AN INSTBUl~,NT TO ,ME'ASUrF THE IWO-DIM2EN3SIONAL

WAVE SLOFP SFECTRUM OF OCTAN CAPILLARY WA\VE

By

CHAlLES SHELBY PALM

A DISSERTATION PR7SNTED 7O -HE CRALIUAE COLUCIL OF

THE 'JNTVE'IET- 1 F r7 ORIDA

IN FLPTI.'L 'FUUF ILv:, P' 0:;'i ;:.' '0 .UIREENTS FOR THE

DEIGBRE OF DOC7F0? ILOSOPHY

UNIVERSITY OF FLORIDA

Dedicated to

JESSE ALBERT GOODE, M.D.

I

ACK',OWLEDGEEINTS

I am thankful for the support provided by the members of my

Supervisory Committee. Special thanks are due to Dr. R. C. Anderson

for suggesting the project, Dr. 0. H. Shemdin for providing the funding,

and Dr. D. T. Williams for providing encouragement and guidance through-

out my stay at the University. Mr. Harry Stroud was instrumental in

material acquisition and was generally invaluable in seeing the project

to a successful completion. Mr. Allan Reece generously applied many

hours of his own time to this project. His suggestions were helpful and

his expertise with the wave tank facility controls and the IBM 370

computer helped to reduce the total project time considerably. I am

also grateful to my wife, Emelyn, for the many hours she spent typing

this paper.

TABLE OF CONTENTS

ACKNOWLEDGEMENTS . . . . . .. . . iii

LIST OF FIGURES . . . . . . . . v

ABSTRACT . . . . . . . . . . . . .. viii

CHAPTER I, INTRODUCTION... ... . . . . . . . . 1

CHAPTER II, DESCRIPTION OF THE INSTRUMENT . . . . .. .

Theory of Operation . . . . . . . 3

Optical Design . . . . . . . . . . 6

Description of Instrument Components . . . . . 9

CHAPTER III, THEORETICAL PERFORMANCE AND OPERATING . .. . 27

CHARACTERISTICS OF THE INSTRUMENT

Introductory Remarks . .. . . . . .... 27

Wavelength Response Characteristics . . .... 35

CHAPTER IV, WAVE TANK TEST ..... . . . . .. .... .56

CHAPTER V, OCEAN TEST. .. . .. ... ... . . . . 61

CHAPTER VI, SUMMARY . . . . . . . . . . 66

CHAPTER VII, CONCLUDING REMARKS . . . . .. . .... 67

REFERENCES . . . . . . . . . . . . 68

BIOGRAPHICAL SKETCH . . . . . . . . . . .. 70

LIST OF FIGURES

Figure 1: Geometry associated with the refraction

of a ray of light at a water-air interface. ..... 5

Figure 2: Paraxial design of the optical system.. . . . . 7

Figure 3: The optical receiver portion of the

waveslope instrument . . .. . .. . . . . 10

Figure 4: Performance curve for the objective lens.. . . .. 12

Figure 5: The imaging lens module.. . . . ... . . 14

Figure 6: Two-dimensional Schottky Barrier Photodiode.. ..... 16

Figure 7: Electronic amplifiers.. . . . . ... . . .18

Figure 8: Block diagram of the electronic amplifier ..... . 19

Figure 9: Schematic diagram of one amplifier channel ...... . 20

Figure 10: Illustration of the effectiveness of the

analog dividers to produce a normalized

signal voltage.. . ..... . . . . . . .22

Figure 11: The oceangoing support structure.. . . . . .. 25

Figure 12: Sketch of the wave slope instrument

during ocean operation.. . . . . . . . ... 26

Figure 13: The calibration set up in the optical

laboratory.. . . . . . . . . . . . 29

Figure 14: A typical plot of calibration data

obtained using the analog divider . . . . .. 30

LIST OF FIGURES (continued)

Figure 15: Offset voltages measured with respect

to the amplifier outputs. . . . . . . . .32

Figure 16: Photographs showing the location of

the optical-tube thermometer relative

to the amplifiers in the imaging lens

module . . . . . . . . . . . . .33

Figure 17: Configuration for the ith-wave

component and the Gaussian

laser bea .. . . . . . . . . . . . 40

Figure 18: Approximation to the Gaussian

intensity distribution used

in the computer error analysis.. . . . . . 44

Figure 19: A/r0 as a function of the parameter k/ . . . . 47

Figure 20: Error in the waveslope measurement,

6, as a function of parameter k/r at wt = 0

for various values of parameter ak.. . . . . . 48

Figure 21: Percent error in wave slope

measurement as a function of

parameter k/c, for wt = 0 and ak-- 5, 45 ....... 49

Figure 22: Error in wave slope measurement, as

a function of wave phase, for

various values of maximum wave slope

and for k/v = 0.7 (X/ro = 6.34).. . . . . . 51

Figure 23: Sample records from the wave tank tests.. . . . 57

Figure 24: Frequency spectra of the mean square slope.. . . .. .59

LIST OF FIGURES (continued)

Figure 25: Photograph of shrimp boat's position

relative to the ocean test site . . . . . .... 62

Figure 26: The instrument in operation during

the ocean test ...................... .63

Figure 27: Sample records of ocean test data . . . . . 64

Abstract of Dissertation Presented to the Graduate Council

of the University of Florida in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philoscphy

AN INSTRUMENT TO MEASURE THE TWO-DIMENSIONAL

WAVE SLOPE SPECTRUM OF OCEAN CAPILLARY WAVES

By

Charles Shelby Palm

August, 1975

Supervisory Chairman: Roland C. Anderson

Major Department: Engineering Sciences

An optical instrument for use in determining the two-dimensional

wave slope spectrum of ocean capillary waves is described. The instru-

ment measures up to a 350 tip angle of the surface normal by measuring

the position of a refracted laser beam directed vertically upward through

the water surface. High quality optical lenses, a continuous two-

dinensional Schottky barrier photodiode, and a pair of analog dividers

render the signals independent of water height and insensitive to laser

beam intensity fluctuations. Calibration is performed entirely in the

laboratory. Sample records and wave slope spectra are shown for cne-

dimensional wave tank tests. Sample records of two-dimensional ocean

tests are presented. A mechanical wave follower mechanism was used to

adjust the instrument's position in the presence of large ocean swell

and tides. Errors due to use of a finite-sized laser beam are discussed.

// / /I /

CHAPTER I

INTRODUCTION

Statistical descriptions of typical shapes of the sea surface and

its changes with time are useful in many studies. The sutdy of the

effective forces governing the interactions between wind and water led

Cox to examine the statistics of snail capillary waves. As Cox points

out, the shape parameter used in the study of small, high frequency

waves is the wave slope. This is so because small wavelength waves have

small amplitudes which become difficult to measure accurately as the

wavelength decreases. However, these same waves may have wave slopes

of considerable magnitude.

The surface slope statistics have been useful in studies relating

to the behavior of surface waves in the presence of internal waves, the

resonant interaction among capillary waves, backscattering and radiative

transfer of electromagnetic radiation in connection with remote sensing

of the state of the sea, and relations between a roughened sea surface

and multipath effects in communications between satellites and aircraft

2-8

flying over the ocean.

Optical methods of measuring waveslopes seem to be popular because

the water surface is left undisturbed by the measurement, and the instru-

mentation required to make the measurement is basically simple, at least

in concept. Some of the optical methods used in waveslope studies are

based on the principle of measuring reflected light. These schemes

include the measurement of light reflected from an artificial source

and light reflected from the sky, and the observation of sun-glitter

patterns on the open sea surface.9,210,11 Other optical methods of

wave slope measurement are based on the principle of refraction of light

at a surface of the water where a discontinuity in the index of refrac-

tion exists. Measurements based on the refraction of light have used

both continuous, extended light sources and highly localized light

sources in the form of a laser beam .112,13

The instrument described in this paper is a second generation devel-

12

opment of the one described by Tober et al.12 The new instrument was

designed for and used in both laboratory and oceangoing field tests to mea-

sure the two-dimensional wave slope at a point as a function of time.

I

CHAPTER II

DESCRIPTION OF THE INSTRUMENT

Theory of GOperation

The orientation of the surface normal of an elemental area on the

surface of a water-air interface may be established by measuring the

angle of refraction of a light ray passing through the interface. The

relationship between the incident ray and the refracted ray is given by

Snell's Law:

(1) n. sin --. = n sin 4-

1 1 r r

where n. is the index of refraction for the medium containing the inci-

1

dent ray, nr is the index of refraction for the medium containing the

refracted ray, and-&. and -- are the angles of incidence and refraction,

1 r

respectively. By defining a deflection angle, j as

(2) -=43 --

and using Snell's Law as expressed in equation (1), the deflection angle,

, may be expressed in terms of n and-O as

(3) csin (n sin

(3) ( = arcsin (n sinde) -efi

where n is the relative index of refraction, defined as n = ni/nr.

Using the direction of the incident ray as a reference direction, the

deflection angle, C, may be used to measure the refraction of the inci-

dent ray. By aligning the incident ray with the surface normal under

calm-water conditions, the angle of incidence, -., may be related to the

angular displacement of the surface normal from the calm-water surface

normal orientation, -, by

(4) =

Figure 1 shows the geometry corresponding to the situation described

above. Thus, substituting equation (4) into equation (3), the expres-

sion for ) becomes12

(5) = arcsin (n sin4-) 6-a

The inverse relation which expresses -- as a function of ) is readily

found to be

r sin 1

(6) e- arctan sin s

n cos

Thus it is seen that the measurement of the wave slope, 6-, results from

a measurement of the deflection angle, C, provided the relative index of

refraction, n, is known. Throughout this paper the value for n is taken

as n = 1.333.

direction of incident ray

and calm-water surface

normal-

plane of incidence

refracted ray

refracted ray

water

surface

7 / elemental

surface

air / area

water

incident ray

Figure 1: Geometry associated with the refraction of a ray of light at

a water-air interface. The wave slope is defined as the

angle *-.

Optical Design

The initial Gaussian or paraxial design of the optical components

for the wave slope instrument was completed in late May, 1974. The

system was designed to measure wave slopes of. magnitude up to 450 using

a United Detector Technology SC/50 Photodiode as a two-dimensional light-

to-electrical signal transducer. Matrix methods of ray-tracing were

employed to give the first-order system design shown in Figure 2.1 The

optical receiver portion of the instrument consisted of three lenses and

the photodiode transducer,

In the ideal case, the laser beam was assumed to be collimated

perfectly and focused to a point at the focal plane by the objective

lens The position of each ray in the back focal plane of the objec-

tive lens is determined using a linear transformation

(7) f = a

where p is the radial displacement of the focused laser beam from the

optical axis, a is a constant which is determined from the lens para-

meters such as surface curvatures, index of refraction of the lens mate-

rial, and the thickness of the lens, and < is the angle of deflection of

the laser beam from the optical axis. An expression for < in terms of

the wave slope, -, was given earlier in equation (5). The azimuthal

angle,', is unaltered in the transformation. Thus it is seen that any

light ray entering the objective lens having a deflection angle f and

azimuthal direction is directed to a point (pt,) in the back focal

plane of the objective lens. This occurs for every ray entering the

optical axis

plane of the photodiode

38.63 ami

I BPP ]

FPP imaging lens: f/1

'T f = 32 mm

186.35 mm

I- BPP

FPP field lens: f/l

f = 140 mm

130,00 mm

_-___i .- FPP objective lens: f/1

Z>- - f = 130 M.m

deflection f =*

angle h = height of objective lens above

laser beam the mean water level

n mean water level

Figure 2: Paraxial design of the optical system.

The laser beam enters the objective lens and is focused

130 mm from the back principal plane (BBP). The front

principal plane (FPP) of the field lens is located at the

back focal plane of the objective lens. The field lens

confines the laser beam to the optical tube dimensions.

The imaging lens images the field lens FFP onto the

photodiode surface.

340 mm

8

objective lens, regardless of the point of incidence on the first sur-

face of the objective lens. Since a unique point in the focal plane

(TY?) corresponds to a unique ray orientation ( an), and since i is

uniquely related to the wave slope, -, via equation (5), it is clear

that measurement of the position the laser beam in the focal plane (PP)

gives unambiguous information about the orientation of the water surface

normal. Thus the problem of wave slope measurement is essentially

solved in the back focal plane of the objective lens.

Although the necessary wave slope information was displayed in the

back focal plane of the objective lens, it was necessary to reduce the

size of the information-laden portion of the focal plane to match the

dimensions of available photodiode transducers. This was accomplished

by using a field lens to confine the laser beam within the dimensions

of the optical tube and an imaging lens which formed an image of the

back focal plane of the objective lens on the active surface of the

photodiode. Each of the lenses shown in Figure 2 was to consist of one

or more lens elements to correct for optical aberrations.

The system described above was never fabricated because a test-

schedule time constraint did not allow enough time to have the elements

ground,tested, and assembled. Therefore, the design was modified so

that off-the-shelf optical components could be used. The imaging-lens

requirement was filled by selecting a high quality 35mm camera lens.

An objective lens was selected after six weeks of testing a variety of

different commercially available lenses. The one selected was a mili-

tary surplus aerial camera lens, a 12",f/2.5 Aero-Ektar. It was found

to be capable of measuring 350 wave slopes, displayed negligible aberra-

tion errors, and had a large enough clear aperture to allow measurements

to be made with the objective lens at a reasonable height above the

water surface. A suitable field lens was located, but an excessively

long lead time for delivery led to the use of a diffusion screen in

place of a field lens. The diffusion screen had the advantages of being

lighter in weight and easier to mount in the optical tube, but had the

disadvantage of scattering most of the light outside of the imaging lens.

The light flux onto the photodiode was thus lowered by approximately two

orders of magnitude, and additional electronic amplifier stages were

necessary to produce a satisfactory signal. The following section

describes the components used to construct the instrument.

Description of Instrument Components

Figure 3a shows a phantom view of the optical receiver portion of

the wave slope instrument corresponding to the original design concept.

Figure 3b shows a view of the final configuration, and Figure 3c is a

photograph of the assembled unit. Brief descriptions of each of the

major components and comments on their operating characteristics are

given in the following paragraphs. Unless otherwise stated, the physical

location of the individual components is indicated in Figure 3.

Objective Lens

For a given height, h, above the water surface, the maximum measur-

able deflection angle of the laser beam, &nax is dictated by the clear

aperture of the objective lens, d, according to the relation

(8) ax = arctan (1/2(h/d))

(a) (b) (c)

Figure 3: The optical receiver portion of the waveslope instrument.

(a) original concept, (b) final configuration, (c) assembled

unit. Components: laser beam, A; objective lens, B;

diffusion screen, C; scattered laser light, D; imaging lens,

E; two-dimensional photodiode, F; tube end cap, G; imaging

module mount, H; optical tube, I; diffusion screen adjust-

ment screws, J; N2 purge lines, K; main support collar, L;

plate glass window, M; field lens, N.

A number of commercially available lenses were tested on an optical bench

to determine f as a function of the ratio (h/d). A military surplus

aerial camera lens, an Aero-Ektar, f/2.5, 12" focal length lens, was

selected as being the most suited to the intended purpose of the wave

slope instrument. A special lens element support was fabricated for the

lens to reduce the total weight from 18.75 lbs to 11 lbs. Figure 4

shows the maximum measurable deflection angle, ,ax as a function of

the objective lens height above the water surface, h. Scales indicating

the maximum measurable wave slope,-9- and the ratio (h/D), where D

max

is the maximum diameter of the lens barrel at the.plane of the first

element of the objective lens, are also shown in Figure 4.

The shape of the back focal surface of the objective lens was deter-

mined by a series of Foucault knife-edge tests. The surface showed con-

siderable curvature at large displacements from the optical axis, but a

plot of the radial displacement of the focal point of the laser beam as

a function of the deflection angle, 0, was linear to a good approxima-

tion. The results indicated that the linear transformation expressed in

equation (7) was accurate to within 0.5% or less and a Gaussian or par-

axial model of the objective lens was indeed a good approximation. This

notion was reinforced later by the results of the calibration tests and

the measurement of errors due to optical aberrations.

Diffusion Screen

The diffusion screen was used in place of a field lens because a

satisfactory lens could not be procured at the time of instrument

fabrication. The diffusion screen served to scatter light from the

focused laser beam, thus allowing a sufficient quantity of light to

35

15

30

max imax

(0) (o)

10

20

5

S10

0 0, 0

10 h (cm) 20 30

0 0.5 1.0 1.5

(h/D)

Figure 4: Performance curve for the objective lens. Symbols are

defined as: h = height of objective lens above water

surface, D = maximum lens barrel diameter, = = maximum

deflection angle, -- = maximum wave slope.

max

13

enter the imaging lens module and activate the photodiode. A number of

different types of translucent and frosted plastic sheets were tested to

find a satisfactory screen material. Scattering lobes for eight differ-

ent potential screen materials were measured and evaluated. Best results

were obtained using a one-eighth-inch-thick sheet of clear acrylic which

was roughened to a uniformly frosted appearance by grinding both surfaces

with a 400-grit abrasive.

The diffusion screen was placed at the focal plane of the objective

lens using a set of adjustable mounting tabs. The mount was made adjust-

able rather than rigidly fixed to allow for accurate and optimal adjust-

ment of the position of the screen in case the instrument calibration

tests indicated that such an adjustment was necessary. Subsequent

calibration tests showed that the initial placement of the screen was

satisfactory and no further adjustment was necessary.

Imaging Lens Module Optics

The imaging lens module consisted of a 35mm camera lens, the two-

dimensional Schottky barrier photodiode, and the electronic amplifier

and signal conditioning components. These items were mounted in a common

aluminum housing and the entire module somewhat resembled a 35mm camera

in appearance. Figure 5 shows front and rear photographs of the imaging

lens module.

An interference filter was attached to the camera lens to reduce the

effect of background illumination on the output signal from the photo-

diode. This filter passed the He-Ne laser line at 6328 Angstroms and

had a bandwidth at the half power points of 100 Angstroms. A 100o band-

width was selected because the laser light scattered from the edges of

F 5>

*1~

WI

-~~~1

$7~: >

(. -

P .-'

:s V

H1

analog

dividers

connector with

leads for power,

signals, and test

points

Figure 5 : The imaging lens module; front view, (a), and back view, (b).

camera lens

interference

filter

SCO '

I

I

I

)~f------L

.,ui

the diffusion screen impinged on the surface of the interference filter

at relatively large angles of incidence diad would not have been passed

by a much narrower bandpass filter.15 The imaging lens was a Super

Takumar multi-coated, 55mm focal length, f/1.4 camera lens.

Imaging Lens Module Electronics

The laser beam position sensing transducer was a United Detector

Technology continuous Schottky barrier Photodiode Model SC/50 with a

3.56 cm x 3.56 cm active area, shown in Figure 6. Similar devices have

been used as radiation tracking transducers, and descriptions of mathe-

matical models for the lateral photoeffect are available in the litera-

ture.16,17 The photodiode may be operated in either a photoconductive

mode, where a reverse bias voltage is applied to the diode junction, or

a photovoltaic mode, where no reverse bias is applied to the diode junc-

tion. In the photoconductive mode the depth of the depletion region

associated with the p-n junction is physically greater than that occurring

in the photovoltaic mode of operation. The photoconductive mode features

a lower junction capacitance and higher junction resistance which gives

rise to a much higher frequency response capability, but also it gives

rise to a greater absolute value of noise due to the reversed-bias shot

noise. The high value of the junction resistance, approximately 68

megohms at 11 volts reverse bias, allows greater input light intensities

to strike the diode surface with no damage due to excessive junction

currents. The photovoltaic mode features a higher junction capacitance,

lower junction resistance, and a reduced noise level due to the absence

of the reversed-bias shot noise. The frequencies dealt with in the

present application were low enough to make the effect of the junction

Ji. ,

I. .

^ ^ l^J ^ilL^ j^L i p 1,1.1

Figure 6 : Two-dimensional Schottky Barrier Photodiode.

capacitance negligible. Both modes of operation were tested, and the

photovoltaic mode appeared to be more stable during calibration tests.

Therefore, all of the subsequent tests iere conducted using the photo-

diode in a photovoltaic mode of operation.

Figure 7 shows a view of the imaging lens module with the hinged

door opened to expose the electronic amplifiers, the regulated power

supply and the analog dividers. Originally, commercially available

amplifiers were used to amplify the photodiode output, but serious

signal drift problems required re-trimming of the amplifier stages every

15-30 minutes. This made practical application of the instrument impos-

sible. The amplifier shown in Figure 7 was a first-generation improve-

ment assembled using low-drift amplifier components. This amplifier

allowed operation of the instrument for periods of up to six hours with-

out incurring any serious errors due to electronic amplifier drift.

Components for a second-generation improved amplifier were ordered but

not delivered in time to be used in any of the tests conducted and

reported on in this paper.

Figure 8 shows a block diagram of the electronic amplifier and

related signal conditioning components. The photodiode produces four

current signals through the four ohmic contacts to the n-type lattice

material. Four transconductance amplifiers convert the current signals

to amplified voltage signals. The sum and difference amplifiers increase

the gain of the signals. The analog dividers normalize each difference

signal to the sum signal; the quotients are the x- and y-axis outputs

which are insensitive to input light intensity fluctuations at the

photodiode. The circuits for the x- and y- channels are virtually iden-

tical, and are shown in the schematic diagram in Figure 9.

u 01 23

r i

I

'f

*^ i-; -i70I

Figure 7 : Electronic amplifiers.

y-axis

output

x-axis

output

Figure 8 : Block diagram of the electronic amplifier.

C

R2

photodiode -

--

R3

opaop

R. E{ 4 R3

R 4 to the

1R2 analog

R. RI R3 divider

---^--- I -typical bias-offset

R5

I- -- I circuits

Figure 9 : Schematic diagram of one amplifier channel. Approximate

values for resistors, R, and capacitors, c, are: Ri, Rj =

3.5 KA R1 .6 Mn R2 .5 Mn, R3 = 10 Kn, R4 = 3.5 Kn ,

R5 = 5 Kmr, R6 = 1 M1., 7 = 50 Kr R8 = 50 KL c = 1000 ;f.

Low drift analog dividers, manufactured by Analog Devices, were

used to perform the signal normalization ;with respect to the input light

intensity. The dividers were trimmed with external potentiometers in

the manner suggested by the manufacturer. It was found that normaliza-

tion accuracy was within 1% when the denominator voltage was in the

range -8 v to -4 v. This performance was somewhat degraded as the denom-

inator voltage increased to -10 volts and also an error of 3o was ob-

served as the denominator dropped to -2 v. The normalization error

increased very rapidly as the denominator voltage fell below -1.5 v.

Therefore, a useful operating range was taken to be any laser intensity

that would produce an analog divider denominator voltage in the range

-2 v to -10 v.

The analog dividers served as a very convenient means of compensa-

ting for variations in the laser light intensity. This was especially

useful during calibration of the instrument, since the analog dividers

served as real time computers performing the signal normalization. The

normalization could have been performed more accurately and over a

greater range of laser beam light intensity fluctuations by recording

digitized voltages of the sum- and difference-amplifiers and performing

the division on a digital computer. However, this method would have

been more cumbersome, more time consuming and would have caused a serious

delay in the development of the instrument. The effectiveness of the

analog dividers in compensating for laser beam intensity fluctuations

is illustrated by the oscilloscope traces in Figure 10. Figure 1Oa

shows 60 Hz fluctuations in the sum-amplifier outputs due to powering

the laser from a rectified 60-cycle power supply. The x-axis denomi-

nator is shown as the upper trace and the y-axis denominator is shown

60 Hz component

upper trace: x-axis; lower

I

150 my fluctuations

trace: y-axis

_5

- 50 mv fluctuations

upper trace: x-axis; lower trace: y-axis

(b)

Figure 10: Illustration of the effectiveness of the analog dividers

to produce a normalized signal voltage. Typical fluctuations

in an un-normalized signal, (a); the normalized signal, (b).

_ __1

as the lower trace. The numerator signals exhibited the same type of

fluctuations with roughly the same amplitudes as those shown for the

denominators. Figure 10b shows the result of normalizing the numerator

signals to the denominator signals with the analog dividers. Notice the

60 Hz fluctuations are virtually eliminated.

The entire amplifier was found to be very sensitive to small changes

in the supply voltage to the amplifier components. A regulated +-12 v

power supply was incorporated to maintain a constant supply voltage to

within 5 my. at room temperature, 75 F, and to within + 20 my at tempera-

tures ranging between 45 F and 95 F.

Optical Tube

The optical tube was fabricated using 0.025" aluminum sheet, type

2024-T3. The sheet was rolled and spot welded, and then sealed with

room-temperature-vulcanizing silicone rubber. The external surface was

coated with a titanium dioxide epoxy resin to provide extra protection

from the elements and also to reflect sunlight and aid in maintaining

a lower internal tube temperature. The internal surface of the tube was

coated with a flat black paint to minimize reflections of scattered

light. The main support collar, shown in Figure 3, was an annular ring

machined from acrylic plastic stock. The support collar was designed to

carry the entire weight of the optical receiver without loading the thin

aluminum tube in a lateral direction. This minimized the chance of tube

flexure and errors introduced by distortion in the optical alignment of

the internal components.

Oceangoing Support Structure

The oceangoing support structure in its final configuration is

shown in Figure 11. The structure was designed to withstand loading by

six-foot ocean waves with deflections that would produce optical errors

no greater than 0.3%. Laboratory tests at three to four times the

design loading showed that deflection errors were on the order of + 0.3%

to + 0.7% with resonant frequencies for various deflection modes ranging

between 1.8 and 3.8 Hz. This indicated that the flexural errors encoun-

tered during ocean operation would be less than the 0.3% design value.

The laser carriage had two sets of adjustment screws to aid in

establishing the optical axis. The optical receiver carriage featured

a double-gimbal support ring to carry the weight of the optical tube.

This made optical alignment in the field a relatively simple task,

usually requiring no more than 15-30 minutes. A bulls-eye type of bubble

level was mounted on the upper horizontal member of the support struc-

ture to aid in adjusting the optical axis to the vertical. The level

was calibrated to provide a measure of the optical axis deviations from

the vertical to + 30 in any direction.

The entire oceangoing support structure was designed to be mounted

on the servo-driven boom of a wave-follower mechanism. The wave-follower

mechanism was designed and built by the Department of Coastal Engineering

at the University of Florida. It is a semi-portable structure that may

be mounted on the ocean floor in thirty feet of water as shown schemati-

cally in Figure 12. The hydraulically driven boom has a dynamic range of

+ 2 meters about the mean water level and operates such that an attached

instrument package maintains a relatively constant height above the

water surface.

-j

optical receiver carriage

double-gimbal support---

mean water level _

laser carriage

location of the wave-

follower support tube

iL^J 1 f^

i ^ ,;I- ""..,

I- .il.

Ij i!A

i l ,, ^ -/

*

;[

bI

K _i n

"i 1^

f1,I

Ii

Jj I

Figure 11: The oceangoing support structure.

optical receiver

capacitance

gauge

laser

servo-driven boom

oceangoing

support structure

wave-follower power

supply and controls

power cable

- ocean bottom

Figure 12 : Sketch of the wave slope instrument during ocean operation.

ocean surface

CHAPTER III

THEORETICAL PERFORMANCE AND

OPERATING CHARACTERISTICS OF THE INSTRUMENT

Introductory Remarks

Calibration of the instrument consists of measuring the amplified

output voltage signals for each of the two orthogonal axes of the two-

dimensional photodiode as functions of the laser beam deflection angle,

q, and the aziruthal angle, '. The deflection angle 9, is then related

to the wave slope, 4, via equation (6) and the orientation of the surface

normal, as shown in Figure 1, may be determined. If the signal normali-

zation process discussed earlier is to be performed on a digital computer,

then four voltages must be recorded for each value of 0 and T i.e. the

sum-amplifier output and the difference-amplifier output for each of the

two orthogonal axes is required. If the analog divider is used to per-

form the normalization, then only two voltages, the analog divider

outputs, are required for each value of I and P Calibration using the

analog dividers was by far the more convenient method of calibration

during the development of the instrument, but it was subject to uncer-

tainties in calibration due to some sensitivity of the analog dividers

to the value of the denominator signals.

Calibration Using the Analor Dividers

Figure 13 shows a photograph of the calibration set up in the

optics laboratory. A 2 mw polarized HoNe laser was mounted at right

angles to the optical bench which was used to support the optical

receiver portion of the wave slope instrument. The laser beam was

directed through a Polaroid filter to a right angle prism used to reflect

the beam parallel to the axis of the optical bench. The axes of the

optical bench and the reflected laser beam were aligned prior to mount-

ing the instrument in an optical bench cradle. The cradle was then

adjusted such that the optical axis of the instrument coincided with the

path of the laser beam. The Polaroid filter could be rotated to vary the

light intensity into the instrument and thus simulate attenuation of the

laser beam due to absorption. The prism was used to simulate refraction

at water surface. Changes in the deflection angle, i, could be simulated

Vy rotating the prism. This was accomplished by mounting the prism on a

leveled turntable that could be rotated about a vertical axis. The

accuracy of the turntable movement was approximately 0.020. Changes in

heights above the water surface could be simulated by sliding the instru-

ment cradle along the optical bench.

Figure 14 shows a typical plot of the calibration data. The x-axis

and y-axis analog divider output voltages are denoted by Qx and Qy,

respectively. Solid lines connect data points corresponding to = con-

stant and broken lines connect data points corresponding to > = constant.

The points were generated by first setting the null signal position, and

then stepping the simulated deflection angle, {, in 2 -increments across

the entire surface of the objective lens. The output voltages from the

analog dividers were read using a digital voltmeter and recorded. The

wave slope instrument

prism mounted

on a turntable

\

-Polaroid filter

r I iL L4r j '\

n:, i* v.jur s Y

I, ^ I f / -

*i ^ 1] i'V

,,~~~ r,^ --. ., --'<

I'.,,- .-. 1 .. f -,

S"" /

i '*'U T

1Ii-I --~i II jI- ~~~ F ..-s,

Figure 13 : The calibration set up in the optical laboratory.

K 7

i i!

100

lines of constant

lines of constant

Figure 14: A typical plot of calibration data obtained using the

analog divider. x and y have units of volts.

31

instrument was then rotated approximately 150 about the optical axis and

the procedure repeated.

A major concern in the application of the instrument is the repeat-

ability of calibration data. It was found that at a constant temperature

the calibration data were consistent to within 2-3%. However, a change

in temperature of 10-150 led to calibration variations on the order of

10-20%. Nearly all of this variation could be accounted for by temper-

ature dependent changes in the offset voltage. The amplified offset

voltages as a function of temperature are shown in Figure 15. The

temperatures were measured inside the optical tube at the physical

location of the amplifiers with the optical tube sealed in its ocean-

going configuration. A gas-tube thermometer with a dial indicator was

used to make the temperature measurements. The thermometer and its

location relative to the amplifiers is shown in Figure 16. Calibration

data for various optical tube temperatures were taken in the laboratory.

In the field, the tube temperature during the test was monitored so that

an appropriate set of calibration data could be applied to the resulting

test data. After correcting for temperature changes, the maximum un-

certainty in the calibration was estimated to be on the order of 7%.

Evaluation of Optical Aberrations

Optical aberrations result when rays from a common point in the

object space enter the lens at different locations and fail to come to

a common point of focus at the location of the image point determined

from Gaussian or paraxial approximations. Any rays which deviate from the

Gaussian approximation are thought of as some sort of aberration.

Lengthy treatments of aberrations may be found in any of several good

32

d

y

+05 wave tank tests

OFFSET '

VOLTAGE

/

0

d

nx

x I -

-0.5 d

ny ocean test

70 80 90

OPTICAL TUBE TEMPERATURE ( F)

Figure 15 : Offset voltages measured with respect to the amplifier

outputs. Numerator and denominator voltages are denoted

by n and d, respectively. Subscripts refer to axes of the

photodiode. Error bars indicate the scatter of data

points; the broken line segments represent temperatures

at which no data points were generated.

33

I~`'

I

( ~

2 13

.w ,-. - .-- ,. . *

--- n ^^_,TW---- [*1-1- ji* -tn- i fi *.-r^*T* A-.

- -

4

yCA

h.<~i..J^e ^e^^^ ,*jm ^.^ ^-.j^ mcA -li~l

Figure 16 : Photographs showing the location of the optical-tube

thermometer relative to the amplifiers in the imaging

lens module.

_I______

~TZ

__

reference books on optics.1,18,19

The effects of optical aberrations associated with the objective

lens were evaluated by an indirect method. Since optical aberrations

may affect both the location of a point in the image plane and the

sharpness of the image, and since any significant aberrations would

lead to measurable variations in the instrument's output signals, the

output of the instrument itself was used to estimate the severity of

optical aberrations.

The laser beam was treated as a "fat light ray." This test ray was

allowed to enter various portions of the objective lens while keeping

the deflection angle, 4, and the azimuthal angle,t, constant. During

these variations the output signals were monitored to detect any signifi-

cant changes. In-practice, the angle 4 was.held constant by avoiding

any rotation of the optical tube after the optical axes of the instru-

ment and the laser beam were aligned on the optical bench as shown

previously in Figure 13. The deflection angle, f, was then varied such

that the laser beam moved over the entire surface of the objective lens.

The x- and y-axls output voltages were recorded and plotted as a function

of f. The distance between the prism and the objective lens was varied

keeping T constant and the deflection angle, 4, was again varied through

its full useful range. Again the x- and y-axis voltages were recorded

and plotted as a function of ". The entire procedure was repeated a

third time at still a different value for the separation between the

prism and the objective lens of the instrument. This process allowed

the beam with a specific orientation, (4,$) to enter the objective at

different radial locations on the surface of the lens. If aberrations

had been important, the data points corresponding to common values of

Sand would have changed noticeably. This was not the case, however.

The voltages corresponding to common values of t and K were repeatable

to within the value of the noise in the analog divider outputs. Thus,

it was concluded that the effects of optical aberrations were negligibly

small and completely masked by the noise in the electronics.

The Frequency Response of the Instrument

The frequency response of the instrument was determined by physically

chopping the laser beam with a rotating mechanical aperture. The laser

beam was positioned to produce a null signal output from the analog divid-

ers when the beam was allowed to enter the objective lens. Blocking the

laser beam with a mechanical stop caused the denominator signals to the

analog dividers to go to zero, thus the divider became saturated at the

maximum divider output voltage. By switching the laser on and off at

high frequencies the frequency response of the entire system, from photo-

diode to divider output, was evaluated. The results showed that the

frequency response of the entire system was flat to frequencies well

above 400 Hz. Since all of the data gathered with the instrument were

filtered to eliminate frequencies in excess of 100 Hz before digitizing

the analog signals, it was concluded that the frequency response of the

instrument was flat throughout its useful operating range.

Wavelength Response Characteristics

Cox has shown that the finite dimensions of the spot on the water

through which all rays must pass on their way to the detector form a

"low-pass" filter to the signal. Measurable wavelengths must satisfy

the inequality A B6.8r where e is the radius of the spot size, and 7

is the resolvable wavelength. This resolution criterion is applicable

to systems in which the light intensity is uniform across the spot. The

following analysis considers the case where the light intensity varies

across the spot in the form of a two-dimensional Gaussian distribution

function.

A first approximation to the wave slope instrument's response to a

finite-sized laser beam assumes that the photodiode is sensitive only to

the centroid of the laser beam focused on the diffusion screen. This

is a reasonable assumption because the spot on the diffusion screen is

usually small and well defined for wavelengths satisfying the Cox criteria.

This assumption, plus the use of the linear relationship between the

radial displacement of a ray on the diffusion screen, and the deflec-

tion angle, j, as given in equation (7) allows the analysis to be simpli-

-fied. It follows that f where is defined as the deflection angle

corresponding to refraction at a plane surface which results in a centroid

located at p.

The variation between i, the measured deflection angle and o, the

deflection angle of the central ray in the laser beam was examined first.

Next, the fact that the photodiode is not strictly sensitive to the laser

beam centroid was dealt with and an error associated with the spreading

of the light intensity distribution about the centroid was estimated.

Errors Due To A Finite-sized Laser Bean

The theory of operation presented earlier was based on the refraction

of a single ray of laser light. For the single ray case, the wave slope,

6-, is related to the deflection angle, by equation (6):

'( sin$

(6) --= arctan s- V

Sn cos /

where n is the relative index of refraction for water.

In the case of a laser beam of finite dimensions, the beam may be

considered as a bundle of rays distributed over a finite region about

the optical axis. Here, the optical axis is placed at the centroid of

the laser beam as it emerges from the laser cavity. In general, rays

penetrating the water surface at different locations will encounter

different wave slopes and hence give rise to-different deflection angles.

A measure of the wave slope at the optical axis is still desired, but

it becomes necessary to work with a mean deflection angle, , rather

than a single deflection angle at the optical axis. A value for the

measured wave slope, is defined as

sin

(9) g- = arctan ------- ---

n cos .

Since 1 will generally vary from the value of the deflection angle at

the optical axis, o the measured wave slope, 6- will generally be

different from the true wave slope occurring at the optical axis,-6 .

0

Therefore, an uncertainty or error in wave slope measurement may be

defined as

S=-&- -e-

m o

(10)

38

It is desirable to describe the behavior of the error, E as a

function of tne laser beam parameters and the configuration of the water

surface at the test site. This is a problem of considerable complexity,

but a first-order estimate of the behavior of the error may be made

following the introduction of a few simplifying assumptions. These

assumptions are

(1) Geometric optics are assumed to be adequate for a satis-

factory description of the system, i.e. diffraction effects

are considered negligible and ray tracing techniques may

be employed.

(2) The coherence properties of the laser beam are assumed un-

important, thus interference effects are negligible and

a light intensity distribution rather than an electromag-

netic wave amplitude distribution is associated with the

laser beam.

(3) The laser beam below the surface of the water is assumed

to be perfectly collimated, i.e. consisting of a bundle

of perfectly parallel rays.

(4) The light intensity distribution in the laser beam is

assumed to be a Gaussian distribution centered about the

optical axis.

(5) The surface of the water is assumed to be accurately de-

scribed by a linear combination of sinusoidal plane waves

of varying wave numbers and with varying directions of

propagation.

39

The fifth assumption allows the surface amplitude to be described

as

(11) z(x,y,t) E a sin ( w t + )

1=o

th

where the subscript "i" refers to the i -component of an infinite series

-a

of plane waves, k is a three-dimensional wave number, r is a radius

vector measured perpendicular to the optical axis, w is the radian

frequency of the wave component, t is the time, and 5 is an arbitrary

phase offset measured at time t=O. The x-, y-dependence is introduced

by the use of a cylindrical coordinate system:

(12) z

x = r cosi'

y = r sin 4

where is an azimuthal angle measured with respect to the positive

x-axis. Figure 17 illustrates the configuration for the i -wave com-

ponent propagating in the I azimuthal direction. The Gaussian laser

beam is centered about the optical axis at (x-O, y-O) and is directed

in the positive z-direction,

To simplify the analysis, consider a single wave component with

wave number k, where the cartesian reference frame has been rotated

such that the wave component travels along the y-axis, and I has been

set to zero. Under these conditions, the surface is described by

(13) z = a sin (ky wt)

40

z

x

optical

axis

i -thwave component

aussian laser beam

Figure 17 : Configuration for the i -wave component and the

Gaussian laser beam.

where the subscript "i" has been dropped. This type of analysis is

consistent with that of Cox in estimating the wavelength response

characteristics.1

The wave slope of this simplified model is then

dz

(14) -= -- ak cos (ky wt)

dy

In cylindrical coordinates

(15) -&= ak cos (kr sin(V ) wt)

The deflection angle, , is found by Snell's Law,

(16) sin( +t) = n sin -,

to be

(17) = arcsin(n sin(-8)) 6-

which becomes, upon substituting equation (15) into equation (17),

(18) (rt,) = arcsin n sin(ak cos(Ir sin(j) wt)))

ak cos(kr sin(t) wt)

The deflection angle, q(r,f), is specified at each point (r,f) through-

out the xy-plane by equation (18). The Gaussiadi laser beam weights

each of these points by a factor

72r2

(19) p(r,t) = e

where the factor & was introduced to make the weight function normal-

ized in the sense that

-to

(20) (p(r,k) dA- 1

-CO

The expectation value of or the mean value of p is T and is

defined by

(21> = (r,t) p(r,f) r dr d .

Substituting equations (18) and (19) into equation (21) gives the ex-

pression

c z 2 2

(22) e e r- arcsin n sin(ak cos(kr sin(t)

0o wt) dr dr

,o 21r 2 2

T r e- r ak cos(kr sin(C) wt) r dr d .

o0

Referring back to equation (17) and Snell's Law, it is seen that the

first integral in equation (22) has no obvious physical significance,

but the second integral in equation (22) is clearly the mean value of the

wave slope relative to the Gaussian weight function supplied by the

laser bean intensity distribution. Integration of the second integral

is relatively straightforward, giving20

(23) fak e- 'r cos(kr si. (1') wt) r dr d' =

0c -k2/2e62

ak cos(vt) e

Since ak cos(wt) is the slope evaluated at the optical axis, the weight-

ed mean slope is simply the slope at the optical axis weighted by the

factor exp(-k2/4,).

Integration of the first integral in equation (22) could not be

accomplished in closed form so that a numerical scheme was necessary for

evaluation.

The numerical integration scheme used an approximation to the Gauss-

ian weight function as illustrated in Figure 18. The central cylinder

and each cf the annular-ring segments carried equal flux and were thus

equally weighted. These equally weighted segments were then divided by

a number of radials such that each of the resulting sub-segments were

equally weighted. The equally weighted sub-segments were then replaced

by equally weighted delta-functions. The delta-functions were located

at the centers of each sub-segment, with the center being defined by

a mean radial distance and a mean azimuthal angle for each of the sub-

segments. The approximation shown in Figure 18 provides a 16-point

approximation to the Gaussian distribution. In practice, 360-point

and 1800-point approximations were used.

Note that the solution to the second integral cf equation (22) as

expressed in equation (23) is a function of three parameters: ak, the

maximum wave slope; wt, the phase of the plane wave; and the ratio k/r .

This suggested a change in the radial integration variable for the first

integral of equation (22) from r to rcr= After making this variable

^4

Gaussian intensity distribution

in the laser beam

Approximation to the Gaussian

intensity distribution

Annular segments carry

equal flux

/ .Sub-segments carry

Figure 18 : Approximation to the Gaussian intensity distribution

used in the computer error analysis.

change and re-defining the locations of the delta-functions in the

Gaussian weight function approximation, the mean deflection angle, ,

became

(24) 2 4 2 --- S ( i,Y ) arcsin n sin(ak cos(kl/

( j Inn i 1 1

i sin(6 ) wt)] ak cos(wt) e-k24

Equation (24) ras then evaluated on an IBM 370 digital computer for

various values of the parameters ak, wt, and k/r- The same computer

program also evaluated -6- as expressed in equation (9) and the wave

m

slope error, 6, as defined in equation (10).

At this point it is convenient to digress momentarily to clarify

the significance of and k/r The laser beam parameter, c is a

measure of the radius of the Gaussian intensity distribution given in

equation (19). The radius of a Gaussian laser beam usually refers to the

1/e2 point of the beam, i.e. the radius at which the intensity reaches

approximately 0.14 of the maximum intensity. Calling this radius ro,

it is seen from equation (19) that may be defined as

(25) = _1/r0

and has units of reciprocal length. Note that since C is a function of

ro, it is also a function of distance from the output window of the laser

cavity for a diverging beam.-

A plane wave of wave number k has an associated wavelength given

by

(26) X= 2Tr7/k

Thus it is seen that the ratio of the water wavelength to the laser

beam radius may be expressed as

21T 4.44

(27) ,/r =

S(k/) (k/r)

Equation (27) is plotted in Figure 19.

The parameter k/a" was convenient to use in the computer analysis,

but wavelength-to-laser beam radius ratios are difficult to visualize

using k,/4 as a parameter. Therefore, whenever k/r is used as a

parameter, the conversion to units of N/r is indicated.

The computer calculated wave slope errors indicated that the

greatest absolute error occurred at phase wt=0, i.e. at the point of

greatest wave slope. Figure 20 shows the error in wave slope for various

values of the parameter ak plotted as a function of parameter k/-

In all cases, the error is associated with the worst-error case occurring

at wt=0.

Figure 21 shows the percent error in wave slope measurement for

ak- 50 and ak4> 450 plotted as a function of the parameter k/r Again,

these values correspond to the worst-error case of wt = 0. It is seen

that a 10% error in wave slope measurement occurs at k/T" = 0.65

(A/ro = 6.83) for ak= 50, and k/ = 0.76 (A/ro = 5.84) for ak- 450

30 r

20 L

o 0.5 k/O. 1.0

dI I I-- I I t I I

0 -.5 -1 -5 -10 -15

PERCENT ERROR (450 WAVE SLOPE)

SI 1 I *'l -

0 -.5 -1 -5

ERROR IN DEGREES (450 WAVE SLOPE)

Figure 19 : /r as a function of parameter k/t- Additional abscissas

are scaled to show the percent error and the error measured

in degrees for the case of a naximum wave slope of 450.

X/r

-3.0 .1- '14. /

.3 14.8

.5 8.9

.7 6.3

.9 4.9

(degrees)

-2.0

50

-1.0

0

0 0.5 1.o

k/

Figure 20 i Error in wave slope measurement, as a function of

parameter k/-r, at wt = 0, for various values of parameter ak.

-18 ---

slope = 50

k16- '\/r0

.1 44.4 450

450

-15

.3 14.8

PERCENT ERROR

.5 8.9

IN WAVE SLOPE

.7 6.3

MEASUREMENTi

.9 4.9

-10 i------------------------- -----

-5I

0 0.5 1.0

k/10

Figure 21 i Percent error in wave slope measurement as a function of

parameter k/r, for wt = 0 and ak : 50, 45

Taking k/g- 0.7 (X/ro = 6.34) as a representative value correspond-

ing to a 10% error at wt = 0, the behavior of the error as the phase of

the wave changed from zero degrees to 900 was examined for various values

of the wave slope parameter, ak. The computer evaluations, of the error,

based on a 360-point Gaussian approximation, are plotted in Figure 22.

The most noteworthy features in Figure 22 are the noticeable inflection

points in the curves for slopes >300, and the actual change in sign of

the error for values of slope approaching 400. The sign change was found

to be a physically observable phenomenon and not an artifact introduced

by the computer analysis. The error in wave slope measurement was

found to be satisfactorily modeled by the expression

(28) E(wt) = E (ak, k/-) cos wt + E2(ak, k/.r) cos 3wt

where the coefficients E1 and 2 are functions of the parameters ak

and k/C'. For certain values of these parameters, the coefficient (2

may become greater than ( I thus shifting a substantial portion of the

error from a frequency w to a frequency 3w. For example, a steep sin-

usoidal wave having a slope of 450, when measured with a Gaussian

laser beam-with radius ro = 0.157A, would produce a 4.3% error at radian

frequency w and a 4.3% error at a radian frequency 3w. The result is an

erroneous indication that the wave of radian frequency w contains a

superposed ripple component of radian frequency 3w. Growth of the co-

efficient 2 is very rapid as the slope exceeds 300, but for slopes of

35 or less, greater than 80; of the error is associated with the coeffi-

cient C the "in phase" error component. Since the Instrument described

3

2

1 450

-1

2 crest

// location

-2

350 450

-4

0 300 600 900

WAVE PHASE (degrees)

Figure 22 : Error in wave slope measurement, E as a function of

wave phase for various values of maximum wave slope

and for k/Va =0.7 ( A/ro = 6.34).

_-

in this paper is not suitable for measuring slopes in excess of 350 at

reasonable heights above the water surface, the problem of errors shift-

ing to higher frequencies may be considered of negligible importance.

The quantity defined by the symbol e should be taken as an error

rather than a correction factor to be applied to the wave slope data,

unless the wave number spectrum of the wave field is accurately known.

This is true because the orbital velocity or the fluid velocity associ-

ated with the gravity waves destroys the one-to-one correspondence

between the observed signal frequency, f, and the wave number, k,

as is discussed presently.

A simple wave field consisting of a single sinusoidal plane wave

propagating through the test site v;uld produce a wave slope signal of

a single frequency, f, related to the radian frequency, w, by the

expression

(29) f = w/2TT .

Phillips gives the radian frequency of a wave as

(30) w = (gk + r k3)1/2

where g is gravitational acceleration, k is the wave number of the wave,

and is a constant determined from the surface tension and the

density of the water.21 He also shows that the wave number of a

short wave changes in the presence of a long wave according to

the expressions

S k

o

(32) r-I o(U/c

where k is ile ;'.e nlirber o: tn, !..:' n.c in the presence of a

long wave, k is the wave nui,.T, r ofh t shoci arve in the absence of

a lorg -;ave, 6 k is the ob'-srved ch~es c~ r -te w -r e number of the short

wave, U it. the fluid velocity of the lrrn "'Ie, and c is the phase

velocity of -dh lorng jve. It is readily loLsn that an approximate

cxpo-cssior for the r dija n requen4cy of c.ho'r wave in the presence

of r: long i:-v.e s

1/2

(33) (gko 3 0ko, 3?ko U/c)1'

The observe slg. al freqtency rray then bo.- written as

i 1/2

(3) f 1" (/2W)(gt, + ko3 + gk o/c -- 3ko U/c)

At. the crest of a lung swve, I is a r.aximn. pcjitive value and the

obs-rvea si-n,"a frequency cfr the short wv-,.e Is greater than the

frolenc .xT'cte ih"n no l"g wave -.s present. When no long wave

3s present, U = 0 an1i equatior (34) ciogenerotes to equation (30).

At tie trough rloation; of the long wave, U is a maxiru negative

value zwn the ctserved frequency is less .ha'n that occurring when

wc lonr; '.- is pres-nt .

Lo-termining the fluiJ v-.Lcity , 5, itihe prese:ncc of' evsral

rand.omliy superlpoed long wv"as of varyini .,-.clengths hecomas complex

and intractable 'cr rractic:-] p,.rross. Thus, the one-to-one corres-

pondence bc;-,een an observed sig nal frcvEroy and a unique wave number

as expressed in equr.tions (I9) .:ld (3C) is .ies'royed by the presence

of long waves. Surface car-rents, whether .unerated by the action of

wind o, the water, tides, or other nccha.isrTs,also complicate the

problem by introducing adlitionl fluid vricity components.

The error tn the wave sloi ;-eseasr.;m-nt, as discussed above, is

not the only source of error due to a fli'ie-lized laser beajit, although

it imay be cornaser ed doTAminnt. A secondary error, 6, is associated

mlith the typo of sensor used, .e. the conitiauous, 'o-dimesFJ.onal,

Scho hy c-rrier phc-ejdidiu. This secondary errer results -';o the

fa't -.haL!- chithi.-de i no t i.truly iscli e in -he centroid of

tihe light. internsty distri ution striki-n, the s-uflce of the photodiode.

That is, a aelto-function di tribuLton and Gausrian distribution,

each having; the sane centroid location and -arryinri the s.,e a-.ount

of flux, will generally produce different ou-put algr-ls front the

photodiodde. Taking the delta-function distribution as a reference

distribution, the error, may be defi.ed as a fractional error

by

(35) ( ref rcf

wh roE S .s thE ou+pu:t signal for a gene' :Listribatjon of ligh- on

the surff.ce rc the photodiode anl S is the output signal for a

rrel

celc.-futior. rlIstrit..ti.o, '.ing u i-s:. t:.. that both distributions

h?a'e the r".o: centroiad ccm ..Oi.

A co.si.rvative estimate to be ia'.niutl- of 5 for the case of' a

SGau.ia'j-" lser 'Ct of radi-; r tra.:tirn with a sinrusoidal plane

wave of wae.lern..th '\ 6.4 amn a sxr!v.m- wave slope of 35 was

foLmud to be 6 0.019, 1,", : -1.9- c-ior. Rei'erring back to

Figure 21, it is seen that the perucl.t erroir due to the error 6 is

app-oxi.ately tor the cae \ 6.3- r and slore = 350 Thus

the e--orroe-_ accounts for morp thtr F:05O of tih! totil error, For the

case of a slo.pe of 3 0 and\ '" ro, thie -error increases to

approximiate y --5.6; and th- 6 -ecrrorr lnc- .:sc-s to approximately -155%.

Thus it is neer that the irflaunce cf the 6-n;r'r bicor.es more and more

apparent az aL-o:h i-,, '/ 0/'C'...a-o, O:r; .-:, significantly to

the total error associated iith the use of a fini e-simed laser beam

as the waveler.gth, A/ approaches tr. dimension cf the laser beam.

,.rAPFEL TV

A\tE TANK TE''C-

The ins3eruTe-' nas usc in a series of un-o-di:i iasional have tank

tests conducted at the University c' Flori-d-i's iave Tank Facility in

late "r-ch, y19?. The larr W. a ;oiLL .-. ho'izosntally, outside of the

wavc tar rouly 35 ca below the ma.-n water level, The laser bean was

diiec;te] through pljdate glass window to a~, unr-.cw.trtr-rmounted prisr.

w.hcore th'e '~oet d, o deflected upward alcm: Wh-' vertical. The wave slope

instruiment was ricLntwid wiit +A'- objective ::Es. 8 ci above the mean water

level. Optical alig,_-nent was F'rfcorned uwler c.al-water conditions with

the y- anr :-xi.. Ch'n :.ls di- 'ir.ct; i.: *L1 (c'rcswind andi downwind iliras-

tions, respectijvly.

F'igui-e 2?3 l:oSs aomplEus of the dournw;i]d charnel signal for the

cases: io :;ie;chnir,.cal wave, wir speed of 11.9 -/sec; 2 second, 2.5 cm

arFiituda rFCchanical wave, win: speed of '11.9 n/sec; and 2 second 5 cm

amplitude mechanical wave with wind speed of S.2 m/sec. In all cases

the upper ti-.e series is the downwind channel output signal a;:d the

lower time scr"s3 is a capacitance ;gaRe output, which gives an indica-

tion of ti; phase of the wae i-chanjcal waves and wind go from right

to left. Some features to note 'ire

(1) the diUvelup:i.ent ofi a 4 Ha wavoe structure due to the action

of t-he wird with no mr hcch;r.i.l wave,

(2) the 24 Fz wird gr-ine:irtd we-;l w: orgoes oh apparent frequency

cicLrease to 4.-% F'- aF t(he crc-st of the, 2,5 cri riocl.anical

140

OC.

0__

0 cr--

-.5 C' ~_

23

0

0---

14c-

142--

0 O

-5 c0i _

c~ic-. wx' i~. u;ls.

58

wave, ana deciea-es to 3.b in zhe trough,

(3) the freqiencn o ... ...i ;lLio of the wind generated wave over

the 5 cm mech:r. ,.1 -' ave. .-: fre]qurcies approaching

10 Hi at th1 rc= and *..L 1h:' in the trough,

(4) the cornen:ra.io; of Tr iae aciv i:it: on the downwind face

of the windd -o d aves -as indica- ed by tho 80-100 Iz

components in the wave slope :.st+r.s-rent's tiLe series

re cord,

(5) maxinmu wave slope of '.11 --i1 in case of the 8.2 m/sec

:;iitd speed, an- 20-20 in l ;e case of the 11.9 m/sec

nwi' spe-ds.

Phillips has ;:,hirn that the ;ive iur,-' d .he i;ave amplitude of short

waves irn.'rcase a3 the crest of lo:jF, a.av-; an. denc '.,n in-the trough.21

This lves !"ri to -raor 2 0-'- sp 7 wi5, hri.er c.0e'^vaj10-h fr-0u'rnci0 -

at the cresre of the lcng wave and lese'r wav- siop.-s with loiter fre-

quencies at the trough. The t i- record- t'or the 5 cm nmchanical wave,

as shown jn Figulre 23, is entirely consistent with 'litlins' prediction.

At high w:ave slopes the la.s:r l:om would lefilcct off the objective

lens and a signal 0roiout occurred. IhcpouL woas de .-ected by a zero

denominator signal. Minimal droLout occc:rr-e dur?'.: these tests and

there are no ex=nples of dropout in thr. sa;rple ree:'ds snotmn.

Figure 2-4 showi tthe 'r-oquLenoy pec i r of' the j-.n square slope for

the thrre sample records plotted on lcg-log ,calcs. The energy units

'.To ,j'aectr.- pr,- cnt-,- werc prc,'; '-.' K /.1c '.n Reece of the

Lepart:ent o' Ccr-ial I. i.Ineerin at tih L:I-.'ers-ty. of Florida. The

sp,,ctr. er-e ,:-..:r-;tep:! -si, ca F",st. -o v-'ir Translft' colpurer program

de-el-pj. bv i obso-.

13

0-1

1-2

17'EPGY

^-2

2'

(radlann )

10

7..-.

~'19'J~,X -t

. r -i

1 0 10 1 10i "

FR';EQU: .iCY, f

10

10-1

1 -2

10-3

10-4

10-5

E nERGY

(radians -sec)

10- 6

4o

Figure ?2 : 5requeiery I"' :.ira. of' thl ;C;.an squire slope.

for the 2.5 cm and 5.0 cm mr -... ic';: war c,'s a:e in rad.i -see

and the appropriate" ordinate a;s.rs to e riZht ,f the figure. The

spetri-m for the case of no ..-tl:anical :'.e '-:s irltip! ied by the fre-

quency in or-der to cm!.npare ;::. the hect h of Cox which is also

shown in the f iguer. T'h, ,i, n -,, r.,' thcte ],-Lter casc s are radians and

the app-opriate ordinate al;p.saLs to !'.e ]eft of the figure. Note the

increase in the 0.5 Hz component as the nchanical wave arplitude in-

creases. Alsc note the decrea'-e in the ;: Hi component as the mechanical

wahe s:rl-uiue increaRses. Pi'-a ab thi: L, due to a frequency spread-

in6 effect c- discussed earli e in cron.eclLcn "wi the modification of

short v;ven hy the fluid velocity associated with lo'ng waves. The

reason for the ordor of' m.gnit-. differc.i. '. b"ctwcei; the Cox spectrum

and the present spectrum I, still under active investigation, Some

possible rc:.-:.c" for the di-'fcrorence ar~e ti- rresence of light oil

slick in the University of Florida W'ave Tank; the Cox fetch was 2.14 m

ohil hethe res.nt tests were conducted it a fcech of 9. i r Cox was

sensitive to light intensity fluctuatio:. n and also nade linearizing

assusipotio:s concerning tho light intensity distribution across the

spot ard the imodulation of the in-'nsizy due to changing wave slopes;

ma. ard sub-rm ripples may have affected ith present casurernents due

to excessive spreading of the refracted laser beam.

( ; ,H A ', .- 9P ,

cThe Jcean .:-VP s!opr: ri.:ut.-!i:ents w:.c r-.Le on 0 9.' ,9. 19?5, at a

.ocatici off thI ccast of rilai ric.rla. A EnTli.p boat carried

tha auxiliary pos;er supply arzits anid The d-.-a acquisition station. The

wave fItliover r=.'h iisi: ,';s:;e. i t.hlirln -c:ri of water. A standpipe

,ith. a'" altch :; ladder was ?:oi.t.J ri'x.I tr the wave-follo0'wer booi to

faci.lLiate the ins:.tf.ilation a :l rcmovai of instrument packages. Figure

29 :.'!;u;' rs'.caph of tie r,; :Dp tio.t's position relative io the test

iT? in;slru-en: carri c :eT.: ate- tica.. r :cr packs for the elec-

[ronric "li'e' t. t.ha Cohc'tnnc R'?i::+-.'": Model 0-2" sub-orsible

laser. W':iav slop? signals wrrc hard-,iircd to the data acquisition

canter. F:re .i shc.;s the instru:.:ent in ;', erati:or -iring the ocean

is .-:. T: i:'miiecdiate teL site Cas prote-.e ftro:: thec effects of sun-

Glittor t ar cpaqcue p1] st.ic shcsr .

Sa.a eLc records of tl; :ceLn icet dLat ar. shown ir Figure 27.

Tro.ces (a) and (b) are the y- end x-axis outputs, respectively. Trace

(c.) l -:,e x--a.i :;enom'riw tor .-i.;nl- whinh w;,; mironritoretd to icdntify

reg"u:u: of ."i.:!-. cropout. Trace (d) i'x L ;':ve staff record indicat']i.

the ric, ad ..-,-",h loc:-Liorns oi' he 60 cn ocean sell.

'i' ocainr. ts data .r'l, r. procss.ed for: s-.vral reasons:

i ) The recrc-inL taL or.te'r a-,. lo+t ri.. pr.le'inary tests

ni.un a star.dpiti su.;pnor'..1. ne in.-trament was snapped by

o0 5': s'crr. '.1 t -. pl :' "s I.n

Fpfiu*e 2- : I -.otooiph otf hF -nr, m L's p.,.tior relati t to the

oC'._i it.Lp iC .

Fi{Gr> ': T? nsrT.sr; an In operation drig the osRan, test.

10

Ox' ,-IPFut.

31

I0 ii

:x:* i'i Id o t l ,

i-2 -

2'- I ,!f x-axjs jSoutput

S r i VOLTS

1 t 1, Ir Xli yt' j -

I i -8

Ii i

i region of djro;tct

v : -- ~i ( x >) d e n c n i n n t l ;: t

-2 .- -10

I I

0 wave sta recorded wa height -60

vo:;' -./' :, !i I I I

-so --- --_1--_----------------- -I 0

{--

+60 +60

cm cm

-60 wave staff recorded wore height f -60

Figure 2? : Ba-.le records of oce-n test dsta.

65

(?) ''hi test sit. :. t catu c. i. ... interference during the

tb. test, Swe!' -,flectei fron the shrimp boat, and the

test site i;.: irecl tly Lo.: 3a s-ide of the shrimp boat.

(3) Th- st:U'. p',- :..v' ." .i 'ci: !.rid:r cws too close to the

te.st site anl .;el~l freCu'ctl'y slapped the ladder causing

sprio'is iwavelici re'lecticn .

( P;) Po -tic.s of the record dirpl ayed ccnsideraule dropout due

to f'sh in the laser t e:n. On -ve ral occasion C a group

of 20 30 s:Da' l' isin toe. rsLi. filing through the laser

b-am. Fish drop:,;ut ; s ccily iden.f'.ed by long periods

of zero dnor.intt<,r signal.

5) 'h;. cc;L of co:,.p'er ti: ? hired d to araly;':e he occan

d i.a w a.s corsi'.i'de S:'Tnc nn direct application could

bP nade of th- -n araly.i< s dW to items (l)- (h) above,

'LL "Ite l..1 n1

S:.T'iA R Y

The instrinent descr:it-u i're has sho.:. severall sigrnificai 'Lvance

in obtaining continuous timc. records of w:veo slopes ocurri:n at a point.

The data re t;:o-di:-.c;itonn.l and connmat.ible irth the nature of the water

:-rqface. I'ic instrumiet has de.onsiratid tr, ability to function as

both a loaorotcrry inltroumerr and a portable field urit. It is insensi-

tive t'., a,.- height ani ligt intensity Cflctuations, thus making it

suiit.able i..r mr ~srng rirpi:.: in Lth presmice of swell in either clear

or tu.;Lii w:.-cr conditions. The i.nsri3'. t hasu an ircreaoed slope r..~suzr-

ng ''.:acitty' an'i a dpronstr1't.d c'. i -i. Ly for .a-time field use not

shui y i.-' ou,:.Ls in!str!raentl:. i ,r, were a-da on eqctio:., t:

;:e:r rTi li:n-r -d, tihus glivirg t. or realistic description of wave-

length 'csol-uiocn that that of Cox.

For the tests conducted, h!i. :.x~m n :: errors due to wavelength reno-

lution :!er- estimated to be -2..t at \ = 1.33 cm, -5% at \ .80 c.,

-j.1.92 at ., c. -a nd -2'.5' at ,; .:- cn.

'uJ.> ITNG .

I. i. T,1 th. the i n'r'-m.int Joscri.r hAre is the nost advanced

wave se slope ililri: instru:tont o' its "t.sc to date. Further i l;rove-

ments, :'-ih ; r c:.,ter I;.ivc 1loe r;a u-'iUnli -.ps.hility, lower tempera-

ture-decpind'd, c..ib-'tion orrors, ania erim.er wiaveieng-th resciution

are -;:]i within: r-och, Cr-ic.ar licld uti!iL. :ayj result from an added

signer igl crnt7' mcaule an' 01 simpi i.ed calir'ra~ion scheme which may

be appli..i in the field i- i'ecez. sry. iTh -e i::tr'cnt's performance is

expected s -'.o inx.cr-ase ma;hcd]y c y incorp'ortying a field len:; in place of

the difi T ora-.. .a"n 'oc : si d aa cc teso-e less '.ine consui'ing

by i:to' cia : the dita aI;qu'isi Ltj s.j.ie:. with a high speed digital

computer. A third--enteration instrmlc nt should provide wave amplitude

nlor.otion in additionn to th w two-i.-li.ei:ijonr l w '-ve slopo information,

thus prov'dini a mire complete ti, e history of the surface at a point.

[I:hi: '-2

I. Co>, lio'F.le:, ,., ";:Ec -... .u of -I" 3_ of Hig!.-FrCquenc.y Wind

Waves," Jo; ,-r, o-' c, .--. ..-,: n :-:', -_, 3, 3c1 5 pp. :99-225.

2. Sillwuell, LD.nci, Jr., O: _1 ^.s A-~ l for the 1971 A1P2 Tower

Ei,-- '::ent, .L Repor -, y, -

3, McGolorica:, L;Iwrenr.ce :', "i ,:o.i:.,. T.. rcions Anona Capillary-

Gravi.tyL Wivs, Jo .-A1 ni In:i r -- c... c 1 2, 1965, pp. 305-3)1.

4.; Valer:'en Je Y. .,, Y, R ,. ; a:l JI C. TDle'y, "Oceain Spectra for

t.he F. 1-i---.r.- nenc- t v e ;J :i't i err-;' from.. A artor.e RadtEir

tes-..ur-yen-- ,." Jo. nt I IC I,:. .- .-- 25. 2, 19/1, pp, 69 84.

.. .. 1i5, 1 19 5 1936.

tjulj:-or'J. ":, i. I y-.'p;.,". 7c- :vo;'- T-Rv;' .- -R--c in Jc? n.aal of

7 ,. Pys G 1-',i" T ., 0-:'o 2. oal W ;.. :.. j 7\naohn A. Guinn, Jr. ,

"+,.l TT-^-__+_..Ci.'_ --.... n.. ..ie n.. I Ocean; Inflr uence

8. il on ., ": -j o ni 0.-A Surface Leecrptors Using Sea

PKhot Ar .'. .; T cAni :.:.- ," ,; :. -crp-rtu 757 -, July, 1973.

7. Wr, An "oSiC'p a u; ; :'; ,.,' .-- io.s of WhiAd- l stmu bed Water

Sui:c-," -o;.:._1 o t:-;.__i' r *jA cic., 1, 7, July,

I'. C-", J ">- -.r Wa '. ..r. 'S :'. *-'+.7t ." the Sea Surface Deriveo

1r1;- L, -; .t- t, iur- c. t.:.L -l WI> c 5, 2, 12, 95', pp. 198 -

,27.

o ,h

11. Co .'. ,: .liv s, -.. .I r ,.u ':. o' '' Fou hness of the

3 Sr-'.crc fr- -. :.: a th- P. 's G: ,. _r of the

ot^ A. J -c' i -._. T -.' , : .rci-7 r -7, J y, 836 850.

S. ... .: : ..... ".he jar Sra tsnt for

-. .. e f... ..... .. i 4. l4, p 1973,

13. Scott, John C.,"An Cptlc i Pro"e fo r 'rjsring Water Iave Slopes,"

Jou-rnal of Phy.ics 2E: .i'.:ifj _Ir=---"-its, 1974, p'. 747 749.

14. IKlin, Mil-s V., Opt0! c, .h: W'iley ," Sons, New York, 1970,

15. Eainr 0nr i.cs Ca-talc.:i ._:t ? ','he aling Corpratiion,

Cambridgo, ':ss., 1973. -, 1-8 -'6 .

16. Allt!, D. A., ''A Anai;,:,is ol' the l'~j,>tic. rackir.n Transducer,"

:.. ": : -.'on '. ie -s. September, 1962, pp 411 416.

17. Lucovsky, -,, "Photo.ffects in Ncin 'r-.fo-':nly Irradiated p-n Junctions,"

Joiurnal of A.pcpli-d Pi-fics, 31, Ju.e 1960, pp. 0983 -1095.

18. Born, Max, id F:.J. Wolf, iPinc~oeI of Optics, Perga-mon Press,

Oxfurd, 1970.

19. Conripdy, A. E. Appriedi Ortiics an0d O.ic- DLsign, Dover Fublica-

tionI. .;ew York, 195'.

20. Gradsi.c' ., f S., _n-' I. '. Ryzhik, :.:.3,s of Inteirals eris,

andr Producl' s. A Jc.lemijc Press, !ei: Yo:., 1 i, .

2 Pnilips., H. :. :.:.ics o1f '.-r Ocean, Cambricge nilver-

sily Press, London, 1969.

22. Ron: ,r., E, A., .1.' I .an.'l1 ?: .. A-.s ly' is wich Mijal

Co':ut:-r Pro'r.ms, iiHoeai.n--ay, N;ew lor, 196u

23. Prot tt;: an, Clinton E2. andl Mary D. r-rm.ak, "i.-.e Variation of the

Ros h OcePa S..r-face a:-. its effect ,n a* incident Laser ream,"

cEi:E Tr:.nsctic.s onr, cscierin Elirctronirc, GE-7, 4, October, 1969,

pp. 23-2 P,

J' KPe hICAL S,"ST'

Charles S. Pal... ts, or. July ?2k, 1903 in ietroit, Michigan. He

spent the fir i-.t 22 yel.s of his life in The Detroit area and gained

industrial expcricnc- as n an.41rentice electrician for the Ford Motor

Company. He remove to Flo-r!id in 1965j and taLrted his college education.

Mr. Pa]m received ao SSE sid on .ASE rat the University of Ylorida in

August, 197c ,>.:. ialch, 19 973, respctively. ;,orlk toward a Ph.D. was

co.uinletcd in: A

-'c. PFlr is in-aied to th, former -~-rh E.elharn Bapgell and has a

three-year-old non, Karl.

i'r. ':ir: ..larins to cocrt'inut his or: in Lhe research and development

c' f' c inr.c:ietaion.

1 .c; ify 'at I Pavoe '.a Tio at'..... . :, in my opinion it

confor'v.. to ?cr:.-L ae st1.0lu ,'-, of schc' -.- ::.,:i{t:L in and is fully

edCC.'a, 'r crpeo qrea;li-7. .s a ris .-r."ticr for the deg-ee of

IDctor of Fbilocophy.

Rolarn C. A'.,ierson, Ciair:.,in

Professo- o.f .c;.;.:ering Sciences

I c.;rtiy th.t I have road this tudy an'd ti-. in iy op inion it

ccenforms to ac:eptale .tndtr ,ls of scholarly presentation and is fully

adequate,. in and o ." A qu~iity, as a di.ssr)'tion f'or the degree of

Doctor of Philosophy.

C /(' /

O tr } i. :t.- ,id.in

Pro;eossou cf engineeringg Sciences

I certify that 1 h- ve rea.I tins .t'r-y and that in m.y opinion it

c.nrforL-, -.: :c:;.. t-ble Stand:d. of .. :l-.Ely ;- rc~tation and is fully

ad,'Lua-+, i o .coe anl.d qi quality, as -- c.i;rci:.i Jn ."'- the degireB of

iDctir of Philosoph y.

Dav i 1T. 'illli7 s

Profss~tor of i-r rineeritng Sciences

I c. tify thAt I have re-t ths stiud, an. that in my opinion it

corfo ur- t accetabae standtrzds of s.hla-ciy pre('entation and is fully

a.deq.t., '. sBco'e an:l quality, as a tdr.scrtatioe fr'r the degree of

M 2w -' 'bi:,.asophy.

Ul[rich Kr-rE.cu:

Assoc:,t : Frofossor of Engineering

Sciences

J certify t.:jt 1 have o'. i r . that in ,O opinion i-

conforTt o accepableai satnd s of r.::tll. l.y presenlation and-is fully

aul' Ue, ir. a co and quail, f --ticn for the degree of

rP-tu;o of I.lO.c'phy.

Tau] r'o

Prof'.:;er. of E~\ ir-onmental

Engeina-e-i.. 3Sciences

This li ,s:rtatio: Was subr'-.tod to .i.' C'r.te .Faculty of the College

of" r ;ierinz *,rid to the C+.'o i -Lite Counr.ci, ael pias accepted as partial

i'i.ln.iment of the relc.i.rue.-lr for- three d:(2. of Dc.;tor of Philo ophy.

Ai.t"u-a, '17

,, i'"oll!e of Lnprncertng

Dean, C' .' '-iJL School