SPECTRAL MEASUREMENTS OF WATER
PARTICLE VELOCITIES UNDER

WAVES

Michael William Bordy

NAVAL POSTGRADUATE SCHOOL

Monterey, California

THESIS

SPECTRAL MEASUREMENTS

OF

WATER

PARTICLE VELOCITIES UNDER

WAVES

by

Michael William Bordy

The

sis

Adv

risor: e. B.

Thornton

March 1972

Approved fan. pubtlc A.eXeose; dLitiAhmtion untimite.d.

Spectral Measurements
of
Water Particle Velocities Under Waves

by

Michael William Bordy
Lieutenant, United States Navy
B.S., United States Naval Academy, 1965

Submitted in partial fulfillment of the
requirements for the degree of

MASTER OF SCIENCE IN OCEANOGRAPHY

from the

NAVAL POSTGRADUATE SCHOOL
March 197 2

ABSTRACT

Simultaneous measurements of the instantaneous sea
surface elevation and water particle velocities were made
using a wave staff and an electromagnetic current meter.
Measurements were made at three elevations in 18 meters
of water at an open ocean site during moderate wave and
wind conditions . Coherence of the wave height and orbital
velocities was computed to be greater than 0.90 through
the range of significant energy-density between 0.05 and
0.22 Hertz. This range contained greater than 90 percent
of the total spectral energy-density which indicated that
the water particle velocities were almost totally wave-
induced. Measured velocity energy-density spectra were
compared to theoretically computed spectra using linear
wave theory formulation. The measured spectra were 27 to
45 percent greater than theoretical spectra indicating
that measured velocities were 5 to 7 percent greater than
the theoretical value. Phase spectra were computed for
the measured wave heights and orbital velocities. They
compared reasonably with first order wave theory.

TABLE OF CONTENTS

I. INTRODUCTION 9

A. BACKGROUND 9

B. RESEARCH OBJECTIVE ■ 10

II. THEORETICAL CONSIDERATIONS 12

A. FIRST ORDER WAVE THEORY AND SUPERPOSITION 12

B. LIMITATIONS OF FIRST ORDER WAVE THEORY 13

III. INSTRUMENTATION 14

A. WATER CURRENT METER 14

B. WAVE STAFF SYSTEM 16

IV. DATA PROCUREMENT AND PRE-PROCESSING 17

A. DATA PROCUREMENT 17

B. DATA PRE-PROCESSING 21

V. SPECTRAL ANALYSIS 23

A. COMPUTATION OF THE ENERGY-DENSITY SPECTRUM 2 3

B. COMPUTATION OF THE CROSS-SPECTRAL DENSITY 24

C. COMPUTATION OF PHASE ANGLE 25

D. COMPUTATION OF COHERENCY 26

VI. ANALYSIS OF DATA 28

A. HORIZONTAL VELOCITY MEASUREMENTS 28

1. Predicted and Measured Horizontal

Velocity Spectra 28

2. Coherence 29

3. Phase Angle 29

B. VERTICAL VELOCITY MEASUREMENTS 31

1. Predicted and Measured Vertical Velocity

Energy-Density Spectra 31

2. Coherence of Wave Height and

Vertical Velocity 31

3. Phase Angle Between Wave Height and
Vertical Velocity 35

4. Phase Angle Error 36

5. Percent Error of Predicted and Measured
Energy-Density Spectra 37

6. Current Meter Calibration 40

C. ANALYSIS OF HORIZONTAL VELOCITY COMPONENT

AND VERTICAL VELOCITY 40

VII. CONCLUSIONS —
A. COHERENCE

45
45

B. ENERGY-DENSITY SPECTRA 45

C. PHASE ANGLE 46

BIBLIOGRAPHY 47

INITIAL DISTRIBUTION LIST 48

FORM DD 1473 50

LIST OF FIGURES

Figure

1. EMCM-3B water current meter mounted to measure
orthogonal components of the horizontal velocity
field.

2. EMCM-3B water current meter mounted to measure
total vertical velocity and u-component of
horizontal velocity.

3. Strip chart record of u-component horizontal velocity,
w-component of vertical velocity and Baylor wave
height.

4. Measured and theoretical horizontal velocity spectra
Depth 4.12 meters, 21 October 1971.

5. Coherence of wave height and u-component of horizontal
velocity. Depth 4.12 meters, 21 October 1971.

6. Phase angle of wave height and u-component of hori-
zontal velocity. Depth 4.12 meters, 21 October 1971.

7. Measured and theoretical vertical velocity spectra.
Depth 5.34 meters, 21 October 1971.

8. Phase angle between wave height and vertical velocity.
Depth 5.34 meters, 21 October 1971.

9. Coherence of wave height and vertical velocity.
Depth 5.34 meters, 21 October 1971.

10. Measured and theoretical vertical velocity spectra.
Depth 8.35 meters, 21 October 1971.

11. Phase angle between wave height and vertical velocity.
Depth 8.35 meters, 21 October 1971.

12. Coherence of wave height and vertical velocity.
Depth 8.35 meters, 21 October 1971.

13. Measured and theoretical vertical velocity spectra.
Depth 10.00 meters, 21 October 1971.

14. Phase angle between wave height and vertical velocity.
Depth 10.00 meters, 21 October 1971.

15. Coherence of wave height and vertical velocity.
Depth 10.00 meters, 21 October 1971.

Figure

16. Phase angle error between wave height and vertical
velocity. Depth 5.34 meters, 21 October 1971.

17. Phase angle error between wave height and vertical
velocity. Depth 8.35 meters, 21 October 1971.

18. Phase angle error between wave height and vertical
velocity. Depth 10.00 meters, 21 October 1971.

19. Percent error of measured and theoretical vertical
velocity spectra and coherence (dashed line) of
wave height and measured vertical velocity. Depth

5.34 meters, 21 October 1971.

20. Percent error of measured and theoretical vertical
velocity spectra and coherence (dashed line) of
wave height and measured vertical velocity. Depth

8.35 meters, 21 October 1971.

21. Percent error of measured and theoretical vertical
velocity spectra and coherence (dashed line) of
wave height and measured vertical velocity. Depth
10.00 meters, 21 October 1971.

22. Phase angle and coherence 5.34 meters. 21 October 1971.

23. Phase angle and coherence 8.35 meters. 21 October 1971.

24. Phase angle and coherence 10.00 meters. 21 October 1971

LIST OF SYMBOLS

a wave amplitude

B magnetic field intensity

C degrees centigrade

C, ~(f) co-spectrum

E induced electric potential

f frequency

g acceleration of gravity

h total water depth

i square root of -1

k wave number

L wave length

m meter

P(T) Parzen window function

Q,?(f) quadrature spectrum

t time

u horizontal component of orbital velocity

v horizontal component of orbital velocity orthogonal
to u

w vertical component of orbital velocity

W amplitude of vertical orbital velocity

x horizontal coordinate

z vertical coordinate

Y2 coherence function

A incremental difference

e phase angle

a angular wave frequency

x lag time

<J>,-,(t) auto-correLation function

(j).-(t) cross-correlation function

^.(f) auto-spectrum

$12(f) cross-sectrum

I. INTRODUCTION

A. BACKGROUND

Measuring instantaneous v/ater particle velocities in the
presence of wave-induced motion has been difficult due,
primarily, to a lack of instrumentation. This is partic-
ularly true for measurements made in the field. Velocity
meters must have a high response time, be rugged to operate
under wave conditions, and have good directional character-
istics, besides the usual requirements of being reliable
and accurate. Inman and Nasu (1956) made one of the earliest
attempts at field measurements. They used a force meter in
shallow water and compared their results to solitary wave
theory. Miller and Zeigler (1964) used both an acoustic
meter (based on the doppler shift principle) and an electro-
magnetic flow meter to measure water particle motion in the
surf zone. They compared their measurements to high-order
wave theory. Nagata (1963, 1964a, 1964b) used an electro-
magnetic flow meter in a series of experiments to measure
water particle velocities in the surf zone and to determine
wave directional spectra. Bowden and Fairbairn (1956),
Bowden and White (1966) , and Simpson (1968) , used an electro-
magnetic flow meter developed at the National Institute of
Oceanography .

Simpson's experiments were performed at an open coastal
site. He employed spectral methods in the analysis of the
orbital velocities and wave height records. The attenuation

of waves with depth determined from the ratios of surface
elevation and pressure spectra, and the ratios of the
velocity and pressure spectra were found to be consistant
with linear wave theory.

Seitz (1971) measured vector water velocities and sea
surface displacement at a depth of one meter in an esturary
and compared the measured velocity spectra to a derived
velocity spectra obtained from direct measurement of the
surface wave field. The conclusions drawn were that the
direct measurement of the velocity components yielded
energy spectra that could be separated into a component
due to background turbulent energy and a second component
due to surface wave induced velocities. Thus the total
water particle motion at depth can be described as the
sum of the turbulent velocities and the surface wave-induced
motion.

B. RESEARCH OBJECTIVE

The scope of this investigation was divided into three
phases :

1. Simultaneous measurements of the instantaneous
water particle velocity and sea surface elevation
were conducted at an open ocean site in intermediate
water depth employing an electromagnetic current
meter and wave staff system;

2. Spectral analysis was performed on the wave and
velocity records to obtain the statistical prop-
erties of the data;

10

3. First order wave theory was employed to describe
the wave and water particle motion spectra and
to compare the measured results with the
theoretically predicted values.

11

II. THEORETICAL CONSIDERATIONS

A. FIRST ORDER WAVE THEORY AND SUPERPOSITION

The linearity of first order wave theory permits the
utilization of the principle of superposition whereby a
random sea surface may be expressed as a summation of an
infinite number of small amplitude progressive waves.

^totai = nT = nx + n2 + n3 + ••• + nn (2.1)

Using the first order expression for nT

oo

nT = Z-f an sin (kn x - aRt + £n) (2.2)

n=l

where a is the wave component amplitude, k is the wave
number, a is the radial frequency, e is an arbitrary phase
angle, and n is the nth harmonic.

First order (Airy) wave theory can be used to predict
the water particle motion under the wave surface. The
water particle velocities are described as a linear process
and therefore are subject to superposition. For horizontal
motion the expression is

oo

ang kn cosh kR(h+z)

/ x V^ ang kn cosn k in-t-z;

UT(t) - Z-r — u i v, sin(knx-at+E:n) (2.3)

1 *—* a cosh k„h " n

n=l n n

where h is the total water depth and z is the vertical

coordinate measured positively upward from the surface.

For vertical motion the expression is

12

Eang kn sinh kn(h+z)
Gn cosh knh cos(knx-at+cn) (2.4)

B. LIMITATIONS OF FIRST ORDER WAVE THEORY

First order wave theory assumes an inviscid and
incompressible fluid, irrotational flow and that the
wave steepness ratio, a/L (amplitude to wave length), is
very small. While a slightly more accurate representation
of a random sea surface can be obtained using higher order
theory, the non-linearity of these theories does not permit
superposition of solutions to the wave equation. This in
turn precludes analysis by spectral methods.

First order theory can provide a good approximation
to the wave processes for small to moderate conditions.
Dean (1968) compared various wave theories and how well
they fitted the dynamic and kinematic free surface boundary
conditions. He showed that for waves with profiles having
small values of slope in intermediate water depths, first
order theory provides one of the better solutions to the
wave equation. These two conditions were satisfied in this
study which lends validity to the analysis performed.

13

III. INSTRUMENTATION

The water particle velocities were measured with an
Engineering Physics Company water current meter Model
EMCM-3B. The sea surface elevation was measured with a
Baylor Company wave staff system Model 13528.

A. WATER CURRENT METER

The EMCM-3B is an electromagnetic current measuring
device which measures two orthogonal components of water
particle velocity through a range of from zero to five
meters per second. Two d.c. voltage signals proportional
to the magnitude of the orthogonal velocity components
are produced. The minimum electrical response time of
the system is 0.2 seconds corresponding to frequency
measurements of 5 Hertz. The output rms noise level is
a function of the output time constant. A maximum noise
level of 22.4 millivolts (-16db) occurs at a frequency of
5 Hertz. The maximum output error is one percent of full
scale reading. The transducer is made of a fiberglass
material. The dimensions of the transducer are 2-3/4
inches in diameter and 36 inches in length.

The current meter operates by Faraday's principle of
magnetic induction. A toroidal magnet imbedded in the
cylindrical transducer produces a magnetic field in the
water. The intensity of the field decreases according
to the inverse square law. A potential distribution is

14

produced in the water flowing through the field and is
sensed by two pairs of electrodes on the face of the
transducer. The vector equation is

■+■*■-*■

E = u X B (3.1)

-> -►

where E is the induced electric potential, u is the

->■

velocity of the motion, and B is the intensity of the mag-
netic field. It has been established that the magnitude
of the induced potential E is independent of the conduc-
tivity of the metered fluid provided the conductivity is
above a threshold value of 10" 5 mhos per meter of fluid.
Both fresh and salt water have conductivities several
orders of magnitude above this threshold. The direction
of the flow is indicated by a reversal of the polarity of
the induced potential E when the flow reverses.

The transducer senses the motion of the fluid in an
area approximately two to three probe radii from the sensor.
This limits the spatial resolution to approximately six times
the diameter or 42 centimeters. A spatial resolution places
a limitation on the frequency resolution. A wave having a
wave length in deep water of 42 centimeters corresponds to
approximately a 0.5 second period wave. This limits the
frequency resolution to 2 Hertz.

To increase the frequency resolution, it is necessary to
use a smaller diameter transducer. The spatial resolution
is the limitation of the response of electromagnetic flow
meters as it is for other meters based on doppler shift or
back scatter of acoustic or electromagnetic (laser) waves.

15

B. WAVE STAFF SYSTEM

The Baylor Model 13528 is a wave profile recording
system which measures instantaneous sea surface elevation
as a function of time. Measurements are accurate within
one percent of actual height. The system consists of
three components, the wave staff, transducer, and recorder.
The staff is constructed of two wire ropes held parallel
by tension and spaced about nine inches apart. The staff
is installed perpendicular to the water surface and is
stretched with sufficient tension to resist movement due to
wind and wave forces .

The transducer unit measures the electrical resistance
of the length of the staff above a short circuit between
the elements of the staff produced by the motion of the
conducting salt water of the sea's surface. A direct cur-
rent voltage proportional to the height of the sea surface
is taken across this fluctuating resistance.

16

IV. DATA PROCUREMENT AND PRE-PROCESSING

A. DATA PROCUREMENT

Experiments were conducted at the Naval Undersea Research
and Development Center Oceanographic Research Tower at San
Diego, California, 21 October 1971. The tower is approxi-
mately one mile off Mission Beach in a mean water depth of
17 meters. The structure is of steel and concrete and is
built into the sea floor. The sides of the tower are inclined
at an angle of five degrees from the vertical. An enclosed
upper deck houses electronic equipment while an open lower
deck provides a working area to place instruments in the
water.

Winch controlled instrument carriages which travel on
rails down the sides of the tower to about one meter above
the sea floor are located on the north, south, and west sides
of the tower. The carriages may be placed at any depth for
data collection.

The Baylor wave staff is installed on the west side of
the tower adjacent to the carriage rails. The west carriage
was chosen for placement of the current meter transducer due
to the location of the wave staff and to the exposure of that
side to the open sea and predominant swell.

The current meter was installed on the carriage as shown
in Figs. (1) and (2). With the transducer mounted vertically,
two orthogonal components of the horizontal velocity field
are measured. When the transducer is mounted horizontally,

the total vertical field and one component of the horizontal

field is measured.

17

Figure 1. EMCM-3B water current meter mounted to measure

orthogonal components of the horizontal velocity
field. The Baylor wave staff is to the left of
the water current meter transducer.

18

Figure 2. EMCM-3B water current meter mounted to measure
total vertical velocity and u component of
horizontal velocity. The Baylor wave staff
is to the left of the water current meter
transducer .

19

Data chosen for analysis were taken during 4 recording
periods each of 20 minutes duration. The instrument depths
z (m) , height of water column h(m), and velocity components
measured for each recording period are summarized in Table I

TABLE

I

RUN

h(m)
depth

z(m)
instrument
depth

COMPONENT

1

18.15

4.12

u & V

2

18.15

5.34

u & w

3

18.15

8.35

u & w

4

18.00

10.00

U St w

The current meter transducer was oriented to measure the u-
component of horizontal velocity at an azmiuth of 27 0° true
and the v-component of horizontal velocity at an azmiuth of
360° true.

Throughout the recording time a bi-modal surface swell
of about 0.3 meters height was visually observed with one
component having a period of about 7 seconds from 285 true,
and the other component with a period of about 5 seconds
from 260° true. The surface was observed to be non-breaking
with a wind capillary system present. The wind was from 310
true at a speed of about 8 knots. A moderate temperature
gradient was present with a temperature of 17.5 q at the
surface decreasing to 15.0°c at a depth of 18 meters.

20

B. DATA PRE-PROCESSING

The velocity and wave height data were simultaneously
recorded on a Sangamo Model 3500, 14 channel FM tape
recorder. The taped data were transcribed to a rectilinear
eight channel Clevite Brush recorder after amplification by
a Krohn-Hite Model 3322 variable filter. The filter was
operated in the 20 decibel gain, low pass, maximally flat
mode with a cutoff frequency of 99.9 Khz. This resulted in
a gain of ten with no filtering or phase shift.

The two velocity components and the wave height were
simultaneously recorded on adjacent channels of the brush
recorder (Fig. 3) to conserve phase relationships during
the digitizing process.

A Calma Company Model 480 mechanical digitizer was used
to digitize the continuous strip chart information at 100
points to the inch. This corresponded to a sampling
interval of 0.2032 seconds. The digitized records were
interpreted by program CONVERT (Lynch, 1970) using the CDC
6500 computer at the Fleet Numerical Weather Central,
Monterey.

21

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V. SPECTRAL ANALYSIS

Analysis of the digitized records was performed on an
IBM 360 digital computer in accordance with the methods of
Blackman and Tukey (1958) .

A. COMPUTATION OF THE ENERGY-DENSITY SPECTRUM

The energy-density spectrum is computed from the Fourier

transform of the auto-correlation function of the record

time series. The auto-correlation function expressed as a

function of lag time t, is defined as

t+T

lim f

<f> (t)= T+ co - / x1(t)x1(t+x)dt (5.1)

t

where x, (t) represents the time series being analyzed. The
Fourier transform is

lim
11

(f) = TSS i /,^1(T)e-i27TfT di (5.2)

T
" 2

A Parzen lag window was applied to the correlation
functions to account for the finite length of the records
in computing the spectra. The Parzen window has the
desirable property of no negative side lobes. This elim-
inates any numerical instability problems that may arise
in the analysis procedure. The Parzen window function is
defined between the limits of + Tm/2 as

23

P(T) -

Tm

r 1 2 ii

f - 6 k\ i1 -^r] l

(5.3)

where Tm = total lag time. The Fourier transform function
of the modified auto-correlation function is

00

$ll(f) = f P(T) <'11(T)e~l27TfT dx (5.4)

— 00

The energy-density spectrum is the statistical estimate
of the energy-density distribution in' the frequency bands
comprising the spectrum. The area under the curve of the
wave height spectrum is proportional to the total potential
energy-density of the sea surface elevation. The area under
the curve of the velocity spectrum is proportional to the
total kinetic energy-density. In both instances, the area
under the curve is equal to twice the variance of the random
variable.

B. COMPUTATION OF THE CROSS-SPECTRAL DENSITY

The cross-spectral density is defined as the Fourier
transform of the cross-correlation function. The lag window
considerations are identical to those for the energy-density
spectrum described above. The cross-correlation function is
similar to the auto-correlation function except that a
second time series record is lagged against the first instead
of a single record lagged against itself. The cross-correla-
tion function indicates the dependence of values of one

24

record on values of the second record for various values

of lag time. The time averaged cross-correlation function

is defined as

T
1 im -i "2"
<J>12(t) = T+°° ± J x1(t)x2(t+T) dt (5.5)

T
1

and the cross-spectral density is

12

00

(f) = irP(T) c()12(T)e-i2TrfT di (5.6)

where P(t) is the Parzen function defined above. The cross-
spectrum can be defined in terms of its real and imaginary
parts

$12(f) = C12(f) " i Q12(f) (5*7)

where the real part of <1> ~(f) is the co-spectrum

C12(f) = 2

/ [4>12(t) + <fr (-t) ] cos (2-rrfT) dx (5.8)

and the imaginary part of $ (f) is the Quadrature spectrum

00

/pi

Q12(f) = 2 /P12(t) " *i2(_T)] sin(2TTfx) dx (5.9)
o

C. COMPUTATION OF PHASE ANGLE

The phase angle of the cross spectral density is a useful
tool in analysis of bivariate systems. The spectral phase
angle represents the average angular difference by which the

25

cross correlated components of x„(t) lead those of x (t)
in each spectral frequency band. The phase angle of the
cross spectral density is

9(f)
e12(f) = arctanf

-

Lc1:

(5.10)

2(f) J

where Q12(f) an<^ ci?^^ are defined above. e,2(f) spans
the values

-180° < E12(f) 1 180°. (5.11)

D. COMPUTATION OF COHERENCY

The coherency spectrum provides a non-dimensional
measure of the correlation between two time series as a
function of frequency. The coherence function is defined
as

[$ (f)]2

Y2 (f) = L J (5 12)

Y12^ *n(f) *22(f) {b'lZ)

The cross spectral density function $,2(f) satisfies the
inequality

|*12(f) |2 £ *n(f) $22(f) (5.13)

which implies that

0 < y2 (f) < 1 (5.14)

For an idealized constant parameter linear system with
clearly defined input and output, the coherence function
will be equal to unity. Bendat and Piersol (1966) attrib-
ute coherency values greater than zero but less than unity

26

to three possible causes; the system is not linear,
extraneous noise is present in the measurements, or the
output response is due to more than one input function.

27

VI. ANALYSIS OF DATA

The data was sampled at 0.2032 second intervals which
gives a Nyquist frequency of 2.46 Hertz. This was suffi-
ciently high to avoid aliasing problems. The maximum lag
time was chosen as 10 percent of the record length which
were 20 minutes each. This gives a frequency resolution
of 0.0042 Hertz.

A. HORIZONTAL VELOCITY MEASUREMENTS

The predicted and measured horizontal velocity spectra,
the phase angle and the coherence of the wave height and
u-component velocity were computed during run number 1.
The total depth of water was 18.15 meters and the velocity
meter was at a depth of 4.12 meters.

1. Predicted and Measured Horizontal Velocity Spectra
The energy-density spectrum of the predicted hori-
zontal velocity was calculated from the wave-induced hori-
zontal velocity field derived from the expression

2
|u(f)|2 „ [f^OShMhj^)] I (f)|2 (fi-1)

1 ' |_2tt smh kh J ' '

The term in brackets is the first order velocity amplitude
response function squared. The measured total horizontal
velocity energy-density spectrum was formed by computing
the energy-density spectra of the measured u and v
orthogonal velocity components and summing the spectra
across the spectral frequency band.

28

The energy-density spectra, Fig. (4), have
approximately the same shape between 0.0 5 to 0.3 5 Hertz
with the measured spectrum slightly wider than the
predicted spectrum. The measured spectrum is from 30
to 50 percent greater than the predicted spectrum across
the spectral band width of significant energy.

2. Coherence

The theoretical coherence between two ideally
correlated records is 1.0. The coherence of the measured
wave and u-component velocity is greater than 0.90 in the
frequency band of 0.05 to 0.22 Hertz, Fig. (5). This
frequency band contains greater than 90 percent of the
total energy-density in the spectra, and will be referred
to henceforth as the band width of significant energy-
density. The high coherence between the wave and velocity
spectra indicates the instantaneous velocities are primarily
wave induced over the region of significant energy-density,
and the incoherent (turbulent) energy contribution to the
spectrum is small.

3 . Phase Angle

The theoretical phase angle between the instantaneous
wave height and wave induced u-component of velocity is zero
degrees. The computed phase angle ranges from 10 to 70
degrees across the band of high coherence with low values
of phase angle at low frequencies increasing to larger values
at the higher frequencies, (See Fig. (6)).

29

X

W

U
<U
•P
CD

a

Figure 4. Measured and Theoretical Horizontal Velocity
Spectra. Depth 4.12 Meters, 21 October 1971

0)

o

c

CD
U

CD

X3

O
U

Figure 5. Coherence of Wave Height and u-Component of
Horizontal Velocity. Depth 4.12 Meters,
21 October 1971.

w

CD
CD

u
On

CD
XS

CD

C7>

C
<

CD

en
ft

Figure 6. Phase Angle of Wave Height and u-Component of
Horizontal Velocity. Depth 4.12 Meters,

21 October 1971.

30

B. VERTICAL VELOCITY MEASUREMENTS

The predicted and measured vertical velocity energy-
density spectra, and the phase angle and coherence spectra
of the wave height and vertical velocity were computed for
depths of 5.34, 8.35, and 10.00 meters.

1. Predicted and Measured Vertical Velocity Energy-
Density Spectra

The predicted vertical velocity energy-density

spectrum was computed from the wave spectrum using the

expression

i ie\ 1 2 f sinh k(h+z) . ,n,2 ,, ~x
lw(fH2=[_2¥ sinh kh j ln(f)|2 (6.2)

The term in brackets is the first order vertical velocity
amplitude response function squared.

The predicted and measured energy-density spectra
are shown in Figs. (7), (10), and (13). The spectral band
of significant energy-density is essentially the same for
the three depths measured, 0.05 to 0.35 Hertz. The magnitude
of the measured energy spectrum is 3 0 to 50 percent greater
than the predicted energy spectrum at each depth. This means
that the actual velocities were 5 to 7 percent greater than
theoretically predicted values.

2 . Coherence of Wave Height and Vertical Velocity

The coherence for each depth are shown in Figs. (9),
(12), and (15). The coherence is greater than 0.9 at each
of the three depths over the range of significant energy-
density. This indicates a high correlation between the
wave and wave-induced vertical velocity spectra which means

31

X

CM

u
d)
+J
a)
S

Figure 7. Measured and Theoretical Vertical Velocity

Spectra. Depth 5.34 Meters, 21 October 1971

en

CD
Q)

^1
tn

Figure 8. Phase Angle Between Wave Height and Vertical
Velocity. Depth 5.34 Meters, 21 October 1971

Figure 9. Coherence of Wave Height and Vertical Velocity
Depth 5.34 Meters, 21 October 1971.

32

IT)

I

O

X

u

0)

-p

Q)

2

measured

a:.-;

SEP

Figure 10.

n

Cn
0)

o

rH

S

w

rd

si

Measured and Theoretical Vertical Velocity
Spectra. Depth 8.35 Meters, 21 October 1971

j

s___

r^wA—

1

■

■

■t ft 1

/

Jul

hz

1* t^J 1 J «J^J

M

Figure 11. Phase Angle Between Wave Height and Vertical
Velocity. Depth 8.35 Meters, 21 October 1971

d)
o
c

0)

u

CD

o
u

Figure 12. Coherence of Wave Height and Vertical Velocity
Depth 8.35 Meters, 21 October 1971.

33

O

X

w
u

+J

s

measured

Figure 13. Measured and Theoretical Vertical Velocity

Spectra. Depth 10.00 Meters, 21 October 1971

u

Q)
T3

r-\

c

<

0)

Figure 14.

Phase Angle Between Wave Height and Vertical
Velocity. Depth 10.00 Meters, 21 October 1971

0)

o

c

0)

u

o
u

Figure 15. Coherence of Wave Height and Vertical Velocity
Depth 10.00 Meters, 21 October 1971.

34

that most of the instantaneous measured velocity is due
to the waves and turbulent energy contributions to the
spectra is small.

3. Phase Angle Between Wave Height and Vertical Velocity
The theoretical phase angle between wave height and
vertical velocity is 90 degrees. The measured phase spectrum
at a depth of 5.34 meters, shown in Fig. (8), corresponds
well to this theoretical value. The phase spectrum at 8.35
meters, shown in Fig. (11), is uniformly greater than 100
degrees and decreases across the band of significant energy-
density. The phase spectrum at a depth of 10.00 meters,
Fig. (14), shifted from approximately plus 90 degrees to
minus 90 degrees across the band of significant energy-
density which was not expected. The coherence through this
range is uniformly high. This phase shift is partly due to
a horizontal displacement of the current meter transducer
from a vertical line to the wave staff. This is discussed
below, but the shift cannot be fully explained. The 10.00
meter run in which this occurred was measured sequentially
between the 5.34 and 8.35 meter runs which did not show a
similar phase shift. All three runs were conducted in a
two hour period; this tends to discount the possibility of
equipment malfunction. This suggests a phase shifting
departing from the theoretical increasing with depth, but
more experiments need to be conducted before any definite
conclusions can be made.

35

4. Phase Angle Error

The rails on which the instrument carriage was
mounted are inclined at an angle of 5 degrees from the
vertical. This inclination caused the current meter
transducer to be displaced horizontally from a line
vertical to the Baylor wave staff at the depths of measure-
ment. The magnitude of this displacement was 0.467, 0.732,
and 1.068 meters respectively at the measured depths of
5.34, 8.35, and 10.00 meters. The displacements resulted
in a phase error due to the time lag between sensing of
the waves by the current meter and the wave staff.

The maximum percentage error was computed by
employing the first order expressions for n and the w-
component

n = a sin(kx-at) = a cos [90° - (kx-at)] (6.3)

w = W cos (kx-at) = W cos [180° - (kx-at)] (6.4)

where W is the amplitude of the vertical velocity. The
bracketed terms in each expression are the phase angles,

e = 90° - (kx-at) (6.5)

and

ew = 180° - (kx-at) . (6.6)

Using the wave staff as a reference, the x term in e is set
equal to zero, so that

e = 90° - at (6.7)

n

and the phase difference Ae is

36

Ae = £w-en = Ae = 90°- kx (6.8)

where x is the horizontal distance between the wave staff
and the current meter transducer. If x is equal to zero,
the theoretical phase angle of 90° is obtained.
The percentage error is

/360\
\2* I

Percent Error = j« |

The percent error of the maximum computed phase
spectra for each depth has been plotted in Figs. (16), (17),
and (18) . The amplitudes of the actual phase spectra are
smaller than those of the computed phase spectra by the
indicated percentage at each frequency. The error is small
at low frequencies and increases at the higher frequencies
where the ratio of the horizontal displacement of the cur-
rent meter to the length of the waves is high.

5. Percent Error of Predicted and Measured Energy-Density
Spectra

The percentage error of the energy-density spectra
for each depth are shown in Figs. (19), (20), and (21).
The error rapidly decreases to low values (25 to 50 percent)
at low frequencies and remains uniform across the range in
which the coherence is high. The average error over the
band of significant energy-density was highest (45 percent)
for the measurement at 5.34 meters. The average error was
approximately the same (30 percent), over the band of signif-
icant energy-density for the measurements at 8.35 and 10.00
meters. The actual error between the predicted and measured

37

M
O

u
u
w

Figure 16.

Phase Angle Error Between Wave Height and
Vertical Velocity. Depth 5.34 Meters,
21 October 1971.

u
o
u
u
pq

0.1

0.2
hz

0.3

0.4

Figure 17. Phase Angle Error Between Wave Height
Vertical Velocity. Depth 8.35 Meters,
21 October 1971.

and

u
o
u
u

w

OP

0.1

0.2

hz

0.3

0.4

Figure 18. Phase Angle Error Between Wave Height and
Vertical Velocity. Depth 10.00 Meters,
21 October 1971.

38

CM

© -

x

o
u

0) i M

w

fd
-P
c
<u
u
n

0)

1 f\

- \

*/•

Figure 19.

302
hz

d:j-

Percent Error of Measured and Theoretical
Vertical Velocity Spectra and Coherence
(dashed line) of Wave Height and Measured
Vertical Velocity. Depth 5.34 Meters,
21 October 1971.

0)

° IT

o

u

c

>-l

(U

0

u

M

0)

M

J3

w

0

/

u

dP

I 1
r?

__ LI

o::

1 "n

- -v-
hz

_£X.

nty

Figure 20.

Percentage Error of Measured and Theoretical
Vertical Velocity Spectra and Coherence
(dashed line) of Wave Height and Measured
Vertical Velocity. Depth 8.35 Meters,
21 October 1971.

Q)

o

u

a

0

0)

u

u

u

0)

w

si

o

dp

u

Figure 21.

Percentage Error of Measured and Theoretical
Vertical Velocity Spectra and Coherence
(dashed line) of Wave Height and Measured
Vertical Velocity. Depth 10.00 Meters,
21 October 1971.

39

velocity amplitudes is the square root of these values
and amounts to approximately 5 to 7 percent.
6. Current Meter Calibration

The current meter was calibrated under steady flow
conditions in a circular tow tank. The boundary layer under
steady flow on the face of the transducer is greater than
that for unsteady flow. The instrument essentially inte-
grates the intensity of the potential which decreases in
intensity as the reciprocal of the square of the distance
from the transducer. The steady flow calibration will
indicate a faster flow than for unsteady flow. This indi-
cates that the measured flow will be over-predicted. The
orbital velocities measured in this investigation were of
an unsteady turbulent nature indicating that a calibration
error exists. The magnitude of this error is presently
unknown. A method of calibrating the current meter under
turbulent conditions is currently being developed at the
Naval Postgraduate School. Re-calibration under dynamic
conditions is expected to reduce the errors computed in
this investigation.

C. ANALYSIS OF HORIZONTAL VELOCITY COMPONENT AND VERTICAL
VELOCITY

The coherence and phase angle of the u-component of ■

horizontal velocity and the vertical velocity were computed

for measurements at depths of 5.34, 8.35, and 10.00 meters

and are shown in Figs. (22, (23), and (24).

40

According to first order wave theory, the horizontal
and vertical orbital velocities have a phase angle dif-
ference of 90 degrees with the w-component leading the
u-component . The theoretical coherence is equal to unity.
The computed values for the measurements at 5.34 and 8.3 5
meters agree quite well with theory. The coherence is 0.9
through the region of significant energy-density. The
coherence of the velocities at 10.00 meters is also 0.9
but the phase angle again exhibits peculiar behavior as
it did between the wave height and vertical velocity
discussed above. More data are required to explain this
phenomenon.

41

CO
CD
CD
U

Cr>

CD
T3

CD
rH

CT>

C
<

CD
W

id

CD

u
c

CD

u

CD
X!
O
U

Phase Angle Between Vertical and Horizontal
Velocity Component. Depth 5.34 Meters,
21 October 1971.

Coherence of Vertical and Horizontal Velocity
Component. Depth 5.34 Meters, 21 October 1971

Figure 22. Phase Angle and Coherence 5.34 Meters
21 October 1971.

42

w

CD
CD

u
tr<
<D

Q)
.H

<

<D
w
(d
A

04

. /

/y

H +l-

•

■

■

•

ajy

hz

ds;

V

u

c

Q)

u

<D
O

u

Phase Angle Between Vertical and Horizontal
Velocity Component. Depth 8.35 Meters,
21 October 1971.

Coherence of Vertical and Horizontal Velocity-
Component. Depth 8.35 Meters, 21 October 1971

Figure 23.

Phase Angle and Coherence 8.35 Meters
21 October 1971.

43

0)
U

Q)
T3

w

0)

o

c:

(D
M
0)

o
u

Phase Angle Between Vertical and Horizontal
Velocity Component. Depth 10.00 Meters,
21 October 1971.

Coherence of Vertical and Horizontal Velocity
Component, Depth 10.00 Meters, 21 October 1971

Figure 24.

Phase Angle and Coherence 10.00 Meters
21 October 1971.

44

VII. CONCLUSIONS

The measurements were made under near ideal conditions
of moderate waves of low steepness and light wind conditions
A bi-modal swell was present during experiments with little
locally generated wind waves . An electromagnetic flow meter
having a response time corresponding to 2 Hz was used to
measure water particle motion.

A. COHERENCE

Coherence of the wave height and orbital velocities
was computed to be greater than 0.90 through the range of
significant energy-density between 0.05 and 0.22 Hertz.
This range contained greater than 90 percent of the total
spectral energy-density which indicated that the water
particle velocities were almost totally wave-induced. The
high values of coherence obtained in all runs over the
band of significant energy-density imply a near linear
process between the wave height and induced orbital veloc-
ities. Under the moderate conditions encountered, non-
linearities were small. Turbulent energy contribution to
the energy-density spectra from both the sea surface and
bottom boundary layer effects were quite small.

B. ENERGY-DENSITY SPECTRA

The energy-density spectra for the measured velocity
at the four depths had approximately the same frequency
bandwidth of significant energy-density. The shapes of

45

the measured and predicted energy-density spectra were
approximately the same at each depth. The magnitude of
the energy-density spectra decreased with depth according
to theory .

C. PHASE ANGLE

The computed phase spectra of the wave height and wave-
induced orbital velocities were consistent with theory in
all cases except in the deepest run at 10.00 meters. The
phase shift encountered in that instance was unaccounted
for in the phase angle error analysis.

46

BIBLIOGRAPHY

Bendat, K. R. and A. G. Pierson, Measurement and Analysis
of Random Data, pp. 234-237, Wiley, 1966.

Blackman, R. B. and J. W. Tukey (1959) , The Measurement
of Power Spectral, Dover Publication, Inc., New York,
New York.

Bowden, K. F. and L. A. Fairbairn (1956) , Measurements of
Turbulent Fluctuation and Reynolds Stress in a Tidal
Current^ Royal Society of London Proceedings, Ser .
A237 (1956), pp. 422-438.

Bowden, K. F. and R. A. White (1966) , Measurements of the

Orbital Velocities of Sea Waves and Their use in Deter-
mining tne Directional Spectrum^ Geophysical Journal
of the Royal Astronautical Society, Vol. 12, 1966,
pp. 33-54.

Inman, D. L. and N. Nasu (1956) , Orbital Velocity Associated
With Wave Action Near the Breaker Zone, U. S. Army Corp
of Engineers, Beach Erosion Board, Technical Memorandum
No. 79, March 19 56.

Lynch, J. L. (1970), Long Wave Study of Monterey Bay, Naval
Postgraduate School, Monterey, California, M.S. Thesis.

Miller, R. L. and J. M. Zeigler (1964), The Internal Velocity
Field in Breaking Waves, 9th Conference on Coastal
Engineering National Council for Wave Research, Lisbon,
Portugal .

Nagat a , Y . (196 3), Observation of Wave Motion by Electro-
magnetic Current Meter, Proceedings Meeting on Coastal
Engineering , Japan Society of Civil Engineering, 1963.

Nagata, Y. (1964a, 1964b) , The Statistical Properties of
Orbital Wave Motion and Their Application for the
Measurement of Directional Wave Spectra, Journal of
the Oceanographical Society of Japan, Vo 1 . 19, pp .
169-181.

Seitz, R. C. (1971), Measurement of a Three-Dimensional
Field of Water Velocities at a Depth of One Meter in
an Estuary, Journal of Marine Research 197 1 , Vo 1 . 29,
pp. 140-150.

Simpson, J. H. (1968) , Observation of the Directional

Characteristics of Waves, Geophysical Journal of the
Royal Astronautical "Society, Vol. 17, 1969, pp. 93-120.

47

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Documentation Center 2
Cameron Station

Alexandria, Virginia 22314

2. Library, Code 0212 2
Naval Postgraduate School

Monterey, California 93940

3. Oceanographer of the Navy 1
The Madison Building

732 N. Washington Street
Alexandria, Virginia 22217

4. Department of Oceanography Code 58 3
Naval Postgraduate School

Monterey, California 93940

5. Commander, Navy Ship Systems Command 1
Code 901

Department of the Navy
Washington, D. C. 20305

6. Dr. Ned A. Ostenso 1
Deputy Director (acting)

Code 480 D

Ocean Science and Technology Division

Office of Naval Research

Arlington, Virginia 22217

7. Dr. Albert D. Kirwan, Jr. 1
Program Director/Physical Oceanography

Code 4 81

Ocean Science and Technology Division

Office of Naval Research

Arlington, Virginia 22217

8. LCDR Jon W. Carlmark (USN) 1
Project Office

Code 485

Ocean Science and Technology Division

Office of Naval Research

Arlington, Virginia 22217

48

9. Professor H. Medwin, Code 61Md
Department of Physics
Naval Postgraduate School
Monterey, California 93940

10. Asst. Professor Noel E. Boston, Code 58Bb
Department of Oceanography

Naval Postgraduate School
Monterey, California 93940

11. Asst. Professor Edward B. Thornton,

Code 58Tr (Thesis Advisor)
Department of Oceanography
Naval Postgraduate School
Monterey, California 93940

12. Lieutenant Michael W. Bordy (USN)
USS Implicit (MSO 455)

FPO San Francisco, California 96601

49

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ORIGINATING ACTIVITY (Corporate author)

Naval Postgraduate School
Monterey, California 93940

2«. REPORT SECURITY CLASSIFICATION

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26. GROUP

REPORT TITLE

Spectral Measurements of Water Particle Velocities Under Waves

DESCRIPTIVE NOTES (Type ol report and, inclusive dates)

Master's Thesis, March 1972

au THORtSi (First name, middle initial, last name)

Michael W. Bordy

REPORT DATE

March 19 7 2

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Approved for public release; distribution unlimited

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12. SPONSORING Ml LI TAR Y ACTIVITY

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Monterey, California 93940

J. ABSTRAC T

Simultaneous measurements of the instantaneous sea surface
elevation and water particle velocities were made using a wave
staff and an electromagnetic current meter. Measurements were
made at three elevations in 18 meters of water at an open ocean
site during moderate wave and wind conditions. Coherence of the
wave height and orbital velocities was computed to be greater
than 0.90 through the range of significant energy-density between
0.05 and 0.22 Hertz. This range contained greater than 90 percent
of the total spectral energy-density which indicated that the
water particle velocities were almost totally wave-induced.
Measured velocity energy-density spectra were compared to
theoretically computed spectra using linear wave theory formulation
The measured spectra were 27 to 45 percent greater than theoretical
spectra indicating that measured velocities were 5 to 7 percent:
greater than the theoretical values. Phase spectra were computed
for the measured wave heights and orbital velocities. They
compared reasonably with first order wave theory.

)D

FORM
i nov es

/N 0101 -807-681 1

1473

(PAGE 1)

50

Unclassified

Security Classification

i-31408

^classified

"" - Security Classif nation

KEY WO RDJ

LINK B

v-ve orbital velocities

Bergy-density spectra

jherence

aase angle

jectromagnetic current meter

| FORM

NOV SS
1 1-807-68?!

1473

BACK)

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Security Classification

2» AUG74
4 APR 7$

2202U

6 5

2 7 003

Thesis
B7145

c.l

Bordy

33857

waves
21 AUG74

4 APR 78

Spectral measure-
ments of water part-
icle velocities under
2 2 0 2 U
2 ^ Sy 6 5
2 7 003

Thesis

B714^
c.l

Bordy

er

133857

Spectral measure-
ments of w=»ter part-
icle velocities under
waves.

thesB7145

Spectral measurement of water particle v

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