 035C 0051 0052 ₀₀₅₃ 0054 ('055 0056 0167 orse 00,54

FINE AVERAGE OF FIRST DATA POINTS SUm(ti..0.000 no 320 1=1,20 .12(11=I1111 2(1)=DFLOAT(JZ1111

320 SIP,C,L=S1'4CPL.Z(I)

SU10AL=SUMCAL/20.000 CO 310 J=2,NCERD READ

(5,17) 111(1),I=1,20/

DO 300 1=1,20 IPJ=I+(J-1).00 JZ(1)=1ZIII

300 1)1PJ).0Finp1(j111)) 310 CONTINUE

FEM.) RES1 CF 256 CARDS NLEF7=CARDS-NCAk0 IFINL:FT.f0.C1 GO IC 329 DO 315 J=1,ALCFT

315 REAC

(5,17) 1.1.1(1),1=1,201

FINE ZERO PC1NT OF CALIBRATICN CURVE

329 DO 33C I=1,RCA1 IF ICABSIZT(111eLT.0.00000100) GO TO 310 CONTINUC WRITE 16014/ GO IC 100 340 VUR0=VI(I)-5U.CAL DO 350 1.1,K14t 350 VIII)=VT(I) VZER1

WRIL-16,11/ NC-AL wRIIE(6,71(17(1),1=1,NCAL) WRITE(617)10.11/.1=1,NCAL)

340

:7=

771 7.71

-77

0n4c nn 200 Ixt,nnly 0361 DO 113 I=1,NCAL 0062 IT=1 0063 OV=VT(1)-1(J) 0064 IF(DV)113,114,118 0065 113 CONTINUE 006f 114 ZIJI.171111 0167 GO TC 115 006E 11R 10.11-1 I 006S CV=V7111/-V111C) 77777 PAGE 0002 7777 77.] 77.3 777

-,F1 ;M1 e 0094

r

900 CALL (yr( END 00,. r .1

,

-, FORTRAN Iv G tCYCL 1-9' GETFSG.-DATE d = 71225-, . 21/00/15 . PAGE 00C3 ,

-..,

T 7 .i*".--"--"^"."...-..7:.-- -..-.1.-'-'NIX*9142.0_ ", OZ=ZT(111-27(1C) 1,-. -. . 0070 ,...

- ----=

0071 Z(J)=(Z(JI*10T(I0))*02/0V+ZT(I0) ,-: 115 ZJ=J 0072 0073 W(J)=11J-11,C01*DELTX.KBEGIN .0074 no coNirniu 1 C C SKIP PRINTS * GO TO 400 GO 7C 40C 0075 WPITF (6,5) 0076 _ DO 375 JSTAR=1,NDIMi6 0077 JEN0=JST6P+5 . 0078 WRITE (6,4) (X(J),Z(J)tJ=JSTARtJENDk... ..-....,...-..-...-...,-..-,...-_-__..:-.-2:,. 007c 375 CONTINUc. 008C 400 C0N7 i NUE -0081 , WRITO (6,9) 0082 LLS1=4+1 _ T.G1 DO 500 NPI1,MPLUSi 0083 0084 N=NP1 -1 0085 CALL COZ:FS

'<ZERO, CUMMY K SUBN On GA N ).

.enfc 0086. P XT-X LER I *Of4GAN/T WOPI 0087 P.P +C.500 ...L. .4 . . F I XPP 0088 0084 P=0cLCAT(LFIKP) -00.9C 0091

VECOFP(NP1)=P CALL FtNOGIN,KetGIN,DELTK,NDIM.,.Y,XT.,-KZERO,F,G,LENGTH,XZEROgal8I-,.-5.aw-,.

--P- _LO 0092 FGS(NP1)=F 0091 FGS (NP1+MPL US 1 =G

.

0094 XEND=TWCPI*F/0,4GAN 0095 ..-7,-77-. 7, WRITt (6,10) N,F,G,PipiENO 0096 500 CONTINUE RETURN 0097 C -A

_ FORTRAN TV 'G LEVEL 19 fANDG DATE . 71223 22/09/04 -A'AGE 0001 3001

SUBROUTINE FANDG(NO(BEGINOELTX,NOIM,Y,XT, [AZEROOF,G,LENGTH,X1FRO,

LP, B) TO FIND F(N,P,O) AND G(NIP,O) FOR A MEASURED LONGITUDONAL CUT SUR= FACE DISTRIBUTION GIVEN BY 1(X) FOR INCREMENTS OF LENGTH DELTX AU) NC THE CHANNEL,

THE NUMBER OF POINTS SPECIFIED IS NDIM.

HENCE,

DATA IS GIVEN OVER THE RANGE X/ERO<X<X1ERO+NUIM*DELTX. REF * EON 27, ONP PROPOSAL. 1970

.

METHOD * INTEGRALS ARE EVALUATED BY SIMPSONS RULE AND NEWTONS 3/8 RULE PLUS ENO CONDITIONS VIA TRAPEZOIDAL RULE.

THE INTEGRAL LIMIT

ARE X1ERO =2*PI*0/0MEGAN TI) X/ERO +2*P1*P/OMEGAN.

SINCE THERE IS

NO CONTROL OVER THE PLACEMENT OF THE UPPER AND LOWER LIMIT POINTS A STRAIGHT LINE FOR /(X) IS ASSUMED BETWEEN THE LAST NEAREST MEAS= URED POINT /IXENO) AND THE UPPER INTEGRAL LIMIT,

AND ALSO BETWEEN

THE LOWER LIMIT AND THE NEXT HIGHEST (GREATER X) DATA POINT B=CHANNEL WIDTH REQUIRES SUBROUTINE SIMPS COEFS

1002 IMPLICIT REAL*8(A=11,0-2,1) U003 REAL*8 AZEROO(SUBN,LENGTH 0004 DIMENSION X(1040),Z(1040),1C(1040),15(1040) . . 0005 COMMON X,1 3006 EQUIVALENCE (101),15(1)) 0007

I FORMAT(' UPP(.R INTEGRAL LIMIT EXCEEDS RANGE OF GIVEN DATA' F=G.Ot 1XEN0='.F10.50 N =',I3)

0008

2 FORMAT(' FAILUFE OF FANDG * EXIT CALLED')

000)

3 FORMAT(' LOWER INTEGRAL LIMIT IS LESS THAN RANGE OF GIVEN DATA, F.- 1G=0, XST,RT=',F10.50 N 4,10)

PI=3.14159265358979300 CALL COEFSIN.A/ERO,DUMMY,KSUBN,OMEGAN)' LOWER LIMIT POINTS, NOTE 1(1) IS THE fiRSf'DAtA POINT

0012 IF (X1ERU.LT.XBEGIN) GO TO 988

-0013 ISTM2=IX/ERC=XBEGINUDELTX 1014 ISTM1=ISTM2+1 0015 ISTART=ISTM2.2 0016 XFIRST=X(ISTART) 4 , 0017 OXFST=OMEGAN*XFIRST 0018 OXSTR=OMFGAN*XIERO 031) TERMA=Z(ISTAKT)*DSIN(OXFST) 0020 CTW0.IZ(ISTART)=Z(ISTM1))/DELTX 0121 CTHREE-.XFIRST=DELTX 0022 CES=T(ISTART=1)+CTWO*IXZERD=CTHREEI 0023 TERMB.CES*DSIN(OXSTR) 0024 TERMCKTWO*(DCGSIOXFSTI=DCOSIOXSTRII/OMEGAN 0025 FFIRST=ITERMA=TERMORTERMCl/OMEGAN 0026 TERMA.IIISTARTI*OCOSIOXFSTI, 0027 TERMB.CESsOCOSIOXSTRI 0028 TERMC=CTWO*(OSINIOXFSTI-OSIN(OXSTRI)/ONEGAN, 000 GFIRST=(-TERMA.TERM(OTERMWOMEGAN 0010 0011

7.1

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