-V

WAVE- TANK STUDY OF INTERNAL WAVES

by

Walter A. Kordek
Lieutenant, United States Navy

Submitted in partial fulfillment of
the requirements for the degree of

MASTER OF SCIENCE

United States Naval Postgraduate School
Monterey, California

19 6 5

Libri

U. S. Nav.il Postgraduate icuooi

Monterey, California

WAVE -TANK STUDY OF INTERNAL WAVES
by
Walter A. Kordek

This work is accepted as fulfilling
the thesis requirements for the degree of
MASTER OF SCIENCE
from the
United States Naval Postgraduate School

ABSTRACT

Internal waves, generated in an approximate two-layer system of mis-
cible fluids by flow over a submarine ridge, were examined. The investi-
gation was conducted in a wave tank 3.0 m long and 0.3 m wide in which
total water depth varied between 7.3 cm and 15.2 cm.

To determine visually the interface deformations, the lower saline
layer was colored with Rhodamine "B" dye. Neutrally buoyant particles,
consisting of a solution of carbon tetrachloride and benzene, were intro-
duced in the vicinity of the pycnocline to indicate particle motion.

Subsequent particle motion and interface deformations were photo-
graphed with a 16 mm motion picture camera. Analysis of the film yielded
particle trajectories and internal wave characteristics such as wave-
length, period, amplitude and phase velocity. The observed motions were
compared with those predicted by internal wave theory. Observed phase
velocities differed from the theoretical phase velocities in all cases,
with a maximum difference in one case of 44 percent of theoretical value.
The discrepancies may be due either to the superposition of the observed
progressive wave on a standing wave or to the observation of non-conserved
wave crests in a dispersive medium. However, observed phase velocities
did decrease in magnitude with a decreasing layer density difference as
predicted by theory.

ii

LIST OF ILLUSTRATIONS

Figure Page

1. Full View of Tank 4

2. Wave- generator Plate 5

3. Wave- generator Power Assembly 7

4. Wave -generator Drive Assembly 8

5. Wave Absorber 9

6. Submarine Ridge Models 10

7. Water Sampling Syringe 13

8. Typical Distribution of Salinity and Density 20

9. Particle Trajectory 24

10. Vertical Particle Displacement vs. Time 25

11. Horizontal Particle Displacement vs. Time 26

Table

1. Data Summary 22

2. Data Summary 23

iv

LIST OF SYMBOLS

h depth of upper layer

h depth of lower layer

T0 period of wave- generator plate

T^g period of standing internal wave

T^p period of progressive internal wave

Lj_ wave length of progressive wave

LQ stroke of wave -genera tor plate

H^ average vertical oscillation of interface (trough to crest)

0 average density of upper layer

P average density of lower layer

tw water temperature

A \ difference of average densities of two layers

hfi height of barrier

L^ length of basin between wave generator and submarine ridge

L2 length of basin between submarine ridge and wave absorber
end-plate

C£Q observed phase velocity of internal wave

C^t theoretical phase velocity of internal wave

C phase velocity

L wavelength

T period

1. INTRODUCTION.

1.1 Background.

Fundamental to a systematic investigation of the oceans are mathe-
matical statements of the physical laws involved coupled with an orderly
collection of data with which to test these laws. Due to the complexity
of the ocean and its vast areas, the collection of sufficient and accurate
data for this purpose is no easy task. The requirement for synoptic ocean
station observations has generally not been met.

A topic receiving considerable attention in recent years is that of
internal waves. Theories have been advanced concerning the environmental
factors required for the existence, generation and propagation of internal
waves. Analysis of available data has supported many of the theories;
however, there remains a constant need for more usable data. With the
advent of more sophisticated instrumentation and increased international
interest in oceanographic research, as exemplified by the increasing num-
ber of trained personnel and research vessels, it is expected that the
data requirements will be met in years to come. In the interim the tech-
nique of modeling can be used to great advantage to test theories of
internal wave motion.

1.2 Advantages of Model Studies.

The major advantage of modeling oceanic processes is the experimental
control that can be exercised. Modeling also provides the advantage of
economy, both in cost and time.

'■

■f

■

2. PURPOSE

The objectives of this investigation are:

(1) The design and evaluation of a model tank and wave generator
suitable for studying internal waves.

(2) The development of experimental procedures for producing inter-
nal waves by causing flow over a submarine ridge model.

(3) To conduct preliminary experiments, collect data, and compare
the results with theory.

3. DESCRIPTION OF LABORATORY FACILITIES AND TECHNIQUES.

3.1 Tank.

The facilities of the Oceanography Laboratory at the U. S. Naval
Postgraduate School were utilized for this investigation. It should be
noted that the wave tank described in this thesis and used in the experi-
ments was originally designed to function as a visual training aid for
demonstrating surface wave phenomena.

The tank is rectangular in shape and constructed of aluminum bottom
and end plates and plexiglass sides. (The full tank is shown in fig. 1.)
It is 3.0 m long, 0.3 m wide and 0.45 m deep with a maximum allowable
water depth of 0.3 m due to structural considerations. The tank is sup-
ported by a welded aluminum I-beam structure. The full plexiglass sides
made observations possible throughout the length of the tank. It is
fitted with a bottom drain and valve near one end plate.

3 . 2 Wave Generator

The wave generator consists of a vertical aluminum plate which
oscillates along the longitudinal axis of the tank (fig. 2). There was
a 0.3 cm clearance between the face-plate of the generator and the sides
and bottom of the tank to prevent metal- to-metal and metal-to-plexiglass
abrasion and friction. A gasket to function in the same manner as a pis-
ton ring was required to reduce flow by the generator. Foam rubber and
rubber "squee-gee" were tried as gasket material but rejected because of
excessive friction. Sheet Teflon, 0.15 cm thick, reduced friction but
permitted some "flow-by" causing mixing and eddies in the vicinity of
the generator. This material was adopted for use.

The wave -generator power system consists of a half-horsepower AC
motor and a hydraulic transmission having speed range of 0-1800 rpm input

-

:

WAVE-GENERATOR PLATE

FIGURE 2

or output (fig. 3). The rotary output of the transmission was trans-
formed to a linear motion to drive the wave generator plate through a
crank arrangement shown in fig. 4. The slot at the end of the generator
driveshaft permitted the shaft to remain horizontal, thereby eliminating
the need for a conventional wrist-pin in the shaft, since the motion of
the shaft must be horizontal at the generator plate. The wheel was
drilled and tapped at four points, 90 degrees apart, permitting a gener-
ator stroke of 3.8, 5.08, 6.35, or 7.62 cm by changing the pin position.
The generator shaft enters the tank through a flanged sleeve screwed to
the tank endplate. A spreader bar attached to the tank support and the
table on which the power assembly was mounted was necessary to prevent
"walking" of the table due to motor- induced vibration.

3.3 Wave Absorber.

The wave-absorbing material lies between two aluminum plates which
are welded together at right angles and are as wide as the tank (fig. 5).
The back plate is 43 cm high and the bottom plate is 61 cm long. The
cavity between the plates and the sides of the tank was filled with alu-
minum shavings and a fine mesh plastic screen drawn tightly over the
ends of the plates, forming an angle of 35 degrees with the bottom and
completing the enclosure. With a water depth of 15 cm the average absorp-
tion distance is 29 cm.

3.4 Submarine Ridge Models.

Two basic types of model were utilized (fig. 6). The first was a
0.3 cm thick aluminum plate cut to the width of the tank. This is shown
at the top of fig. 6. The second type consisted of a metal sheet formed
over templates, as shown in the lower part of fig. 6. The dimensions
were such that the top of the ridge was a sector of a circle with a radius

<5

LlI

en

Z>

Id

CD

ft:
O

oO
OQ

<

LU

lO
LlI

or
O

00

_l

LU
Q
O

LjJ

o

Q
CH

ill

z

or
<

D
00

UJ

q:

0

10

equal to the height of the ridge and the base equal to four times the
height resulting in a slope of 1:2. The sides of the model ridge were
sealed with plastic tape. The single-plate model fit snugly against the
sides and bottom and needed no sealer.
3.5 Formation of a Two-Layer System.

The fluids used in the two layers were fresh water and a brine solu-
tion, which was formed by dissolving rock salt in a vat. These fluids
were selected because their miscibility permitted exchange of salt between
layers, thus simulating mixing by internal waves as it might occur in the
sea. Additionally, the density of the brine was easily controlled and
the rock salt required is inexpensive and readily available. Finally,
the use of brine added to the simplicity of the problem in that the total
salt in the system was conservative. This is a distinct advantage over
the use of a temperature difference between layers as described by Crom-
well j_l_], where heat exchange with the environment would give a non-
conservative system.

The two- layer system was formed by filling the tank initially with
fresh water to depth equal to the desired depth of the upper layer. The
siphoning was accomplished by two methods. In one method the brine
entered the tank at the bottom drain through a 0.95 cm plastic hose. A
small metal plate was placed over the drain to direct the flow horizon-
tally. The valve permitted a controlled rate of flow. By this method a
bottom layer with a depth of approximately 8 cm could be introduced in
one hour, maintaining a sharp pycnocline.

The second method consisted of placing the outlet of a 0.63 cm rub-
ber tube against the bottom of the tank. The rate of flow was controlled
by a "C"-clamp on the hose. The advantage of this method was that it

il

required no permanent tank installation. The disadvantage is that the
rate of flow must be maintained at only about 50 percent of the former
method without excessive mixing at the pycnocline.

The brine solution was colored with Rhodamine "B" dye prior to
siphoning into the tank. Thirty drops of a 40 percent dye, 60 percent
acetic acid solution per 25 gallons of brine yielded the best results.
This quantity permitted discrimination of the internal wave form on black
and white film while providing sufficient trans lucence to observe par-
ticles of contrasting fluid later introduced.
3.6 Water Sampling and Density Determination.

Water samples were withdrawn with a 30 cc syringe using a 10 cm long
needle and a 0.24 cm diameter brass rod 50 cm long (fig. 7). The inlet
depth for the needle was determined by taping the rod to the syringe and
sliding the syringe along the rod until the desired distance between the
bottom of rod and the needle inlet was obtained. Thus when the tip of
the rod was held firmly against the bottom of the tank, the sample depth
was known.

Two methods of density determination were evaluated. The first
method utilized a Bisset-Berman Corporation, Model 621, portable sali-
nometer. Use of this method proved to be a lengthy process with a result-
ant accuracy in excess of that required for the experiments. The second
method utilizing Kahlsico salinity hydrometers was adopted for use in the
investigation. The hydrometer method offered the advantages of simplic-
ity of operation while providing sufficient accuracy for the purpose of
the investigation. The technique consisted simply of placing the water
sample in a 3.8 cm diameter glass cylinder, inserting the hydrometer,
which is graduated in units of salinity, and recording the salinity. A

12

LlI

O
z

cr

O
z

-J

Q_

<

00

u

I

L±J

o

LL

13

water sample volume of approximately 90 cc was required to float the
hydrometer bulb. Corrections were applied for temperature variation and
the corrected salinity was then converted to density by use of tables
provided with the hydrometers.

3.7 Formation and Introduction of Neutrally Buoyant Particles.

Carbon tetrachloride and benzene were combined in solution such that
the resulting density was approximately 1.025 grams per cubic centimeter.
A drop of this solution was inserted into the upper layer of the two-
layer system with an eye dropper and its motion observed. The particle
either settled to the bottom, rose to the surface, or imbedded itself in
the sharp pycnocline, the desired position. In the event either of the
former motions occurred, benzene or carbon tetrachloride, as appropriate,
was added to alter the solution density; and another trial particle was
injected into the layered system. Usually, one admixture was required
to obtain the proper density. Because of the initial near -homogeneity
of the two layers, attempts to suspend a particle near the mid-depth of
each layer were unsuccessful.

Methyl violet dye was mixed into some of the particle solution and
proved to be an aid in tracking an individual particle, particularly in
the lower dyed, translucent layer. The undyed-particle solution was
satisfactory for particles just above the pycnocline.

3.8 Photographic Data Extraction and Analysis.

Wave and particle motion was photographed with a 16 mm motion pic-
ture camera at the rate of eight exposures per second. Kodak black-and-
white, plus-X reversal film was used and proved satisfactory. All motion
was photographed through an acetate grid, on which lines were spaced at
2 cm, taped to the front of the tank. A stopwatch was placed on the grid

14

field to provide a time reference. All photography was accomplished with
the camera approximately one meter from the side of the tank, with top
lighting to provide sufficient differential refraction through the par-
ticles for identification. Best results were obtained with a white back-
ground approximately 25 cm on the other side of the tank.

Analysis was accomplished by projecting the film with a motion pic-
ture projector onto graph paper. The graph paper was aligned parallel
to the projected grid,, and the 2 cm grid reference marks were drawn on
the graph paper. Limitations of the projection equipment did not allow
frame -by- frame viewing. The time interval between frames examined was
approximately one-half second. The particle position was marked on the
graph paper with an appropriate time mark for successive positions. In
this manner particle trajectories were obtained.

Wave lengths were obtained in a similar manner by tracing the color
discontinuity which occurred at the interface on the graph paper and
measuring the distance between appropriate wave phases. The stopwatch
readily provided the time interval required for computing phase veloci-
ties.

15

4. INTERNAL WAVE THEORY.
4.1 A Two- Layer Model.

Internal waves may exist in any stratisfied fluid (Defant [2]). In
a two-layer system, maximum particle motion occurs at the internal bound-
ary surface. Defant further shows that the ratio of the surface wave
amplitude to internal wave amplitude is Ar/ P > where A.\ is the den-
sity difference between the two layers and \. is the density of the
bottom layer. The phase velocities for the surface wave and the internal
wave of a two -layer system are given by

C,as ±- tanh k(h+U') <D

k

Cf= ±- till . (2)

k P coih kh -»- ?' coth kh'

4.2 Generation.

Rattray \_3J has developed a theory which shows that internal waves
may be generated by the surface tidal current or wave impinging on a con-
tinental shelf. Weigand's f4 J model -tank study at the University of
Washington has supported the theory. As early as 1912, Zeilon r5_j de-
veloped a theory and obtained qualitative model- tank results wherein an
internal wave could be generated by a current passing over a submarine
ridge. Proudman [6 J provides a mathematical treatment of internal wave
generation by a current flowing over a ridge.

16

5. GENERAL EXPERIMENTAL PROCEDURES.

Several hours prior to conducting an experiment, water was drawn
into the tank and the vat in which the brine was to be mixed. About one
hour was required to dissolve sufficient rock salt to yield salinities
of 20 to 35 parts per thousand and to allow equilibrium temperature to
be reached. After the brine was formed, the Rhodamine "B" dye was added
and thoroughly mixed. The brine was siphoned after the fresh water in
the tank had drained to the approximate depth desired for the upper layer.
A check on the rate of mixing at the interface was made by observing the
diminution of the vertical extent of the clear layer, and the rate of
siphoning could be adjusted to keep mixing at a minimum. After termina-
tion of the siphoning, the water was sampled to determine the vertical
density distribution. Sample depths were selected about 1 cm above and
below the apparent pycnocline and at intervals of 2 to 3 cm thereafter.
The dyed particles were then introduced. They were placed along the
centerline of the tank between the absorber and ridge at intervals of
approximately 2 to 5 cm. This particle interval was sufficiently large
to permit particle discrimination, yet small enough to show the inter-
action between particles.

When the lighting and camera were in position, the wave generator
was activated. During the first two cycles the period of the generator
was precisely adjusted to that desired for the experiment. This was nec-
essary since the period of rotation of the generator varied up to three
seconds during initial strokes. The generator was then timed for 10
cycles and from this figure the average period of the stroke determined.

17

6. RESULTS.

6.1 Capabilities and Limitations of the Installation.

Progressive and standing internal waves were generated in an approxi-
mate two- layer system by generating an oscillating current over a sub-
marine barrier. Introduced as controlled variables were the generator
period, height and form of the barrier, and the depth and density of the
two layers. The wave absorber was removed for Run 4A to determine the
standing wave characteristics of the basin.

The generator was operated at periods ranging from 5 seconds to 23
seconds. The period of the drive system tended to decrease during the
running of an experiment. The change in period was more pronounced at
high periods, where the maximum observed change in period was 2 seconds
from the beginning to the end of an experiment of 12 minutes' duration.
Near the low period range, the deviation was approximately one-half
second.

The "flow-by" around the generator plate, which was previously
described, continued throughout the course of this investigation. Visual
observation of the basin between the barrier and the generator disclosed
the presence of eddies, particularly near the bottom of the generator
plate, apparently a result of relatively high velocity "flow-by" which
occurred on the backstroke. These eddies did not appear to contribute
significantly to water motion in the region of the tank where the wave
data were taken.

Of the two types of submarine models evaluated, viz., the vertical
plate and the sheetmetal form, the latter provided more satisfactory re-
sults. Eddies in the vicinity of this formed barrier were minimized,
and at large generator periods, eddies were not observed. The major

18

difficulty encountered in the use of the sheetmetal barrier was that pre-
fabrication and subsequent assembly within the tank were required. This
was due to structural members along the top of the tank which provided
an access width 10 cm less than the required width of the model. The
design adopted for use in the tank is shown in fig. 6. Plastic tape
placed around the edges of the model provided the required seal against
water seepage around and under the model.

An additional difficulty was encountered in that simultaneous siphon-
ing into both basins was required when the sheetmetal model was used since
insertion of the model required a dry tank. During the siphoning it was
important to increase the bottom layer depths at the same rate in order
to prevent a current between the basins at the interface. When the plate
barrier was used, siphoning of the entire layer could be effected with
subsequent insertion of the barrier.

An approximate two- layer system was effectively formed for all the
experiments. A typical profile of salinity and density is given in fig.
8. The vertical profiles readily provided the respective layer depths
and densities.

Mixing at the interface rapidly destroyed the sharp pycnocline as
shown in fig. 8. The degeneration of the two-layer system was addition-
ally recognizable in that, after approximately 2 to 3 minutes of wave
action, changes in wave length were observed indicating oscillations of
other modes. Due to the mixing, the system after 3 minutes is considered
to be a poor approximation of a two- layer system. It is noteworthy that
mixing at the interface caused an increase in salinity in the upper layer,
although a corresponding color gradient in the upper layer was impercep-
tible to the author. It would appear that a threshold of color response

19

CD

<3-

oo

O

X ^

i §

2

ce cc

^^^z^

O lj

ry*^*^ ^- ^*

fa Ll

a <

/ /

O <]

/ /

/ /

(■ 4

\ /

\ /

\ /

/ /

/

/

1 A

/ ]

/

^ /

/ /

/ /

/ /

/ /

-^ /

I-* m ,— ■*^~ /r\ •n y^

Vj " ™ t/ l-r

O

o

T"3

^_^

CV)

b

>-

u

h-

if) O)

to

o~

z

1-

LU

co

■o

z

c

lu

n>

Q

>

h-

O

o

Z

Q

_l

-^

1^

-J-
CD

-o

s

*- oo

00

z
O

i-

CO

cc

I-

Q

<

y

cl

>-

or

Z>

O

o

(ujd) lAlOllOQ 3AQ9V 1H9I3H

of the human eye to the Rhodamine "B" dye tends to generate an apparent
discontinuity in the vertical dye distribution. Despite this particular
characteristic of the dye, the technique of utilizing a dyed layer con-
tributed significantly to the determination of the internal wave form.
6.2 Data Evaluation.

Data were obtained from 600 ft of film which was used to photograph
approximately 45 minutes of wave motion. A data summary is given in
Tables 1 and 2.

A qualitative analysis of the data was performed in order to deter-
mine some of the characteristics of internal waves which were generated
by the flow over the barrier. Two methods of analysis were conducted.
One method consisted of tracking a neutrally buoyant particle at the
interface at approximately one-half second intervals and plotting a par-
ticle trajectory. The second method consisted of tracking the crests of
the internal waves and determining the wavelengths, periods, amplitudes
and phase velocities.

The objective of the particle analysis was to determine particle
trajectories. A particle trajectory over a period of 64 seconds was de-
termined in Run 2A. The trajectory for the first 32 seconds is given in
fig. 9. During the period of observation, the internal waves were pro-
gressing from right to left. Except for two small-diameter orbits, all
orbits had a counter-clockwise direction of rotation, as would be expected
from theory for a particle in the lower layer near the interface. With
data from fig. 9, graphs of vertical and horizontal particle displacement
versus time were obtained as shown in figs. 10 and 11. Analysis of fig.
10 disclosed that the vertical particle motion appeared to have a perio-
dicity which consisted of two periods of approximately five seconds and

21

Run 1A IB 1C 2A 3A

h (cm)

0

7.2

7.6

7.1

6.9

h (cm)

15.2

8.0

8.5

8.1

9.0

h+ h' (cm)

15.2

15.2

15.2

15.2

15.9

T0 (sec)

23.0

21.0

21.0

6.0

13.0

Tis (sec)

26.3

9.6

4.0

Tlp (sec)

5.0

4.9

6.3

L0 (cm)

5.1

5.1

5.1

5.1

5.1

Lip (cm)

22.0

16.0

17.0

^ (cm)

0.8

1.2

1.8

1.0

f'(g/cm3)

0.999

0.999

0.999

0.999

f (g/cm3)

0.999

1.025

1.024

1.022

1.017

Ap (g/cm3)

0.026

0.025

0.023

0.018

t» <°F>

61

63

63

62

61

hB (cm)

7.6

2.5

7.6

7.5

7.5

LL (cm)

136

136

136

85

85

L2 (cm)

160

160

160

193

193

clo (cm/sec)

6.8

5.0

3.3

2.7

Cit (cm/sec)

6.4

5.3

4.8

DATA SUMMARY
TABLE 1

22

Run 4A 5A 6A 6B 7C

h (cm)

6.9

7.8

3.8

3.3

3.8

h (cm)

9.0

4.0

3.5

4.0

4.0

h+ h (cm)

15.9

11.8

7.3

7.3

7.8

TQ (sec)

12.0

10.0

5.0

8.0

12.6

Tis (sec)

6.8

9.0

5.0

10.0

Tip (sec)

5.1

13.0

5.0

7.0

6.3

L0 (cm)

5.1

5.1

5.1

5.1

5.1

Lip (cm)

34.0

30.0

40.0

21.0

Lis (cm)

26.2

Hi (cm)

2.7

1.5

1.3

1.0

1.0

?'(g/cm3)

1.007

0.999

0.999

0.999

f (g/cm3)

1.014

1.021

1.020

1.011

A? (g/cm3)

0.007

0.022

0.021

0.012

tw <°F)

59.5

62

62

62

65

hB (cm)

7.5

7.5

3.0

3.0

3.0

L^ (cm)

85

85

88

88

88

L2 (cm)

196

193

199

199

199

Ci0 (cm/sec)

2.6

6.0

5.7

3.4

Cit (cm/sec)

8.2

5.6

4.0

DATA SUMMARY
TABLE 2

23

$

o

4!> •

/ o ^^

ml ^f\$

—CM

2~ V-£ / g

cvi

7^ SI

9 - in

X Ifi

/o

-^r

J

1 &

r <J}

-<o

E

cm* \

u

*" \

z

>-

\ Q

O

-flDt

O

1-

\ "

(/)
O

ill

->

^>v v. ^

a.

<

<n

ok \ 9 \ v

•N A ^7 A— Q

_i
<

Z

0

a:
W

Ul

cc

O

Q VjB'CM 1

-9 5
o

X

i—

DC
<
0.

Ll

<b~^^ i

>»W-£)<Q

i l\ i~ N

Toi \ w

—.CM

*

Iff 4 o ^-*-°* cm

Q-t

d>™l ci

*-* \ o cm

- V^

_£

o_&

cm

m

1

I ■ 1

o

(0 <£> rf

T-

(WO) WOllOa 3A09V1H9I3H

-r-

en

^o-^

_h-

C\J

•

_<M

UJ

1

t-

>

i—

^_^

z

UJ

° 9

u

1

UJ

S

% o

.CNJ^"

_J *~

«""

Q. UJ

O

to oa

UJ
Q.

0

<

-1 LL

-rv-J

<

UJ

o

\-
q:
LJ
>

-C\J

UJ

_i

-O

O

1-

i.

1

1
10

1
CO

1 1 .1 . 1

o co cd ^j-

N

ro

CO N r-

*""

d

d

0 0-

(UJD)

NOIIOH

IVDlldBA BlDliaVd

CM

— r
oo

- r
CD

CM

CM

ro

N

LU

C\J

1

V-

>

CM

CVJ

Z
UJ

2

u

u

N <2

u

<

_J

LJ

2

CL
I/)

Q

T—

~. •-

III

CM

_J

cz

Q

<

D

111

1—

(0

I/)

Z

Q.

O

U_

<

M

<^H

CC

U

o

u

_l

CM

U

1-

o

cc

2

(^jocIsudj; ssai)
(wd) NOIlOlAi IVlNOZIdOH BIDIlUVd

ten seconds, respectively and in phase at the time the particle attained
its lowest vertical position.

The analysis of horizontal displacement versus elapsed time shown in
fig. 10 was complicated by an apparent mass transport of 23.3 cm in 64
seconds in the direction of wave progress. In order to examine only the
periodic horizontal displacement, the mass transport was assumed to be
linear; and for each data point the transport distance was subtracted from
the total displacement. The resultant periodicity of horizontal particle
motion shown in fig. 11 is similar to that for the vertical motion shown
in fig. 10.

For the segment of time for which the above particle motion analysis
was conducted, the period of the progressive wave was determined to be
4.9 seconds by timing the movement of wave crests past a reference mark.
Additionally, the period of the standing wave was observed to be 9.6
seconds by timing the horizontal oscillations of a column of water be-
tween a surface particle and a particle resting on the bottom. The peri-
ods of oscillation determined by the two independent methods compare
favorably and indicate that the particle motion was the result of a pro-
gressive wave in a mass which was oscillating with a greater period.

The item of significance of this segment of the investigation is
that some success was achieved in developing a technique by which the
particle motion of an internal wave could be observed and analyzed.

The second method of analysis involved comparing observed phase
velocities with those for the theoretical two-layer model. The wave
characteristics which were directly observed, in addition to the phase
velocity, were the apparent wave length and period. A phase velocity was
computed from the apparent wave length and period using the relationship

27

C = L/T (3)

and compared with that phase velocity which was directly observed. The
determination of the observed phase velocity by two methods provided a
check against inaccuracies of observation. The observed phase veloci-
ties determined by the two methods, for any given run, differed by less
than 10 percent. The average value of the two observed phase velocities
is shown in Tables 1 and 2.

The theoretical phase velocities were computed using equation (2).
Within the range of variables introduced, equation (2) shows that the
phase velocity of the internal wave is most responsive to changes in the
density difference between the two layers. Comparison of the observed
and theoretical phase velocities shown in Tables 1 and 2 indicates a
maximum difference between the two equal to 44 percent of the theoretical
value. Considered significant, however, is the fact that for Runs 1C,
2A, and 3A the magnitude of the observed phase velocities decreases with
a decreasing density difference, which is what would be expected from
theoretical considerations. A similar trend is observed between Runs 6A
and 7A. The discrepancies between the theoretical and observed phase
velocities may be a result of the observation of "apparent" phase veloci-
ties in a dispersive medium. In all cases conditions were such that the
waves are not long; since more than a single frequency is present, there
is dispersion and the crests cannot be expected to be conserved. Addi-
tional factors which may contribute to the discrepancies are that the
progressive wave crest which was observed was superimposed on a standing
wave of higher period and that the physical system deviates from the two-
layer model.

28

7. CONCLUSIONS.

In an approximate two- layer system, an internal wave may be gener-
ated by a current flowing over a submarine ridge where the height of the
ridge may be greater than, less than or equal to the depth of lower
layer.

The theory that the phase velocity of an internal wave is directly
proportional to the density difference of a two- layer system has been
qualitatively supported.

Particle trajectories as predicted by theory may be observed by
using neutrally buoyant particles of carbon tetrachloride and benzene.

29

BIBLIOGRAPHY

1. Cromwell, T. Pycnoclines created by mixing in an aquarium tank.
Journal of Marine Research, v. 18, no. 2, I960: 73-82.

2. Defant, A. Physical Oceanography, v. 2, Pergamon Press, New York,
1961: 517-570.

3. Rattray, M. On the coastal generation of internal tides. Tellus
XII, 1960: 54-62.

4. Weigand, J. G. A Model Study of Internal Waves, Thesis, University
of Washington, 1962.

5. . H.O. Misc. 15360, Effects of Weather Upon the Thermal

Structure of the Ocean. U. S. Navy Hydrographic Office: 36-37.

6. Proudman, J. Dynamical Oceanography, Methuen and Co. Ltd., London,
1953: 349-351.

30