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RESEARCH AND DEVELOPMENT REPORT4Z

REPORT 1135p
4 SEPTEMBER 19624
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MEASUREMENT OF ATTENUATION OF
LOW-FREQUENCY SOUND (5-8 KC/S)
IN SMALL SAMPLES OF SEA WATER

P. G. Hansen

U.S. NAVY ELECTRONICS LABORATORY, SAN DIEGO, CALIFORNIA
A BUREAU OF SHIPS LABORATORY

THE PROBLEM

Develop equipment for measuring the sound attenuation
at low frequencies in small samples of sea water. The
attending disturbance of the sample from its natural state
should be kept as low as possible.

Correlate the results with other oceanographic variables
and explain the correlations obtained.

RESULTS

1, Equipment which utilizes cavity resonators with acous-
tically soft side walls was developed. The effects of wall
contamination on the measurements are therefore kept to
aminimum. The attenuation can be measured at discrete
frequencies between 5 and 8 kc/s, provided it falls in the
range from 10 to 200 db/kyd.

2. Natural sea water with suspended particulate matter
and high oxygen content showed attenuations in the whole
range the equipment was capable of measuring (10-200
db/kyd). Sea water free of particulate matter never
showed any measurable attenuation. The attenuation of
clean sea water is about 0.5 db/kyd at 5000 c/s, and this
is below the lower limit set by the equipment. It is
suggested that the excess attenuation may be explained by
the presence of a large number of very fine bubbles or
bubble nuclei.

RECOMMENDATIONS

1. Include an investigation of small bubbles and nuclei in
future work, Give special attention to plankton, since air
spaces inside a living cell will have an especially marked
effect due to the surrounding organic matter. Use cavity

wun WM VA

0301 0040560

li

methods for this work, since the effect of a single bubble
or a small quantity of plankton then can be determined.

2. Raise pure cultures of living plankton in the laboratory
to ascertain the effect of living plankton and dead organic
matter separately.

ADMINISTRATIVE INFORMATION

Work was begun under SR 004 03 01 (NEL L4-1) and
S-F011 02 09 (NEL L1-7) and completed under the new
problem designation SR 004 03 01, task 0580 (NEL L4-4).
The manuscript was completed in January 1962 and sub-
mitted to the Graduate School of the Agricultural and
Mechanical College of Texas in partial fulfillment of the
requirements for the degree of Doctor of Philosophy. This
report is a slightly modified version of the doctoral dis-
sertation. It covers work from March 1959 to November
1961 and was approved for publication 4 September 1962.

The author is especially indebted to:

Dr. G. H. Curl for the discussions and advice during
the period of the investigation, and his reading and criti-
cism of the manuscript.

Dr. E. C. LaFond for his interest and support during
the measurement phase and his assistance in the procure-
ment of necessary equipment and facilities.

Dr. E. G. Barham for his assistance with the phases
of marine biology involved.

Professor R. O. Reid, my committee chairman, for
his help and advice with the theoretical parts, as well as
help and guidance with the editing of the dissertation.

Mr. S. A. Burnett for his interested and skillful help
in changing ideas into hardware during the development of
the cavities and transducers utilized in the project.

CONTENTS

LIST OF SYMBOLS USED IN TEXT...page iv
ie NR OD CAM ONE yr 2
Il. METHODS OF MEASUREMENT...3
Field Methods... 3
Laboratory Methods: Plane Wave Propagation... 4
Laboratory Methods: Three-Dimensional Wave
Propagation... 6
Soft Wall Cavity Resonators... 8
Influence of Air Bubbles... 9
III. DESIGN CONSIDERATIONS... 11
Cylindrical and Spherical Cavities... J1
Box-Shape Cavities...13
IV. THEORETICAL DISCUSSION. ..25
Acoustical Wave Equation for Viscous Fluids... 26
Some Special Solutions of the Wave Equation...41
V. EXPERIMENTAL PROGRAM... 60
Observational Platform... 60
Experimental Procedure... 62
Discussion of Results... 73
VI. SUMMARY AND CONCLUSIONS... 97
IPieiliaaelioy IRVSSWULISS 5 42) %/
Suggestions for Future Studies... 98
REFERENCHES...101

ILLUSTRATIONS

1-2 Schematics of preliminary and final cavity designs...page 15
3 Sound transducers (preliminary and final models)... 17
4 Diagrams of measuring systems...19
5-7 Reverberation curves obtained with Bruiel and Kjaer recorder... 21-23
8 Photograph of data-collecting equipment... 24
9-14 Calibration data recorded under various conditions... 67-69
15 Reverberation curves using data of figures 5, 6, and 7... 71
16-32 Attenuation values as determined from measurements on
sea water samples under various conditions... 77-86

iii

iv

LIST OF SYMBOLS USED IN TEXT

A Surface area; Ay amplitude parameters.

B Arbitrary constant.

b Coefficient of thermal expansion (volumetric).

Cy Specific heat at constant pressure.

C Sound speed (A, / 0, )2,

@* Complex sound speed.

D Clearance between free surface and top of cavity
(figure 9).

Dy, Deformation (strain) tensor (4 =1,2,3;%=1, 2, 3).

# Acoustic energy (intrinsic plus kinetic).

e Base of natural logarithms.

PF Body force.

a Frequency (c/s); or functional notation.

t Integer suffix (t=1, 2, 3).

ai Unit imaginary number.

K Isothermal compressibility coefficient.

Ky Thermal conductivity coefficient.

K, Complex wave number corresponding to coordinate Ls»

i Integer suffix (s=1, 2, 3) independent of 7.

ks Real part of wave number.

ds Dimensions of rectangular cavity resonator.

L Gross scale factor (volume/surface area).

m Mass per unit area of flexible wall.

n Unit normal to boundary (the components n; being the
direction cosines of the boundary).

Ns Integer designating the mode of oscillation.

ttotal

Sik

Pressure.

Acoustic pressure (anomaly from hydrostatic value
ort JP).

Quality factor (see page 11).

Resistive (real) part of boundary impedance Z; also
bubble radius (page 90 only).

Bounding surface.
Particle displacement vector (components S,).

Absolute temperature (ae mean value, E anomaly
from mean value); also surface tension (page 90 only).

Time

Reverberation time for cavity with pure water.
Reverberation time for cavity with sample sea water.
Fluid velocity vector (components v;).

Magnitude of velocity vector (v® =v,* + v,* + v,*).
Volume.

Components in a Cartesian coordinate system (i =1, 2, 3)

Boundary impedance (Z; and Z\, , t=1, 2, 3 correspond
to values pertinent to the six faces of a rectangular
cavity resonator).

Spatial (or range) coefficients of attenuation (db/kyd).

Temporal coefficient of attenuation (db/sec); see
page 57 for special designation.

Parameter controlling viscous loss, (2u, +), )/\,.
Transmission coefficient for energy flux.
Effective thickness of walls (n/p, ).

Unit diagonal tensor; unity for i=hk, zero for 1 Fh.

Anomaly of specific entropy from equilibrium value
(ergs/gm °K).

Dilation, V ° Ss.

Thermal diffusivity.

vi

Second Lame parameter.
Second viscosity coefficient.

First Lame parameter.
First viscosity coefficient.
Bo IIS) 6.6

Density; p, is mean density.
Element of surface area.

Total stress tensor (normal components being
tensile stresses if positive).

Phase parameter.

Complex frequency.

Real frequency (radians/sec).

Gradient operator.

Standard symbol for repeated multiplication.

Standard summation notation.

|. INTRODUCTION

It is well known thatthe propagational characteristics
of acoustical energy depend upon the type of material
medium and the distribution of inhomogeneities within the
medium. An abrupt change in the ocean medium as occurs
at the sea surface or sea bed produces reflection or rever-
beration of acoustical energy. Variations of sound speed
in the sea produce refraction effects. The presence of
air bubbles or suspended material, including plankton,
will produce scattering of the sound energy. Viscosity,
diffusion, certain thermochemical effects, and other non-
reversible processes can lead to absorption of sound energy.

The biota of the sea can also influence the acoustic
properties to a marked degree. Schools of fish not only can
give rise to echoes, but can also impair the transmission
of a sonic beam of energy from point to point by scattering
the energy out of the beam. The best known effect of this
sort is perhaps the deep scattering layer. It appears ona
fathometer record as a false bottom above the true bottom
and in view of its vertical migration during the day, is
thought to be caused by a layer of organisms.

In recent years there has been an interest in the prop-
erties of low-frequency acoustic energy in the sea. Theo-
retically, the viscous attenuation coefficient of acoustic
energy in a plane sound wave is directly proportional to the
square of the frequency. However, in view of other sources
of attenuation, particularly the scattering and absorption
associated with air bubbles and plankton, it is expected
that anomalous attenuation can occur. The present report
is concerned with the problem of isolating some of the
factors producing attenuation of low-frequency acoustic
energy in sea water,

Several of the above mentioned inhomogeneities of the
medium are almost always present at the same time. This
makes it difficult in field measurements to separate the
different effects of temperature and salinity gradients,

fish, plankton, and properties of surface and bottom. It
is therefore desirable to employ a measurement technique
that requires a relatively small sample when studying the
medium since difficulties from inhomogeneities then can
be avoided, and cause and effect can be established. It

is also desirable to be able to undertake the measurements
as soon after sampling as possible, to keep changes in the
water ata minimum. The first requirement was met by
using a soft or compliant wall cavity method developed at
the U. S, Navy Electronics Laboratory, and the second
requirement was satisfied by using NEL's oceanographic
tower off Mission Beach, San Diego, California, as the
supporting operations facility.

ll. METHODS OF MEASUREMENT

A great amount of work on the determination of sound
absorption in liquids exists in the literature and it is
clearly necessary to limit the discussion of methods of
measurement to those aspects which are most pertinent
to the present investigation--specifically, the influence
of biological entities and air bubbles on sound propagation
in sea water and the methods of measuring them. The
references cited here have been chosen with the aim of
presenting a representative cross section of the recent
work in this field. There exist several articles and texts
covering the broad aspects of the problem of sound attenu-
ation.+~* (See list of references at end of report. )

Attenuation can be measured by (1) field methods,
taking place in the ocean where the disturbance of the
sample can be kept at a minimum; and (2) laboratory
methods, which involve taking samples out of the ocean
to perform the measurements.

FIELD METHODS

The direct(and the simplest)method consists of intro-
ducing two hydrophones in the ocean and performing trans-
mission measurements. The inherent difficulty is that the
measurements are influenced by a large number of vari-
ables, and it is difficult to unscramble the combined effects
so as to achieve dependable conclusions about the influence
of the plankton population.

For clear sea water the attenuation is small and it is
therefore necessary to use long distances between trans-
mitting and receiving hydrophones. Moreover, the influence
of thermal stratification and of reflections and losses at
surface and bottom, as well as the intensity changes due
to spreading of the acoustical beam, must be accurately
known. The direct method has been used to determine the

sound velocity, and it has also demonstrated that the atten-
uation in sea water is extremely small, when the water is
free of contamination. By way of illustration, Horton*
states that the attenuation coefficient in pure sea water is
approximately 0.5 db/kyd at 5000 c/s.

Studies carried out by the direct method*~” clearly
demonstrate the difficulties involved. Chief among these
difficulties is that of assessing the influence of plankton,
Plankton is far from homogeneously dispersed in the ocean.
It occurs frequently in quite sharply defined layers, and is
often very patchy within the layer. To determine the
acoustic attenuation by plankton from an interpretation of
measurements obtained by the direct method is therefore
virtually impossible and has not been attempted.

A few methods classified here as laboratory procedures
have been modified for field use. A method of velocity
measurement by spherically propagating sound waves has
been adapted for field measurements.” However, in this
system, the path length involved is too short for obtaining
any useful information about attenuation. Adaptation of
the reverberation tank to field use is more pertinent to the
attenuation problem.?

The direct field methods have served mainly in identi-
fying certain propagation anomalies in a gross sense and
in emphasizing the need for more detailed studies. Such
anomalies as the deep scattering layer, effect of ships'
wakes, ''quenched water," "knuckles, '' and ''black-outs
have been detected by direct methods.

11 eK

LABORATORY METHODS: PLANE WAVE PROPAGATION

Laboratory methods require only a relatively small
liquid sample and therefore lend themselves to measure-
ments of the acoustic properties of more or less exotic

*Ref, 1, p. 81
3K
IRIS Ub Ta

liquids of high purity, which can be obtained only in rela-
tively small quantities. Such measurements are of great
interest to the physicist, since they offer one of many use-
ful approaches to the investigation of the nature of matter.

The laboratory methods divide naturally into two
groups. The first utilizes one-dimensional (plane wave)
propagation, and the second utilizes three-dimensional
wave propagation.

The instruments belonging in the first group include
acoustic interferometers, and equipment resembling inter-
ferometers, but utilizing pulsed rather than continuous
signals. Plane waves are usually obtained by using trans-
ducers with a flat actuating surface, large compared with
the wavelength of the sound. For this purpose a quartz
crystal is usually employed, especially for the high-fre-
quency acoustic waves.

Interferometers measure the sound velocity by utilizing
a reflector and create a one-dimensional standing wave sys-
tem. A change in the distance between reflector and trans-
mitter by an integral number of half-wavelengths will not
disturb the standing wave pattern. The sound velocity can
therefore be determined from the frequency and the half-
wavelength, The detection of the standing wave pattern can
be done by one of several methods: (a) measurement of
the drive current to the transmitting crystal at constant
voltage, (b) employment of a second crystal as a reflector
and also as a receiver, and (c) utilization of a suitable
Opticallsystend. yon ae

The sound velocity can also be evaluated by impressing
pulses of the desired frequency on the transducer and
measuring the time of flight from transmitting to receiving
erystal.'* A transistorized version of a pulsed interferom-
eter is being used for sound velocity measurements in the
Mell, =

Both types of equipment (those using continuous and
pulse signals) are also suited for attenuation measurements,

but only when the measuring frequency is high. The main
limitation is that the sound beam must be long enough to
cause a reasonable amount of attenuation. This places

a practical restriction on attenuation measurements

with this equipment to frequencies above 1 Mc/s. ie

Plane waves can also be obtained by letting the wave
propagation take place inside a tube of a material witha
much higher specific acoustic impedance than the fluid
under investigation. This requirement can be met easily
for gases, since pc of steel is about 10° times larger than
oc of air (p being the density of material and c the sound
speed). The same ratio for steel and water, however, is
only about 25.

Tube interferometers have nevertheless been success-
fully used on liquids and mixtures of liquids with water as
the main constituent, but only under special circumstances,!*?
These methods would not be suitable in the present problem,
since the attenuation in many cases is too small to allow
the utilization of a tube of reasonable length. Furthermore
a difficulty is encountered in the presence of air bubbles
on the wall. This is the same difficulty explained later
under resonating cavities. In the case of fluids which con-
tain a high density of bubbles within the fluid itself, the
presence of the bubbles may actually enhance the use of
the tube interferometer since the acoustic impedance of
the fluid is thereby lowered while the absorption coefficient
is increased.+?

LABORATORY METHODS: THREE-DIMENSIONAL
WAVE PROPAGATION

There are two subgroups in this division: reverberant
chambers and resonant chambers. The reverberation
methods all utilize a 'diffuse'' sound field, i.e., a very
large number of normal modes are energized at the same
time, and these modes must bec lose together in frequency.
The volume of the sample required is therefore quite large,

if measurements at low frequencies are desired. The con-
tainers should not be of too regular shape, in order to
insure a ''diffuse'’ sound field with minimum volume of
sample. Spheres, as used by Moen, 2° and cubes have too
many degenerate modes to be well suited. Bubbles settling
down on the walls may actually be helpful under special
circumstances by coupling different modes together.

In contrast to the réverberant chambers, the resonant
cavity involves the excitation of a pure normal mode,
characteristic of the particular cavity shape and size. The
resonant cavity method is the only practical method that
permits attenuation measurements at low frequencies on
relatively small samples.

The reverberant chamber method was employed by
Knudsen for air-acoustic investigations.°+ It has been
extended for use with water by Leonard, and by Wilson
and Liebermann, Mulders used the method near 1 Mc/s,
and needed 2 to 3 liters of liquids for the measurement. **
The smallest absorption he could measure was of the order
of 100 db/kyd. Moen*° employed bottles of spherical
shape ranging from 1 to 12 liter capacity for attenuation
measurements of water. The smallest frequency investi-
gated by Moen was 140 kc/s.

Glotov’ has suggested a reverberation method for use
in the field. The volume of sea water required is 500 liters,
and the lowest frequency utilized is 15 kc/s. Clearly it
would be difficult to obtain homogeneous plankton samples
of such large size, even though it is quite possible to utilize
this type of equipment to measure the sound attenuation in
natural sea water. Glotov indicates that even for this type
of equipment, the bubbles on the walls pose a problem,
since the equipment has to be conditioned at a depth of 60
meters before every measurement.

There are essentially two types of resonant chambers:
the hard wall cavity and the soft wall cavity. The hard
wall cavity has been employed extensively for determining

sound attenuation in pure liquids at low frequencies. The
shape of the container is almost always spherical, since
this shape eliminates losses due to shear at the boundary
for the radial modes. The slightest contamination of the
cavity wall, however, will upset the results, and it is
hardly possible to get reliable measurements with anything
but carefully purified and degassed liquids. A development
of the theory for the spherical resonator can be found in
reference 23. The practical application of this method is
demonstrated in references 11, 24, and 25.

Since a few bubbles on the hard wall will completely
negate the results, it is virtually impossible to use this
method for natural sea water, where the measurements
must be performed rapidly and cleaning must be relatively
easy. The most promising solution for sea water appears
to be the use of soft wall cavities. The surface of water
is an excellent reflector of sound waves, and a small
bubble trapped in the surface film will have no effect on
the reflection of sound. Making a box-shaped cavity of
thin sheet stock will give almost complete pressure release
at the boundaries, since very little force is required to
bend the thin metal sheet. A bubble attached to the boundary
will therefore have only a minor effect.

SOFT WALL CAVITY RESONATORS

Soft wall cavities were first constructed by W. J. Toulis
at the Navy Electronics Laboratory in the summer of 1953,
and the best available in 1957 had a Q factor of approximately
2000 at a frequency of 5 kc/s. This implies a reverberation
time* of approximately 0.9 second. This method was con-
sidered very promising for the present investigation. Fur-
thermore, some equipment of this type was already avail-
able, so a development program of this method was under-
taken, as described in the preceding section.

*The reverberation time is the time in seconds it takes the
sound intensity in a cavity to diminish 60 db.

In order to gain a better understanding of the soft wall
cavity an attempt has been made in section IV to develop a
foundation for the theory describing some of the basic
characteristics of the system. An approach to this prob-
lem had been made by Toulis.*° The theory developed in
section IV follows the procedure for room acoustic theory
discussed by Hunt,*”’*°, Morse,*° and Morse and Bolt.°°
However, it was necessary to employ the wave equation
for a viscous medium, ‘In this connection the phenomeno-
logical approach suggested by Markham, et. al.* and
Skudrzyk** served as a guide.

The wave equation utilized in section IV considers the
first and second coefficients of viscosity as independent.
The question of the relationship between the first and
second coefficient of viscosity has been much debated
since Stokes' original paper on this problem*? and it can-
not be claimed to be fully resolved.**’°° This question,
although pertinent to the quantitative evaluation of the
theory, does not have any particular bearing on the qual-
itative nature of the results.

INFLUENCE OF AIR BUBBLES

The presence of air in water has a significant effect
only when in the form of bubbles. Dissolved air changes
the sound velocity by less than 10 parts per million®* and
the attenuation coefficient at ultra-high frequencies is
virtually unchanged within the tolerance of measurement

error. oy

The anomalous attenuation and change in propagation
speed for frothy liquids was noted in 1911 by Mallock.°*
A recent survey of the subject is given by Devin.°° He
considers the damping caused by a single bubble in water.

“Ber, Ws, jo, O70
Pet. 2 i), (RAG. TIeoUOs

10

Bubbles should have the greatest effect when the sound
frequency matches the resonant frequency of the bubble.
The resonant frequency of a spherical bubble is given by

f,D = 652

where PD is the bubble diameter in cm and f_ is the reso-
nance frequency in cycles per second. A bubble resonat-
ing at 5.2 kc/s accordingly has a diameter of 1.25 mm
and is therefore easily visible. This is pertinent to the
interpretation of the measurements discussed in the later
chapters.

The conditions under which air bubbles may exist in
the ocean are discussed in a number of references. °°" °°

lil. DESIGN CONSIDERATIONS

Several materials and cavity shapes were investigated
before arriving at the final design. Both cylindrical and
spherical chambers were investigated, as well as rectangu-
lar prism forms of various relative dimensions. Some of
the more important characteristics of the types of chamber
which were considered, and reasons for selecting the
rectangular or box-shape form in the final design, are
discussed in the following sections.

CYLINDRICAL AND SPHERICAL CAVITIES

In the cylindrical cavities the walls were too rigid
because of the circular cross section. Hence the bubbles
which inevitably formed on the inner wall had a very marked
influence on the measurements. The wall could have been
made more compliant by introducing corrugations, but this
would make it difficult to clean. Moreover, it was found
that the seam, produced when a cylinder was fashioned out
of a sheet, would disturb the symmetry of the container so
much that a poor resonant cavity resulted. Values of @ of
the order of 2000 could be obtained from this type of cavity
if seamless containers were used, and if proper care were
shown. The term Q referred to above represents the qual-
ity factor of the cavity response as defined by

Fo
Af

where f, is the resonance frequency for a given mode of
oscillation and Af is the bandwidth of the response spec-
trum for uniform power input as measured at the 3 db
level below the resonant peak level. The Q value is in-
versely proportional to the attenuation coefficient for the
cavity or in other words, directly proportional to the

11

12

reverberation time. The quality factor characterizes the
cavity plus the contained liquid. The above value refers to
the cavity filled with nominally pure water. However, it
should be noted that any contained bubbles on the walls will
tend to produce a low estimate of the Q which would apply
for completely degassed water.

A cylindrical cavity in the form of a shallow pan was
also investigated. This has the advantage that the free
surface and bottom represent the major fraction of the
boundary, thus minimizing the area of rigid surface. Good
Q factors were obtained, since bubbles do not tend to stick
to the bottom and have little effect at the free surface.
Bubbles which cling to the cylindrical surface are so far
removed from the center that their effect is negligible,
since the main reflections take place between top and
bottom of the cavity. The main difficulty was that the
slightest vibration caused surface waves to be set up.
This created difficulties in tuning the cavity, as well as
irregularities in the way the sound died out, when the
generator was disconnected.

The spherical cavities were obtained by cutting off the
neck of commercially available, round-bottom boiling
flasks made of Pyrex. Higher Q's can be obtained with
these bottles than by any other readily available method,
but they suffered from the same limitations as the cylin-
drical containers to an even more marked degree. The
opening in the flasks is so small that cleaning the inside
surface is virtually impossible. This type of resonator
has been used extensively for research on pure liquids and
gases, but was considered unsuited for measurements on
sea water with the naturally occurring particulate matter
suspended in it. The cylindrical resonators were likewise
given up for the same reasons.

BOX-SHAPE CAVITIES

Cavity Properties and Construction

The rectangular prism resonators with free surface
have several advantages relative to other forms mentioned:

1. The boundary surfaces are made of flat sheets, and
therefore are relatively easy to wipe clean.

2. The boundary will allow pressure release by bending
in contrast to the spherical and cylindrical cavities, where
pressure release must be accomplished by stretching of
the boundary.

3. The sound field in the corners of the box is very weak
and theoretically zero in the corner itself. The corners
are therefore well suited for support of the cavity.

4, The seams where the sheets are united will naturally
fall in the corners, where the effect from them will be at
a minimum.

5. The corners allow loose coupling of the transducers,
and the coupling can easily be changed.

6. Finally, this type of cavity is relatively easy to manu-
facture.

The square-cornered cavities therefore seemed to
offer the greatest promise and the largest amount of work
was devoted to their development.

Cavities were made of a great variety of materials,
and in many different sizes. The materials ranged from
plastic films and thin shim stock of copper, brass, bronze,
and stainless steel, to heavier sheets as well as readily
available plastic containers. It was found that stainless
steel was superior to any of the other materials. The
power loss within the steel sheets is small and the steel
alloy is chemically inactive when exposed to sea water.
However, stainless steel is not easy to weld or solder.

13

The original stainless steel cavities had the corners
spot-welded to angles made of the same material as the
sides. Watertightness was then insured by soft soldering
the corners (fig. 1), The supports consisted of four SOAB*
rubber blocks, placed as close to the corners as the stabil-
ity of the cavity would allow. The transducers were im-
mersed in the water as close to the corners as the hum
and noise level permitted.

Several attempts were made to improve the construc-
tion of the corners of the cavities. It was assumed that
the spot-welded and soldered joint would increase the
losses of the cavity. Also, the soft-soldered corner had
a tendency to leak in time due to electrolytic action, when
the cavities were filled with sea water. No really satis-
factory solution was found until the NEL machine shop
succeeded in developing a technique whereby the corners
were inert arc welded. It was not possible to weld sheets
thinner than 24 mils, and the final cavities were there-
fore made of this thickness (fig. 2).

The presence of a few bubbles on the wall did not affect
the measurements. A bubble with a resonance frequency
close to the measuring frequency of the cavity, however,
did have a marked effect. As noted in the introduction, a
bubble resonating at 5 kc/s is somewhat larger than 1 mm
in diameter and can be detected quite readily and removed.
The above frequency corresponds roughly to that of the
fundamental mode for the cavity dimensions employed in
the sea water measurements reported in section V. A
large number of evenly distributed small bubbles will be
produced on the walls if air-saturated water is allowed to
heat up in a cavity without disturbance, and they will have
a marked influence on the measurements. This problem
is discussed further in section V.

“Previously made by B. F, Goodrich Rubber Co., Akron,
Ohio.

TRANSDUCERS

SOFT SOLDERED

A A SPOT WELDED

ZANE

Figure 1. Schematic of preliminary watertight cavity.

TRANSOUGERS
GLASS
a a PROBE

WELDED SEAMS,

SECTION A-A

SOAB RUBBER BLOCKS

Os ALUMINUM ANGLES

Figure 2. Schematic of final cavity design.

Five different box cavities with welded corners were
constructed. The dimensions (all in inches) are as follows:

Length Width Depth

IN@s i Ly 1 14
No. 2 12 8 10
No. 3 83 5S 7
No. 4 6 4 5
INO; 5 4z 2+ 35

Cavity No. 2 was used almost exclusively during the
data collection.

16

Cavity Supports

The rubber supports shown in figure 1 were not very
satisfactory. The cavities had to rest on a large area of
the support in order to assure stability. Wire supports
were attempted and these resulted in the improvement of
the Q factor but made cleaning of the side-walls quite diffi-
cult. A further difficulty was that the whole cavity would
oscillate like a pendulum at a low frequency, and thereby
set up wave motion on the rather large free water surface.

The supports finally adopted, and used throughout the
data collection, did not have any of these shortcomings.
They consistedof four pieces of angle aluminum. The line
of support was approximately 4 inch from the extreme
corner of the cavity (fig. 2). No increase in the total
cavity loss occurred if the supports were moved away from
the corners by as much as an additional § inch. This was
regarded as evidence that the loss due to this type of sup-
port was insignificant. In any event, the residual loss
caused by the supports if present is taken into account in
the calibration of the cavity with distilled water. The
aluminum angle supports were used throughout the data
collection phase of the work reported here.

Power Supply to Cavity

Different methods of driving the cavities were tried in
order to minimize the losses due to the transducers, and
at the same time obtain as convenient an instrument as
possible, Provision of power input direct to the cavity
wall was attempted by gluing a ring of radially polarized
barium titanate to it and touching the side with a point
driven by a specially constructed electromagnetic system.
However, the cavity 2 was degraded seriously by this
procedure. The simplest driving methods proved to be
the best. Two small transducers made from barium
titanate cylinders were immersed in the water close to

two corners (fig. 1 and 2). The amount of coupling could
then be controlled by moving the transducers closer to or
further away from the corners.

The transducers first utilized (shown in cross section
in fig. 3B) were found superior in performance to the
usual type (fig. 3A). However, the losses introduced by
the transducers were still significant, since it was possible
to improve the over-all Q of the resonant cavity by moving
the transducers closer to the corners. An effort was there-
fore made to improve the transducers.

BARIUMTITANATE CYLINDER
RUBBER WASHER RUBBER PLUG

C DIPPED NEOPRENE COAT

Figure 3. Sound transducers. A, B, preliminary models;
C, final design.

The final transducer design is shown in figure 3C, The
greatest improvements were accomplished by omitting all
soft material between the cylinder and the metal parts, and
by using an exterior coating of dipped Neoprene rubber.
The soft gaskets used in the transducer (fig. 3A) were re-
placed by a very thin film of epoxy cement, and it may be

IL

supposed that the cylinder is virtually clamped to the metal
end pieces. All motion therefore takes place in the barium
titanate cylinder, and this ceramic material has a much
smaller mechanical loss than that associated with motion

in the rubber gaskets. This improvement is achieved at

the cost of a lower sensitivity of the transducer, but experi-
ments indicated that this loss was offset by the fact that the
transducers could be coupled closer without affecting the
cavity performance.

The Neoprene-rubber solution used for dipping the
transducers is made so thin that bubbles are easily seen
and can be eliminated before the rubber sets. Also, it is
possible to prevent bubbles from forming if care is exer-
cised during the dipping process. Four or five coats are
apparently sufficient, since the transducers have been in
use for almost a year without any significant deterioration.
The loss introduced by the transducers can be neglected
for all conditions except where the total losses are an
absolute minimum,

The block diagrams in figure 4A and 4B show two
possible measuring arrangements that were investigated.
Figure 4A is a ''sing-around" circuit, where a signal pres-
ent in the cavity will be picked up by the receiving hydro-
phone, amplified, and reintroduced in the cavity. The
sing-around frequency will be a natural resonance frequency
of the cavity, if the electronic circuitry is properly adjusted.
The transmission loss from the input terminals of the trans-
mitting hydrophone to the output terminals of the receiving
hydrophone is approximately 80 db. The amplifier must
therefore supply at least 80-db amplification to establish
a sing-around condition, and experience has demonstrated
that it is extremely difficult to keep the system functioning
properly under field conditions. The equipment would fre-
quently sing-around on some spurious resonance frequency,
and very critical adjustments of the tuned circuit and trans-
ducer positions would be required before the measurement
could be obtained.

Figure 4B shows a much simpler solution. The gen-
erator drives the transmitting cylinder, and the received

signal is measured by a vacuum tube voltmeter. The cavity
must now be tuned to resonance manually. However, it is
possible to ''probe'' the cavity with a small captured bubble
and ascertain when the desired mode is obtained, by deter-
mining the position of the nodal lines with the bubble. This
procedure would almost invariably stop the sing-around
circuit in figure 4A from oscillating. The frequency can

be changed in small increments by the cycles increment
dial on the generator and the Q factor can be evaluated in
this fashion, when the losses are large.

80DB
N AMPLIFIER

Figure 4, Diagrams of measuring systems. A, preliminary
system; B, system employed in final tests.

19

20

The best method for lower losses was to record the
reverberation process with a Briiel and Kjaer recorder.
Examples of records obtained in this manner are given in
figures 5to 7, The permanent record obtained can be
studied in detail later on. Small undulations of the free
water surface during the measurements will not be too
detrimental, since only the envelope of the signal is meas-
ured and the frequency at any moment is immaterial. The
relationship between the bandwidth A f in c/s and the re-
verberation time t in seconds is expressed by tAf=2.2.
This relationship was deduced by assuming that the response
curve for the cavity is the same as that of a simple electric
resonant circuit. This relationship was checked in many
instances, where both t and Af could be obtained independ-
ently and no significant discrepancy was ever observed.

The reverberation time, however, has been used exclusively
for the data-taking phase.

A few comments are appropriate in respect to the hum
and noise problem, The input voltage to the transmitting
hydrophone was approximately 50v. The output voltage
measured by the VTVM (fig. 4B) was approximately 5mv
as a maximum value, and voltage levels 50 db below this
value were measured when the reverberation curve was
recorded, Careful shielding and grounding techniques
were essential. The tuned resonant circuit shown in fig-
ure 4B was necessary to overcome the remaining hum and
structural noise as far as possible and to increase the
signal sufficiently for the measurements to take place.

The Q of the electrical tuning circuit in figure 4B was
measured independently. It was 60 to 80, depending on
the frequency used. This corresponds to a reverberation
time of about 1/10 of the shortest reverberation that the
equipment could record, and was considered insignificant.

“‘pAx/ ap ‘pAx/qp

a 7G, Ung COR Bee “A ts WOO = 9°Z] ‘Co ‘UOT}eNUAa}}e foaS [8 °Z

‘f ‘Aouenbarj aoueuoseat atqqng ‘9 ‘9UIT] UOTJeLaqtaAet *S/9 9ZTG

"(Vy ‘sI} JOj Se owes 9y} ATTeUuTMIOU Al KHIM SOUAOS ee 8D). BL

SUOT}IPUOD T9UjO) ToJaUIeIp WW-C °Z 17e@ J91}eM Jind suIsn [-]-] epoul

jo a[qqnq 11e ue jo aouesaud 0} anp JO} §}S9} UOTLBIGI[eO WIOTJ pautezqo

QUIT} UOTJELIGIIAVI UO BdUENTJUL “q aAINO UOTJeLaqgiIaAas TeoIdAT, “VW
*QUIT] UOTJE1aqIaAeI UO aTqqngq peitnyded Jo sjyoajjq °G aInsTy

ee eee

21

*aTqqngd jNOUJIM WG eINST} JO SUOT}IPUOD Jo UNJaYy °g

(°VG ‘sTIJ oJ Se aues ay} ATTeuTWOU

SUOTIIPUOD TayIQ) *S/2 0099 ‘f‘S/2 PSIG ‘°L ‘psH/apP

9°Or oth) SDS 06°O “37 ‘de}eWeIPp WUW-T jo stqqnq jo

a0uasaid 0} anp dW} UOTJeLaqIIAaI UO BdUENTJU ~W
*QUIT} UOTJELaqIaAeI UO aTqqng paanyded jo sjyoojJq °9 JunsIy

22

(“VG °SIj] Joj] se awes ay} ATTeuTUIOU SUOT}IpUOd

RUNS) “DSi/eD M°ES WD S988 HON 4 H/o Ooo “es & L
‘1°61 ‘2 {8/9 6E1g¢ ‘ f ‘tajemeIp Ww-Z"Q Jo aTqqnq jo
gouasaid 0} anp aWIT} UOTJeLOqsaAat UO adUANTJUL *) eINsTy

SS

=

aS
aS

23

24

Figure 8 is a photograph of the equipment used during
the data collection in the summer of 1960.

Figure 8. Data-collecting equipment.

IV. THEORETICAL DISCUSSION

The theory of an acoustically resonant chamber is
examined here, to develop pertinent relations for the natu-
ral frequencies and the particular attenuation factors which
are related to the chamber geometry and the contained
fluid. For a chamber which contains an ideal, inviscid
pure fluid, free of any trapped gas phase, the primary loss
of acoustic energy occurs at the walls and free surface by
radiation to the environment. The amount of such loss
depends upon the configuration of the chamber, the flex-
ural properties of the walls, and the acoustic impedance
of the environment compared with that of the fluid.

For a chamber containing a complex fluid mixture
like sea water, the additional attenuation which occurs is
closely related to the irreversible processes associated
with viscosity, heat conduction, and diffusion. In the
present analysis only the viscous loss is considered,
However, from a phenomenological point of view, all
internal losses can be expressed in terms of an equivalent
viscosity effect. For sea water the effective viscosity can
be several orders of magnitude greater than the actual
viscosity.

The notation in the ensuing discussion is patterned
after that of A. Sommerfield.°° The development starts
with the formulation of the wave equation for acoustical
disturbances in a viscous fluid. Vortex motion and gravity
wave modes are excluded at the outset by adopting the
conventional approximations of acoustic theory.

295

26

ACOUSTICAL WAVE EQUATION FOR VISCOUS
FLUIDS

Fundamental Equations*

The Navier-Stokes form of the equation of motion for
a viscous fluid is

aD ‘i ‘ fe s
ae ONO) O = 4, Vm = (UPL (Yor) ce Wl? = pire (1)

where op is fluid density, bd is the velocity vector, P the
pressure, / the body force per unit mass, U, and i, the
first and second viscosity coefficients and V7 is the general
(three-dimensional) gradient operator. The velocity and
density must also satisfy the continuity equation

se ++ (pb) = 0 (2)

In addition it is presumed that p and P are related by an
equation of state, which can be expressed in a purely for-
mal way as

P= 56 (9) (3)

Relation (3) in effect ignores any temperature dependence;
however, this is unimportant for liquids since the tempera-
ture variation can be considered virtually nil. This matter
is explored in more detail in pages 37-38.

*A list of symbols is given on page iv.

Assumptions

The additional stipulations regarding the nature of the
disturbances are the following:

(a) The changes in P, d, and p associated with the
acoustic disturbance are regarded as sufficiently small
that the usual linearizing approximation can be made.
Specifically, the anomalies of p and p are regarded as
small compared to their respective mean values within
the chamber. Also dd is regarded as small compared with
the speed of sound for the fluid occupying the chamber.

(b) The motion is regarded as irrotational,

(c) It is presumed that there is no net translation of
the medium.

(d) The body forces (including gravity) are considered
to have negligible influence on the acoustic disturbances,

Linearized Relations

In view of conditions (a), (c), and (d), relations (1)
and (2) can be approximated by

ab @ ¢ 08 £
% se 7M Vv d= Wei rY Wy v)+ Vp = 0 (1a)
ap Pp
— + ve =
Sra ey ieee (2a)

where o, is the mean density and p is the departure of the
pressure from hydrostatic.

27

28

Following usual acoustical procedure, a mixed Lagrang-
ian-Eulerian system is employed in which the particle dis-
placement vector § is regarded as a field variable. Specifi-
cally 7 + & represents the position vector at time t of that
particle whose equilibrium position is 7. In view of the
condition of small displacement implied by condition (a) it
follows that

— =D (4)

Accordingly (2a) can be expressed as

1
(=)

fo) nw

eta. Wood) =

5g (o+p, V-S)
or

(>On) = 6, Vee = Ha, © (5)

a

where © is a convenient notation for the dilation V °s.

Equation (3) implies that for small changes
_ ah z
AP = do Ap=hk (6)

where i, is the second Lame parameter (or reciprocal
compressibility), Using (5) and taking AP as the acoustic
pressure anomaly (p) yields

p= -xX, © (7)

which is an adequate approximation for acoustic waves
whose intensity is not excessive.

The Stress Tensor

In the absence of viscosity the acoustic stress tensor™
in a fluid is given by

where 6,7, is the unit tensor (which has unit value for i = 4%
and vanishes for i 4%). Relation (8) is a compact statement
for the isotropy of the normal stress in an inviscid fluid.

For an elastic solid which is in a state of strain relative
to an equilibrium state, the associated elastic stress is
given by the non-isotropic tensor

gps Rs Wag IN, 1) Oa, (9)

Here Dy i, is the strain (or deformation) tensor

where s; is the components of the vector displacement §
relative to the relaxed(equilibrium) state and U, is the
first Lame parameter. The elastic tensor is symmetric
in the sense that 1;, = 1,;, which implies that there are
basically three different shear stress terms and three
different normal stress terms. Relation (9) would be

*The stress tensor 1; as employed here is such that tensile
Stresses T,,5 Tao» Tg, are positive or pressures negative.
The conventional indicial notation is employed, where i, &
ean take on values 1, 2, 3 independently.

29

30

pertinent to the cavity walls, assuming that the latter could
be considered elastic.

For a visco-elastic medium, relation (9) should be
supplemented by the viscous stresses which depend upon

_the rate of deformation and rate of dilatation of the medium.

The complete stress tensor in this case is

which retains the symmetry property mentioned earlier.
The Navier-Stokes equation for a viscous fluid as given by
(1) employs (11) with u, = 0 and -)\,® replaced by the fluid
pressure P.

As required by the second law of thermodynamics, the
viscous stresses can never lead to a decrease of the en-
tropy ina closed system. Eckart* has shown that this will
be assured under all conditions of deformation, if and only
if

Ww, = O acl, = -s Uy,

It may be remarked that the traditional stipulation regarding

the second viscosity coefficient (namely }, = -% U,) just
barely satisfies condition (12). This is important from the
standpoint of loss of acoustic energy associated with the
irreversible processes within the system. The rate of
conversion of the dynamic (acoustic) energy to thermal
energy is directly related to the rate of increase of the
entropy due to internal processes. Clearly the latter is
enhanced if }, exceeds the lower limit imposed by (12).

*Carl Eckart, unpublished class notes on Principles of
Hydrodynamics, Scripps Institution, University of Califor-
nia, 1948.

(11)

(12)

The Acoustic Wave Equation
In view of the identity
VY Wo) 2 H+ VX (XD) (13)

it follows that for irrotational motion (condition b), the
equation of motion (la) simplifies to

a

of - Qu, +4) F B+ vp = 0 (14)

Po 5

Moreover, if (4) and (7) are employed then the latter rela-
tion is rendered in the form of a wave equation for the
displacement 8:

a°é

aa
Do at?

ot

SNe Wee (Uy ae Ne (15)

The basic form of the wave equation remains unchanged
for a visco-elastic medium; the only modification is the
replacement of \, by 2uU, + 4,. However, the boundary
conditions appropriate to the visco-elastic problem are
certainly more elaborate than those required in the viscous
fluid problem.

In the special case of an inviscid fluid, (15) reduces

immediately to the canonical form for simple waves

a

So Vs (15a)

wihencenc = ay /p, represents the wave speed.

31

32

Relation (15) holds equally well for d, 8, or p. The
choice of dependent variable depends largely on the nature
of the boundary conditions. In any event, d, ©, and p are
readily determined in terms of &.

Boundary Conditions

The dynamic and kinematic boundary conditions require
that continuity exist in respect to the displacement and
stresses at the boundaries of the system. Specifically,
if \ denotes the difference in a given quantity across the
boundary of the system then it is required that

As = 0
(16)
Mp tap) = O
where n;, denotes the direction cosines of the normal to
boundary surface.
If the boundary is a liquid-air interface, and if both
fluids are considered inviscid, then it is sufficient to stipu-
late only
AS, = 6)
(iL)
iio = 0

If the air is neglected altogether, then the second condition
is sufficient and implies simply that p = 0 on a truly free
surface,

In the case of a thin flexible sheet separating two in-
viscid fluids, where the sheet can support significant tensile
stress, then the stress condition of (16) can be replaced by

the condition that the change of p across the sheet is propor-
tional to the curvature and tension in the sheet. However,
the latter in turn is dependent upon the distortion of the sheet,
which requires the solution of a separate elastic problem for
the sheet, with its attendant end conditions regarding rigidity.

Associated Energy Equation

In many cases one can bypass some of the mathematical
complexities involved in the detailed analysis of the dynamic
problem by considering the energy budget alone, or at least
as a supplementary relation. The real utility of the energy
relation is associated with its inherent appeal from the
standpoint of physical understanding of the problem.

In order to form the energy relation associated with the
acoustic wave phenomena, relations (2a), (6), and (14) are
convenient as a starting point. Using the definition of },
stipulated by (6), the continuity relation is readily rendered
in the form

1 ap ;
= L+y -g=0 18
wy D (18)

By forming the scalar product of } on relation (14), multi-
plying (18) by p, and then adding the two resulting equations,
the following quadratic relation is obtained

3 i ‘ ‘ 4
BL aoe" Eos - (pd) = (2u, +i, ) b+ VB (19)
(e)

where v is the magnitude of the particle velocity. This is
the energy relation associated with acoustic disturbances
(of sufficiently small energy) in a viscous fluid.

The terms p,v* /2 and p*/2i, represent, respectively,

34

the kinetic energy and intrinsic energy of the dilatational
oscillations, both expressed as energy density (energy per
unit volume). The intrinsic energy is really a measure of
the departure of the internal energy from that at equilibrium
due to compression or dilatation of the fluid.

The term pd represents the energy flux or transmission
of energy per unit time through a unit area. For a trans-
ducer which radiates acoustic energy outwards in a fluid
medium, the acoustic power of the transducer is given by
the integral

[pee
S

which is evaluated over the exterior surface of the transducer,
the surface element d6 being associated (in direction) with
the outward normal to the surface of the transducer.

The viscous term on the right side of (19) can be re-
written in several different forms depending upon the nature
of the flow. Under the approximation of irrotational flow as
employed in (14) and in view of (18), it can be shown that the
viscous term of the energy equation can be expressed as

Q

(2u, +A, ) é be
X ° St

{e)

2
sel 2 |
ho

Q

Substituting the latter expression in (19) yields the
following energy equation appropriate to conditions of
irrotational acoustic disturbances in a viscous fluid:

2
2 -=|2
pw | 54 (20)

where
3 = ———_ (2hi0)

The latter parameter has the dimensions of time and can
be regarded as a characteristic relaxation time associated
with the damping action of viscosity on acoustic oscillations.

In order to appreciate the full significance of relation
(20), it is instructive to consider the integral of this rela-
tion for a finite fluid volume / bounded by the closed sur-
face S. By employing the divergence theorem, the integral
relation takes the form

Se ag o

where # is the total acoustic energy in the volume /. If the
surface S is not simply connected, then one can regard the
surface integral as a sum of integrals over the pertinent
exterior and interior surfaces. The normal component of
velocity v, in every case is taken positive if the flow is
towards the surface from the fluid volume concerned.

For an inviscid fluid, (22) reduces immediately to

Sool [mee (22a)
Ss

which stipulates simply that the change in acoustic energy
in the volume / equals the net rate of influx of energy
across the closed surface S’ which delineates this volume.
For free vibrations, the acoustic energy level can be
maintained only if there is total reflection of energy at

35

36

the bounding surface, This requires that v, = 0 or that

p = 0 on the surface, a situation which is approached only
if a large difference in acoustic impedance (pc) exists
across the bounding surface.

In the case of the viscous fluid, even if p or v, vanishes
on the bounding surface of the medium, the energy of free
acoustic vibrations will decay at a rate depending upon the
square of the time rate of change of p. If the disturbances
are nearly simple harmonic (except for a slow decay) then

= y® Dp

where w is the frequency (radians/second) and the bar
indicates a time average over an integral number of oscil-
lations, Moreover, the energy # can be considered (on the
average) as half kinetic and half intrinsic. Accordingly,

it follows from (22) that, for total reflection at the bounding
surface S,

ee 2p

ee ae E 22b

<7 Bw ( )
or

we -8 2

Pape” © (23)

If energy does radiate through the bounding surface at
a rate proportional to the mean energy density (#/V/) within
the medium then the damping modulus 8w* in (23) must be
replaced by

(60? +, 7)

where c is the wave speed, 4 the surface area, / the volume
and y; a transmission coefficient dependent upon the imped-
ance mismatch between the medium and its environment,
For small volumes, the second term (radiational loss) will
dominate, while for sufficiently large volumes the viscous
term will ultimately dominate.

The above deductions, which have been obtained ina
somewhat heuristic manner, will be derived more rigorously
in the development which follows.

Effect of Temperature Fluctuations*

The question of temperature fluctuations associated
with acoustic disturbances was avoided at the outset by
adopting relation (3). Actually the equation of state for
any pure substance should involve a third thermodynamic
variable, the absolute temperature 7 being a possible
choice.

Thus formally one should stipulate that for the particu-
lar medium

= G2, 2) (24)
The differential counterpart of this is
1
Bien ee ae (25)

where & is the isothermal compressibility and } the(volu-
metric) coefficient of thermal expansion. Relation (3)
and its differential counterpart (6) can be consistent with

*The following section as well as that on energy considera-
tions was suggested by R. O. Reid.

37

38

(24) only if the change in 7 is proportional to that of P or
that the temperature is constant. The second condition
would require that \, * be identified with the isothermal

compressibility, a deduction which is known a posteriori

to be incorrect for acoustic waves in gases.“ The other
possibility requires that

where 4 is an appropriate coefficient. The three parametric
relations implied by (26) represent the differential equations
for a curve in the p, P, 7 diagram. Clearly this curve must
lie on the equation of state ''surface' given by (24), Except

for this constraint, the curve is arbitrary.

In order to remove the ambiguity regarding the thermo-
dynamic path, it is evident that a further relation between

the thermodynamic variables is needed. Sucha relation
could be provided a priori by stipulating that the path is

adiabatic. This requires a definite relation between P and

T which is given by Kelvin's equation

where ©. is the specific heat at constant pressure. This
implies in turn that 4~* is the coefficient in (27) and that

2
nee Bee aie
pC

*This would lead to the Newtonian expression for the speed

of sound, which is known to be less than the measured
values.

(26)

(27)

(28)

which represents the adiabatic compressibility. For liquids,
the value of X is much greater than the second term in (28)
and hence it makes little difference, in respect to the sound
speed, whether one uses the adiabatic or isothermal com-
pressibility (except in problems involving refraction of
sound rays).

The physical justification of the assumption of adiabatic
changes of state lies largely in the notion that acoustically
induced heat conduction, like viscous generation of heat,
will be of second order compared with the change in intrin-
sic energy which occurs in an acoustic vibration. When
one neglects both viscosity and heat conduction, the resulting
acoustic theory predicts reasonably accurate propagational
speeds. However, no information is obtained regarding
the attenuation which is associated with irreversible proc-
essieis.

A good approximation of the viscous damping is achieved
for acoustic waves by ignoring vorticity, in spite of the fact
that viscosity introduces a flux of vorticity inwards from the
boundaries, if shearing motion occurs at the boundaries.

The approximation is justifiable for acoustic disturbances
since the rate of dilatation is much larger than the vorticity.
A similar sort of approximation can be made in considering
the effect of heat conduction. Although the latter causes a
change in entropy of the medium, the process can be con-
sidered nearly isentropic.

If there exists a small departure of the specific entropy
(ergs/gm °X) from its mean value, then relation (7) should
be replaced by

p=—— n-’ (29)

where 7, is the mean temperature, 7 the anomaly of specific
entropy and \, hasthe meaning implied by (28). For isen-
tropic conditions this obviously reduces to (7) as a special
ease. Let the flux of heat be given by -A V T, where X is

40

the conductivity and 7, the anomaly of temperature from
the mean value 7. The linear approximation™ for the
change in n is accordingly given by

roa] IG 2
= VY op
ot Ode 1

(0)

Now since the process is considered nearly isentropic,

it is reasonable to suppose that (27) can be employed as an
approximation for ie in terms of p, with 7 and p taken as
their mean values and 7, and p identified with d7 and dP
respectively. This approximation renders (30) in the form

2
op = ne Fork 20 = X 30
ot OG CXC
Oe
where « is the thermal diffusivity (X/9, 0 Relation (32)

implies that the plot of p versus ® en upon the char-
acteristics of the acoustic waves (e.g., the frequency) as
well as the properties of the medium. Moreover, p and ®
are not exactly in phase as implied by (7).

If (32) is employed in place of (7) then the wave equa-
tion (in terms of p) takes the form

nD° T
io) 5 B+ g op
Cp 3t

*There will exist a steady increase of n with time associ-
ated with the attenuation; however, this is of second order.

op
Ones

= 2
p aly wi

(30)

(31)

(32)

(33)

Thus the attenuation factor 8 associated with viscosity is
supplemented by an additional term related to the thermal
conductivity of the medium. For water the magnitude of
the attenuation associated with conductivity is about 1/10
of that due to viscosity under normal conditions. In view
of this, the neglect of conductivity seems to be justifiable.
However, for greater accuracy, the effect of conduction
can be included within the framework of the viscous theory
alone by employing the quantity

KD° To

“p

(Au, sr oF
as an effective viscosity in place of (2u, +), ).

SOME SPECIAL SOLUTIONS OF THE WAVE
EQUATION

Plane-Attenuated Sound Wave

The particle displacement for a simple harmonic,
plane sound wave, ina fluid with viscous losses and prog-
ressing in the x-direction, can be written

s = 8, Bie n8)) Cy s5

where j =/-1, & is the wave number, w the frequency and
8, is the amplitude of excursion of the fluid particles. The
last term containing a; is the attenuation term. This can
be included in the propagation constant by allowing % to
assume complex values. The complex wave number 4* is
accordingly

[5 IG = sl Cb

(34)

(34)

41

42

Hence, (33) takes the form

Ss Sp ed Wt -A*x)

(35)
For simple harmonic waves it follows that
Ree jws (36)
and therefore the scalar counterpart of (15) takes the form
as = 2 (1+ jw8 ) = (37)

where 8 is the viscous time factor defined by (21), Itis
convenient to introduce a complex sound speed

N al,
ox -(=) (1+.jw8)* (38)

and thereby recover the canonical form

DIF

=e (39)

CS fee = 0 (40)
where w is regarded as real. Thus the complex wave number
is given by
MACE “4
is 2 fite. = 0 eR (1+ jwB ) (41)
[o)

Expansion of the term on the right as a power series in w8
and separating real and imaginary parts gives

(42)

where ¢ is the wave speed / i, [Or . If w8 is much less than
unity, then the above relations can be approximated by

(43)

These approximations apply as long as the viscous compo-
nents of the stress tensor (11) are small compared with the
elastic part. Thus, to this order of approximation there

is no influence of viscosity on the wave number-frequency
relation, while the spatial attenuation coefficient is directly
proportional to the viscosity. The above attenuation factor
differs from that given in the energy considerations in two
respects. First, the factor c enters since ay is an attenua-
tion coefficient per unit length along the wave ray. Second,
the factor = enters since a; pertains to the amplitude atten-
uation coefficient rather than the energy attenuation. Bear-
ing this in mind, the result (43) is entirely consistent with
the earlier deductions.

If one takes as a lower limit, \, = -2u, / 3, waem te is
readily shown that

22
a as 3 (44)

which is the result obtained by Stokes. However, for some
fluids the numerical factor in (44) may be significantly
greater than %, in view of conditions (12).

43

44

Cavity Theory - Inviscid Fluid

The resonators used for the measurements have been
discussed in a previous section. For the present develop-
ment it will suffice to know that they are made of stainless
steel sheets in the shape of a square-cornered tank. In
the following computations an orthogonal Cartesian coordi-
nate system is employed with the three axes labeled (~, ,

%z, “,). The tank is taken in the positive octant with a
corner in (0, 0, 0). The three coordinate planes will then
coincide with three sides of the tank. The corner diagonally
opposite (0, 0, 0) is labeled (1,, 4,, 2, Ne

The wave equation for loss free fluids is given by (15a).
A possible solution of this wave equation is given by

} j -Ob
4 74, cosh (K,x,+ ;) sinh Kjx%7,* 7) sinh (Ay, x7 + Vz) € (45)

where the subscripts t, j, i are cyclic. The value of the
subscript designates the particular component, in the same
sense as the coordinate designation. Each of the quantities
‘Ala Ian the and 2 can take on complex values.

Thus

Ky = ag - Shi
(46)
Q = at = jw
etc. The wave equation is satisfied provided that
sai
a 2 i 2
(=e Ks (47)
1

In regard to boundary conditions, two simple cases
will be considered first. These correspond to total reflec-
tion of energy at the boundaries of the tank. A more gen-
eral boundary condition is considered later,

Rectangular Cavity with Hard Walls

If all of the walls are considered absolutely rigid, as
a limiting case, then (with i = 1, 2, 3)

where 4 , 43,4, denote the dimensions of the cavity.
Relations (45) will satisfy the six boundary conditions
stipulated by (48) provided that

cosh ¥; = 0

and

cosh (Kt; +3) = 0

The possible roots of these relations yield

and

LO eee

(48)

(49)

(50)

46

where n; and m; can take on any integer value independently.
The set nan, Mes 5 Me ) designates the particular mode of
oscillation; a fundamental mode of oscillation corresponds
to the particular set (1, 1, 1). The second of relations (50)
demands that 4; be pure imaginary and hence

as = 0
TT
Moreover, for an inviscid fluid c is real and hence o® must
be negative so that the temporal attenuation factor a4 is
zero and the eigenfrequencies of the system are given by
$
Si ey) Cay ene Oe in Coa en (52)
Rectangular Cavity with Soft Walls
The opposite extreme of the previous case is a cavity
for which complete pressure release occurs at the bounda-
ries. The pressure is readily evaluated in terms of the
relation
p= oko s (53)
if the displacement components & , Sg, S$, are known.
Using (45) the relation for p is
3 3
p= fal) soft JIL sinh (ii; 20, ty, )e (54)
1 aL

Now if p> 0! on the’ six planes 7) = 0) 275) then itis required
that

sinh |. = 0
t
(55)
sinh (¥,t; + ,) =70

LOGEC —a Ae Sp monic inetuicneydelds Gi > O) eliavel

lig = n,(m/L4) (56)

Wineme 7} = 05 1, As ooo03 Elbe IESE ONE Ol Ne 9, ChikiSies) ieOrEA
zero for non-trivial solutions. The relation for the wave
numbers and eigenfrequencies are thus formally the same
as before.

It is possible, of course, to have a mixed boundary
condition for which the displacement vanishes on some of
the walls and p vanishes on the others,

So far the amplitude factors 4; have been regarded as

arbitrary. However, since the vector displacement must
satisfy the basic relation

2

s Re
Po aa ow (57)
it follows that (45) will be valid only if

Al =) ISIC (58)

where 4 is a single arbitrary constant. Accordingly the

47

48

relation for p takes the form

3

=©
—_ 2 7
DO) 18. 0) sinh (Kix, *+V )e

al

v (59)

where (47) and the identity c = /\,/9, is employed.

Rectangular Cavity with Partially Yielding Walls

For the more general case, the boundary condition
can be expressed in terms of the unit area acoustic imped-
ances of the walls. The boundary impedance 4 is expressed

by

ZR eae (60)

where v, is the normal component of velocity into the
boundary from the interior of the system. It is understood
that the various quantities are evaluated at the boundary
concerned, Using the relations for displacement and
pressure given by (45) and (59) it follows that at x, = 0

s Q
Z, (Q)) = (a. 7 tanh V,
al
or
— 1 Z, (OK,
ys = -tanh Soran (61)

On the other hand, application of (60) at the boundary
Gr me ypelcls

1

Wp (Oey

je oer tanh (eta) (62)
al

The parameter |, can be eliminated between the latter two
equations by use of the identity

tanh a + tanh 8

paral CRYO) Se Se
SULIT (ENS) ra aan a saan

The resulting relation is

tannin SSPE PSE Te (63)
aL al

which represents an implicit relation for Ky in terms of ()
and the boundary impedances 7, and ee 3 This relation can
be augmented by the corresponding Hes equations for the
coordinate directions 2 and 3. These three equations,
together with

BS Ao 2 2 2
OP =e (x, ae 21°) (64)

allow an evaluation of the 4; and © in terms of the boundary
impedances.

The real parts of 0 and 4; correspond to the temporal
and spatial attenuation coefficients and these are regarded

as small compared to the imaginary parts. Consequently
the approximations

a 2 °
iit = 185 Mig) Cg (hs

Q? = -w* tf 2 Wie

49

50

are quite adequate. Making use of these in (64) leads to
the approximate relations

(65)

(b) ae Se ) yt

It will be noted that the temporal decay factor a+ in the
present case is related entirely to energy leakage through
the walls of the tank. It will be shown that this loss is
about tenfold greater than the viscous losses when the
cavity is filled with pure, gas-free water. However, it is
much less than the attenuation as measured with most of
the sea water samples.

It remains to consider the transcendental relations of
the type (63). The limiting situations for which 7-0 and
Z+-0 both lead to

tanh EO, = 0

which corresponds to the special case of total reflection at
the walls and zero attenuation, Actually 7 = 0 corresponds
to the extreme soft wall condition and 7~* = 0 to the extreme
hard wall discussed in the previous sections.

For the case of compliant walls for which |Z| is small
compared with the characteristic impedance of the fluid
0,¢, the quadratic term in Z can be neglected in (63)giving

tanh Kt, = — (Z 49) (66)

Moreover, since the departure of LG from the value ii (0)

for zero boundary impedance is small, the expansion of (66)
leads to the approximation

_*, (0) = t
(AK, )e = a (4 +2 ‘|

where AX, is the departure from X, (0) = Jit W ho lati
relation can also be expressed as

Ea cone
(AK, )4 = satay Z| (67)

where #, (0) and w(0) are the zero order estimates of the
wave number (for direction 1) and the frequency under the
condition of zero boundary impedance (hence no loss).

The unit area acoustic impedances of the boundaries
can be estimated by assuming that the walls behave like a
flexible rectangular membrane with the edges rigidly
clamped. The tank employed in the experimental work
reported here employs stainless steel sheets of such
dimensions that the free resonance frequencies of the walls
are much less than the resonance frequencies correspond-
ing to the cavity modes, The load of the wall can there-
fore be considered as a combination of a resistive and
mass load such that

Bo ears a (0) 16 (68)

where 7 is the loss resistance per unit area and m is the
mass per unit area.

Setting (2, +24) =r, + jw m, im relation (67) and
separating the real and imaginary parts yields

ol

tell yn Gis p w(O)e 2
(69)
-i_ (0)
b) Ak =———m
( 1 0 tb, al
Qa
Relation (69a) shows that the spatial attenuation factor
a, is directly proportional to the resistive factor r,, the
proportionality factor being dependent upon the resonant
mode and cavity dimensions. Using ‘, (0) =n, T/L, yields
T n,
— 70
Oo p, 0 (0) ee rs ( )
where 7, is the integer characterizing the cavity mode in
the x, direction.
Relation (69b) expresses the change in %, (0) due to the
fact that the wall takes part in the vibration. The equation
can be expressed in the form
k, (0)
Nis earn Os (71)
al
where At, =m, /0, which represents the thickness of a
layer of the contained fluid, which has the equivalent mass
per unit area as the stainless steel sheet. Using the pre-
vious expression for #, (0), the corrected 4, is then given
by
n,7
i SS (72)
Be) ea ar ab,

in which the conventional relation for 4, is employed with
the actual 4, augmented by the correction At,.

52

An example calculation of the fundamental mode of
vibration for cavity No. 2 will serve as an illustration of
the application of the above relations. The inside dimen-
sions of the cavity are 12 x 8 x 10 inches, the latter num-
ber being the depth. The cavity walls are stainless steel
sheets of 24-mil thickness. Using a specific gravity of
stainless steel equal to 7.9, the correction A 4% for each
wall is 190 mils or 0.19 inch.

The cavity is open on the top and was employed with
the water level always 0.25 inch below the rim. The effec-
tive cavity dimensions (using \% = 0 for the free surface)
are accordingly 12.38 x 8. 38 x 9.94 inches. For the
fundamental mode of vibration (1,, n,, n, being 1, 1, 1
respectively), the effective wave numbers for the cavity
are accordingly

af ep AU delet
One WD We ia, aaa OO

in units of radians/inch. This yields

2 = 5 NT egal ae
ce 0 SD) Ti 5. 69

Bi) ©

and from the first of relations (66)

me
Oi 2a eaters
or
f = 0.08787 ¢

For the temperature and salinity conditions of 60°F and
34 gms/kg, the value of c from standard tables is 4935 ft/sec

54

and hence

f = 5204 c/s

This value is in good agreement with the experimental
results for cavity No. 2.

An estimate of the acoustic loss suffered by radiation
into the surrounding air can be determined by assuming

OEM WS tbe Iorvel wig 4} ACOUSHIE Omens (lyon My = iy = Fa = SE
acoustic ohms). The generalized counterpart of (70) is
a+ = iN ey aces
Y p, ue @
Combining this with relations (65) yields
Le ey
(eG =
A iG Mee
it i =
ie

The latter quantity gives the temporal attenuation rate in
neper/sec. The corresponding spatial attenuation factor
Ct St /c, Using the above relations with the assumed
value of 7 and the previous values of k, gives a_ = 1. 66

neper/kyd, which corresponds to 19.2 db/kyd.

This is of the same order of magnitude as the losses
determined experimentally for the cavity filled with pure
water. The viscous attenuation due to the pure water, for
the same size cavity, is only of the order of 0.5 db/kyd as
estimated from the relations in the following section.

Cavity with Internal Losses plus Radiation

The theory of a resonant cavity developed in the pre-
ceding is only applicable when the fluid is essentially

(73)

without frictional losses. An approximate solution can be
obtained, if it is assumed that the losses are low. The
cavity is, in other words, strongly resonant. The highest
attenuation that can be measured with the present equip-
ment corresponds to a cavity Q of 470, and the 2 with clean
water in the cavity is of the order of 7500, so the losses
are actually quite small when compared with electrical
resonant circuits.

The wave equation (15) for a viscous fluid is employed
in the present analysis. The solution for the components
of displacement as given by (45) is still pertinent; however,
the expressions for the complex wave numbers and fre-
quency will differ from the previous case. Substitution of
(45) in (15) yields

as

where c =/hg/p and B = (2u, +A, )/Ag as employed pre-
viously.

The evaluation of the characteristic equations is
formally the same as in the previous case except that ¢c*
is to be replaced by c* (1-8) in relation (64), Thus the
counterpart of (64) is

3
Se (ae) ie

1

The auxiliary relations relating 4;, © and the boundary
impedances are formally the same as (66), except that the
K, and ) are now dependent upon 8 as well as the boundary
impedances. However, the form of (66) demands that the
first order effect of attenuation on A; is due to the boundary
impedance. Any effect of 8 will be of second order in
respect toX;. Consequently relations (69) and their
counterparts for coordinates x, and x, are still valid.

(74)

(75)

55

56

The primary effect of the viscous parameter 8 enters
in the evaluation of az. By separating relation (75) into its
real and imaginary parts and neglecting terms involving
Bae cia and a,” the following approximate relations are
obtained:

3
(ei) OF = ) k,*
1

Bw®

DoH

oe :
(b) at = Ry ) ashy a
1

where &, anda, (i=1, 2, 3) are given by relations of type
(70) and (72) with appropriate changes in subscripts.
Specifically, the allowable values of %; depend only on the
cavity geometry and the equivalent thickness of the walls.
The values of a1; depend upon the cavity geometry and the
real part of the boundary impedance.

Relations (76) differ from (65) only in the addition of
the viscous attenuation term toa;. It will be noted that
the resonant frequency is not influenced by viscosity in the
present approximation. Moreover, relation (76b) shows
that the effects of internal (viscous) losses are additive in
respect to the attenuation coefficient a+. In other words
there is no first order coupling of the internal and external
(radiational) attenuation phenomena. This is an important
result, since it allows the possibility of evaluating the
internal losses in a fluid sample by measuring the total
loss and subtracting the radiational loss. The latter is
determined experimentally by the calibration tests with a
virtually inviscid fluid (air-free, pure water).

It will be noted that relation (76b) is consistent with
the result anticipated in the considerations of the acoustic
energy decay (page 36). It was shown by a somewhat
heuristic chain of reasoning that the exponential decay
factor (per unit time) for energy could be expressed in

(76)

the form BA
Y TT + 8B w*

where y is a transmission coefficient which gives the frac-
tion of incident energy flux which leaks through the boundary,
The ratio //A defines a characteristic dimension / for the
cavity (V being the volume and 4 the surface area), Thus

the above relation can be written

Bearing in mind that this should correspond to 2 a+, where
az is the amplitude decay modulus, it is seen that this is
consistent with (76b) provided that

V=2L— \ ap, (77)

The quantity on the right can be shown to be directly pro-
portional to the impedance ratio r/p,e as should be expected
provided that the ratio is suitably small. The factor of
proportionality depends upon the shape of the cavity and the
particular mode of oscillation.

Practical Relations Implied by the Theory

The result (76b) for the temporal decay factor pertinent
to a resonant cavity consists of two parts

(a4) = (ag) + Ge), (78)

total

where (a+), is the attenuation factor associated with the
particular properties of the cavity and (a+)_ represents
excess attenuation associated with the contained fluid.

ot

58

The above factors, as they stand, represent attenuation
in units of neper per unit time. In measurements the rever-
beration time is determined. This represents the time that
it takes the sound level to die down 60 db, measured in
seconds. Thus if t, denotes the reverberation time as
measured with the cavity filled with pure water and 7¢,.4.)
the reverberation time as measured with a sea water
sample, then

1 1
(az) = 60 | -4| (79)
vex ttotal t,

the units being db/sec.

In practice it is the attenuation per unit length along
the propagation path which is desired. From the considera-
tions of the plane sound waves it is evident that the conver-
sion to the spatial attenuation factor, a;,, for the fluid is
simply a, = a+/c. For sea water of 34 gms/kg salinity at
60°F ec = 1.645 kyd/sec; hence the excess attenuation
expressed as db per kyd is given by

az, = 36.5 | | (80)

This relation has been used in the evaluation of the data.

It is theoretically possible to determine the sound
velocity from the resonance frequency, but the geometrical
dimensions of the cavity are not very stable, since it is
made of thin sheet metal with welded corners. The accu-
racy of the frequency measurements were only as good as
can be obtained by reading the scale on the tone generator,
and the determination of c on this basis is not precise.
The temperature range encountered was from about 10°C
to 20°C, and the sound velocity changes about 2 per cent
due to this temperature variation, but it is not known how
the presence of plankton may change c. It was therefore
decided to use a constant value for c, to be able to give a;

in units that can be easily interpreted, instead of db/sec.

Some Final Remarks Regarding the Theory

The most general boundary conditions considered
required the determination of p and v, at the boundary.
The pressure p was determined from equation la, with
the simplificationthatu, and\, were zero. These terms,
however, can still be disregarded as long as the viscous
part of the pressure tensor is insignificant compared with
the elastic part. This same approximation has been used
in the preceding analyses.

In the more general case, the velocity at the boundary
must now satisfy one further condition, beyond the condi-
tion imposed by the acoustical impedance of the boundary.
This condition is that the component of the velocity tangential
to the boundary must vanish. To satisfy this boundary condi-
tion, there must exist a transverse frictional wave in the
immediate neighborhood of the boundary, since such waves
experience an extremely high attenuation with distance.

This will add to the boundary losses, but these are deter-
mined experimentally in any event. It must be borne in
mind, however, that the reference liquid should have approx-
imately the same shear coefficient U, as the liquid under
investigation, and approximately the same sound velocity c.

59

60

V. EXPERIMENTAL PROGRAM

OBSERVATIONAL PLATFORM

The equipment described in Section III requires a
reasonably stable supporting surface to work properly. It
would not be possible to use this equipment on a ship. On
the other hand, the time delay suffered if samples were
taken at sea and brought back to the laboratory would be
highly undesirable.

The NEL Oceanographic Research Tower off Mission
Beach, San Diego, is well suited for cavity measurements,
and a program was therefore initiated to develop equipment
suited for this research facility.*°’*?

The tower is situated 0.8 nautical mile off the shore
in approximately 59 feet of water. The shoreline off
Mission Beach is completely unobstructed from the influ-
ences of the open ocean, The influence from man-made
contamination is therefore at a minimum. The main deck
of the platform is 24 feet above the water level, and acts
as foundation for an instrument house of 13 by 13 feet in
plan section. This leaves an outside passageway around
the house about 4s feet wide. Space inside the house
was not available, so the instrumentation had to be placed
outside in the passageway. The tower is quite stable, but
the action of waves on the tower supports causes a slight
oscillation. The main influence was from the wind, which
often would spoil a reverberation measurement by creating
ripples on the surface of the water in the cavity. Measure-
ments were impossible whenever very 'noisy'' equipment
was in use such as compressors and heavy winches, but
the cavities worked well when the conditions were favorable.

The present field data collection began May 1960 and
continued through the summer until the tower was closed
down in September 1960.

The instrumentation on the tower permits a large num-
ber of oceanographic variables to be measured. It is neither
reasonable nor practically possible to try to correlate every
available parameter with the attenuation measurements. In
the present experiments the plankton count and oxygen con-
tent of the water are regarded as the primary factors. More-
over, these variables together with temperature and salt
content describe the state of the sample (at atmospheric
pressure) in a direct manner, independent of the source and
procedure of sampling.

The viscosity coefficients of the water partially deter-
mine the attenuation. It is known that these coefficients are
functions of temperature and salinity for clean sea water,
but the effect is very small compared with the attenuation
measurable with the cavity equipment. The temperature
was measured for every sample, but the salinity was not
considered. The salinity does not change appreciably
around the tower, since there are no fresh-water outflows
close by and there is fairly easy exchange with the open
ocean water.

Samples were taken anywhere from the surface to the
bottom. Some samples would therefore have undergone
considerable pressure changes before the measurements.
The pressure change would not have any appreciable effect
on the sea water itself, but it can well have a marked effect
on the suspended particulate matteras well as on the dis-
solved gases. The sampling depth was therefore recorded
for all samples. However, it should be borne in mind that
the plankton counts and oxygen content were determined for
the sample at atmospheric pressure, The in situ conditions
of these variables and the quantitative influence of pressure
are a separate problem and beyond the scope of the present
study.

The tower is equipped with a hydrophotometer and
measurements are recorded as the per cent transmission
of that in pure distilled water. The path length of this
instrument is meter. Sea water samples at depths be-
tween surface and bottom have an up-and-down motion due

61

62

to surface waves and internal waves. The hydrophotometer
reading will likewise change considerably, and it is not
possible to ascribe a very reliable light transmission read-
ing to any particular sample. The readings on the hydro-
photometer are a function of particulate matter (including
plankton) and air bubbles suspended in the measuring path
of the light beam, but there are no functional expressions
available describing this relationship. The hydrophotom-
eter readings were recorded when available, However,
these measurements are to be regarded purely in the nature
of qualitative information about the general nature of the

in situ conditions of the region being sampled.

As mentioned earlier, the greatest emphasis in the
analysis is placed on the dependence of attenuation on the
plankton count and oxygen content as measured from the
sample.

EXPERIMENTAL PROCEDURE

Sampling Methods

Surface samples were at first obtained by using a
bucket at the end of a line. Deeper water was obtained by
using standard Nansen bottles. However, it soon became
clear that either method was highly unsatisfactory. It was
almost impossible to fill a cavity from the bucket without
working a large amount of air bubbles into the water, and
it was very slow and cumbersome to obtain enough water
with Nansen bottles.

A small submersible centrifugal pump was then pro-
cured and mounted on the cart carrying the hydrophotom-
eter. A hose carried the water up to the main deck
where the cavities were. The pressure in the discharge
line from the pump was approximately 12 psi higher than
the hydrostatic pressure at the level from which the sample
was taken. Cavitation was therefore not likely to occur.

The discharge of the hose was always kept below the water
level in the cavity, and no bubbles were ever observed
coming out of the hose. This sampling method was used
exclusively in all of the subsequent work, but it was felt
that a less ''violent'’ sampling method would be preferable.

Selection of Cavity for Measurements

Five cavities were available for the data collection, as
mentioned earlier. It became apparent however, that only
cavity No. 2 was suited for measurements on the tower.
Cavity No. 1 was too big for one person to handle, consider-
ing the rather limited space available. The difficulties
encountered when cleaning the walls and stirring the content
to insure homogeneity were also considerable. The cavities
smaller than cavity No. 2 were likewise difficult to clean
because of their small size and also because the walls were
not sufficiently flexible.

The influence of the transducers was also marked for
cavities 3, 4, and 5 probably partly because of the greater
transducer dimensions relative to the wavelength for the
cavity. This is also partly because the transducers could
not be decoupled to as great an extent as with cavity No. 2,
since the thickness of the stainless steel sheet corresponds
to a greater fraction of a wavelength.

It was decided to use cavity No. 2 exclusively for the
data collection. This implies that it is not possible to
deduce very much about the variation of the attenuation
with frequency. However, even if the same sample had
been used in all five cavities, the frequency range is not
great, Moreover, transferring the sample from one cavity
to another would undoubtedly have a very marked influence
on the properties of the sea water. An attempt was made,
however, to get data at different frequencies by measuring
the reverberation time of not only the fundamental mode,
but also two higher modes. If the coordinate axes 1, 2, 3
are identified with length, width, and depth of the cavity,
then the modes measured can be labelled:

63

64

(ake 30)
(RD ao a) = he By i)
(29h leary)

Before a consistent data collection could be undertaken,
it was necessary to find a method for keeping the side walls
of the cavity clean. Of the several methods which were
tried, a squeegee mounted on a thin tube appeared the most
satisfactory. This device made it possible to insure abso-
lutely clean walls, with the only drawback that of the un-
avoidable disturbance in the water.

Evaluation of Oxygen and Plankton

Special precautions were taken in drawing plankton and
oxygen samples from the water in the cavity. A 50-ml
pipette with the tip removed was used to draw the samples,
and it was rinsed clean and thoroughly wetted before each
sampling. The plankton was drawn from the center of the
cavity, where its influence on the acoustic properties of
the sea water would be most pronounced. To avoid the
inclusion of water too far removed from the center, only
about 1 ounce was drawn. The sample was immediately
preserved by adding 2 cc of 40 per cent formaldehyde solu-
tion to the sampling jar.

The plankton count was obtained by shaking the sample
thoroughly, filling a Sedgwick-Rafter counting cell with
the sample, and counting the organisms by 100x in a micro-
scope.

In drawing oxygen samples, great care was taken not
to work bubbles into the water at any time. The pipette
was allowed to fill by itself without suction, and the tip
was touched to the inside of the 09 bottle held at a slant,
so that the pipette would empty slowly and smoothly. The
preservation and titration of the sample followed the stand-
ard procedure used in connection with Winkler determina-
tion of oxygen. Standard 150-ml oxygen sample bottles
with ground glass stoppers were used throughout.

The per cent saturation of oxygen was computed in
accordance with reference 42.

Outline of Experiment Procedure

The procedure for taking a measurement can be
summarized as follows:

(1) Fill cavity with sample.
(2) Wipe side walls and bottom with squeegee.

(3) Tune generator frequency and the tuning circuit to
obtain maximum reading on vacuum tube voltmeter (fig. 4B).

(4) Decouple transducers until barely full reading is
obtained on recorder, and with VTVM sensitivity as high
as possible.

(5) Start recorder at paper speed chosen (100 mm/sec
or 30 mm/sec), and interrupt power from the signal gen-
erator.

(6) Tune generator to the two higher modes, repeating
SOS 3, 4, Bo

(7) Take plankton and oxygen samples, and measure
temperature.

(8) Record sampling depth and hydrophotometer reading.

Calibration of Cavity

The cavities were calibrated before they were used on
the tower and again after the data collection with no change
in their acoustic properties indicated. Visual inspection
did not disclose any change in the cavities apart from a
slight surface discoloration of the stainless steel sheets.

The calibration was done with the cavities filled with
distilled water. The temperature was brought down early

65

66

in the morning by adding sufficient distilled water ice, and
the calibration was performed while the cavity was heating
up. The cavity walls were carefully wiped whenever a
desired temperature was reached, and a series of rever-
beration curves were then taken for different distances, JD,
between the rim of the cavity and the surface of the water.
A series at any one temperature always began with a full
cavity, and the walls were wiped only at the beginning of a
series. The difference in the water temperature between
the beginning and the end of a series was always less than
1°C, and the temperatures given are the average of the
series.

The calibration data are given in figures 9to 14. The
attenuation coefficient for the cavity, a,, is plotted as a
function of D and of the resonance frequency Js Figures 9
and 10 apply to mode 1-1-1, or the fundamental mode.
Figures 11 and 12 correspond to mode 2-1-1 while figures
13 and 14 pertain to mode 1-2-1. Figure 9 shows «, (D) for
different temperatures. There is a discouragingly large
spread between the different points, and it appears that the
case D equal to = inch below the rim is particularly poor.
Replotting the same points, but this time using the resonance
frequency as the independent variable, led to figure 10. The
scatter has to a great extent disappeared. An investigation
showed the cause of this behavior to rest with the spacing
between the cavity and the supporting table. The cavity is
approximately 14 mm above the table, and a quarter wave-
length in air is about 18 mm at 5000 c/s. The distance
between the table and the cavity could be increased by block-
ing the cavity supports up, and decreased by inserting a Z
inch aluminum plate between the cavity and the table. It
was found that the distance actually improved the cavity Q.
Moving the supports as far as ¢ inch away from the corners
did not affect the losses of the cavity, however.

In regard to the calibration data for modes 2-1-1 and
1-2-1 (figs, 11-14), the attenuation due to the cavity is
generally greater than that associated with the fundamental
mode (at least for frequencies greater than 5.2 kc/s).

In addition, the attenuation varies greatly with frequency,

50

4

50
dyD

40

30

20

fe)

@,\D B/KYD
x

a

OH

ao (TEMP, DEPTH) ]
MODE I-I-l

O+ Dx
BR

a

°

a

5/16 10/16 ~—D (IN)
A ! 1 ! 1 t | n i

to53.6 to 54.4

ne
F
|)
ik
|

x

a@,(TEMP., FREQ.)

MODE I-I-| x 8.5°C
ao 10.8°C
+ 14.5°C

© 18.5°C |

f,(KC/S)
5.1 5.2 5.3 5.4 5.5

Figure 9. Calibration tests for mode
1-1-1, showing attenuation coefficient
(a,) vs. distance (D) between free sur-
face and cavity sill, using pure water
at four different temperatures as in-
dicated.

Figure 10. Calibration tests for mode
1-1-1, showing attenuation coefficient
of figure 9 replotted vs. the resonant
frequency (f,).

67

1 ame =
a, 0B/KYD MODE 2-I-I
TEMP. 18°C
40 —|
30} |}
20 =]
1O/e =
Figure 11. Calibration tests for mode
2-1-1, showing attenuation coefficient
avi) vs. D, using pure water at 18°C (no
o 2/8 aya 6/8 2/e__4/8 correction for attenuation by the water).
50r
MODE 2-I-|
Per DBAKYD TEMP. 18°C
40 =|
30;- =
aula 4
10 I
L Figure 12. Calibration tests for mode
2-1-1, showing attenuation coefficients
Ht te hi of figure 11 plotted vs. resonant fre-
0 | | | quency f,. Temperature 18°C.

68

SOT
MODE |!-2-1
d,~DB/KYD
on =|
of mal
20- us
10 =
R Figure 13. Calibration tests for mode
1-2-1,showing attenuation coefficient vs.
ee D, using pure water at 19°C (no correc-
DIN) °
i pyeteeavele eal | _| tion for attenuation by the water).
St)
W4o,DB/KYD MODE |-2-I
TEMP. 19°C

Figure 14. Calibration tests for mode
DISS) 1-2-1 showing attenuation coefficients
i of figure 13 vs. resonant frequency.

69

70

particularly for mode 2-1-1. The data accumulated for
these two modes have therefore not been corrected for the
loss in the cavity itself.

The block diagram utilized for the calibrations and the
captured bubble measurements were the same as shown in
figure 4B, except that a quartz crystal controlled electronic
counter was connected in parallel with the output terminals
of the generator. The frequency was therefore known with
an accuracy of +1 c/s. The circuit used for the measure-
ments of the sea water samples is identical to that shown
in figure 4B.

The glass rod shown in figure 2 is very useful as a
probe of cavity modes. Any particular mode can be identi-
fied by passing a captured bubble through the cavity. The
reading on the VI VM instrument will be a minimum when
the position of the bubble coincides with a pressure maxi-
mum in the cavity. This method was always used to identify
the cavity modes, whenever it appeared necessary. The
mode with the lowest resonance frequency is of course
always mode 1-1-1.

A few measurements on a captured air bubble are
included as an illustration of the capability of the cavity
used for the data collection. The original reverberation
curves obtained from the Bruel and Kjaer recorder are
shown in figures 5 to 7, and the results(discussed in a
later section)are plotted in figure 15. It should be noted
in passing that the diameter of the bubble for each test was
estimated by comparison with the divisions of a metric
scale, the bubble being held by the glass rod. The accu-
racy is accordingly rather poor.

Character of the Reverberation Curves

The reverberation curves do sometime show evidence
of ''double'' moding. The reverberation level diminishes
through the first 10-15 db at a faster rate than encountered

50

BUBBLE MEASUREMENT
MODE |-I-1
f=MEASURING FREQUENCY
WITHOUT GLASSROD f,= 5126 C/S
BATS @,, DB/KYD —

30 f= 5154 DIAMETER OF BUBBLE Z|

SS RESONANT AT 5100 C/s
\4
x
“= 5139
20 =
10 =|
£25132
/
f= 5127 y BUBBLE
x DIAMETER
1 2 3 (MM)
0 | | =

Figure 15. Excess attenuation coeffi-
cient for mode 1-1-1 due to presence
of air bubble vs. bubble diameter
(using data of fig. 5, 6, 7).

later on. If the output from the receiving hydrophone is
recorded as a function of frequency when a double mode
prevails, it is always found that there are two resonance
peaks close together in frequency. The peak with the lower
level is always narrower in bandwidth than the peak with
the higher level. This would imply that the lower peak
corresponds to a normal mode more loosely coupled to the
transducers, but with a smaller damping than the higher
peak. It was usually possible to eliminate the mode with
higher damping by altering the effective coupling between
the transducers and the cavity. The reverberation time
measured on the mode with lower damping does not change
by this procedure. The mode with the longest reverbera-
tion time was therefore used in the few cases where the
above corrective method did not work. However this was

71

12

done only if the reverberation trace displayed a straight
part of at least 25 db. Otherwise it could not, with cer-
tainty, be differentiated from the part where the reverbera-
tion curve merges into the noise level.

Possibility of Cavitation

The maximum sound pressure level in the cavity is not
known, but it will be a function of the cavity Q, The highest
sound pressure is found in the center when the 1-1-1 mode
is utilized, and in the corresponding points for the higher
modes, The possibility therefore exists that the water may
be cavitating in the pressure maximum or maxima, The
earphones (fig. 4B) were therefore always used when a
measurement was taken, and it was never possible to detect
any indication that cavitation occurred. Reverberation
curves taken in rapid succession always gave essentially
the same result, and they were always straight with the
above mentioned exception. It was concluded that the
results were not explainable as being caused by cavitation.

Anomalous Observation

A peculiar phenomenon occurred twice during the data-
taking period. The cavity had been filled with plankton-rich
water and left standing. No measurements were completed
because a brisk wind created fairly large ripples on the free
water surface and made the water in the cavity turn over
slowly. After 5 to 10 minutes the plankton had disappeared
from the water, and a few strings of gelatinous matter,

3-5 mm in diameter and 20-50 mm long, were found floating
in the surface. The gelatinous matter had a very large
number of air bubbles embedded in it, and later micro-
scopic investigations showed it to contain large quantities

of chlorophyll. This has some bearing on a possible source
of bubbles which is explored in the discussion to follow.

DISCUSSION OF RESULTS
Accuracy of Attenuation Coefficients

The greater part of the data, especially the part which
showed large attenuation, was obtained at frequencies above
5.1 kc/s, and the changé in cavity loss is fairly small in
this range. The few points taken at frequencies lower than
5.1 kc/s were obtained for the cold oceanic water, and
these samples had a very low attenuation.

The values of the attenuation rate indicated in the cali-
bration curve of figure 10 were computed from the relation

8655
Coane eae db/kyd

The reverberation time t was obtained from recordings

like those shown in figures 5 to 7 by matching a straight
black hairline, engraved in a piece of clear plastic, with
the reverberation curve. By having the curves read several
times by different observers, it was found that the accuracy
with which the reverberation time could be read was:

ne

WV

<
At = + 0,02 sec for ¢ = 1 sec ora, = 36.5 db/kyd

and

At = + 0.05 sec for ¢ = 3 sec or a, = 12 db/kyd

The relationship between At and Aa, can be obtained by
differentiation of the above relation fora,. This yields

CS

74

This relation gives the following:

Te

Aa, = + 0.2 db/kyd for a, = 12 db/kyd

&

u?

I+

1 db/kyd for a, = 40 db/kyd
J k y

The

I+

20 db/kyd for a, = 200 db/kyd

The above computation considers only the accuracy
with which the reverberation curves can be read. Spurious
resonances caused by inhomogeneities in the cavity equip-
ment or in the liquid sample will give rise to additional
inaccuracies in the measurements. The encircled points
in figures 9 and 10, marked 18.5°C, were recorded on
two different days, and show that the measurements can
be repeated within a few db, at least in the part of the
curve of interest for the data reduction. Inspection of
figure 10 shows that the measured points are essentially
contained in the interval Aa, = 45 db/kyd for the part of
the calibration curve above 5.1 kc/s, which is the part of
greatest interest for the data.

The excess attenuation, Xo, is computed as the differ-
ence between a and a_, and Aa 2 must therefore be
larger than Aa,. It has been assumed that excess attenua-
tion is present in a sample, when the measured value for
Ao, is greater than 10 db/kyd.

Remarks Concerning the Bubble Measurements

The original measurements on a captured bubble are
shown in figures 5 to 7. The transducers were adjusted
with no glass rod in the cavity and not changed during the
measurements, The value for a obtained by this setting
was used as the value of Oo and the excess attenuation,

Oey,» Was determined on this basis. The attenuation with
the flooded glass rod suspended in the center of the cavity
was the same as without the glass rod, within the accuracy
of the measurements. By sighting along the reverberation
curves in figures 5 to 7, it can be seen that they are not
absolutely straight. The straight part with the smallest
attenuation is, however, long enough to give a reasonably
accurate determination of the reverberation time. The
points obtained have been plotted in figure 15, and the
vertical dashed line in this figure corresponds to the diam-
eter of a free bubble resonant at 5100 c/s. The glass rod
will undoubtedly change the resonance frequency as well

as the attenuation from the corresponding values for a

free bubble, but the agreement is still quite good.

A vibrating bubble radiates a spherical sound field,
which will represent a scattering loss if the bubble is
situated in a plane progressive sound field. This same
spherical sound field will not cause a loss when measure-
ments are carried out in a cavity with the bubble suspended
in the center, but it will cause a change in the resonance
frequency of the cavity.

The Sea Water Data

The attenuation values as evaluated from the measure-
ments on the sea water samples are contained in figures 16
to 32. The group 16 to 20 pertain to measurements for
mode 1-1-1; group 21 to 25, to mode 2-1-1; and group 26
to 30, to mode 1-2-1. It will be recalled that the three
different modes correspond to the following frequency bands:
5010 85, 655 wo Go 7, cine 759 tO 759 els (ine Gack wali
of the resonant frequency for each mode being dependent
mainly upon the depth of water). Figure 31 illustrates the
variation of attenuation with time for a series of samples
taken over a period of about 16 hours. Figure 32 isa
sequence of measurements taken in rapid succession for
given samples over a period of about 100 minutes.

76

The three groups of five figures corresponding to the
different frequency modes are arranged in the same order.
The first graph in each group is a scatter diagram of attenu-
ation versus relative oxygen content (as per cent saturation),
The remaining four graphs in each group show the attenua-
tion versus particle count for the following particle types:
Gonyaulax, Peridinium, broken cells (with chlorophyll) and
all particles collectively. As noted earlier, the attenuation
is given in terms of excess relative to that of the cavity
for the first group only (mode 1-1-1). The total attenuation
is given in the second two groups in view of the somewhat
uncertain behavior of the cavity attenuation for the modes
2-1-1 and 1-2-1,

The most striking thing one observes in these results
is the large scatter of value of attenuation, a feature which
seems to be inherent in respect to a complex fluid mixture
like sea water - whose properties cannot be uniquely de-
fined in terms of temperature, salinity, and pressure alone.
The graphs indicate that even oxygen content and particle
count are insufficient as additional degrees of freedom for
establishing a reproducible relation for attenuation. It is
evident that one is confronted with the problem of a random
variable in dealing with attenuation in sea water. However,
it is tempting to anticipate that the statistical behavior of
the attenuation bears some definite relation to the statistics
of the particulate matter, oxygen content, etc. There is
some evidence for this in the graphs shown here, particu-
larly those indicating attenuation versus per cent saturation
with oxygen. However, in view of the rather limited sample
of data, there has been no attempt to carry through a de-
tailed statistical analysis.

Some of the scatter in the present measurements may
be related to the method of sampling, effects of stirring,
and other factors characterizing the experimental procedure
These possibilities are discussed below.

200

150

100

50

200

150

100-

LexDB/KYD

dp ,DB/KYD

dg, (% SAT.)

* py (GONYAULAX )
MODE I-I-1

% SATURATION

Figure 16.

Excess attenuation coeffi-

cent for mode 1-1-1 vs. per cent of
saturation with oxygen, based on meas-
urements with sea water.

tt

300 NUMBER/ML

Pa Spek Ne

50 NUM

2 eS
BER COUNTEd
La)

Figure 17. Excess attenuation coeffi-
cient for mode 1-1-1 vs. concentration
of Gonyaulax expressed as number per
milliliter.

a

200;

Wey DB/KYD* sits
dex (PERIDINIUM )

I5O0-

dl |

50° el
: a : Figure 18. Excess attenuation coeffi-
ig aw: 50 A 190 150 NUMBER/ ML cient for mode 1-1-1 vs. concentration
Tia mercy etna mm cmma ame “ay OR ae a 5 i
0 i ; i 1 _"? NUMBER COUNT, of Peridinium.
2005
. |
eae
dey DB/KYD | a,,(BROKEN CELLS CONTAINING)
ex CHLOROPHYLL
| MODE I-1-I
150. 3 Z 4

|
|
|
|
|
|
|
|
|
100 7

|
: :
|

|

|

|

6 fs
5 0}— / ° =|
/ . .
“0 / 6 -
: / oh gS Figure 19. Excess attenuation coeffi-
° Uf 5 See : .
es Bs ue cient for mode 1-1-1 vs. concentration
[es hoz 10, = 2003 + 300 NUMBER/ML. of broken cells which contained chloro-
Baio aeo 30 40. 50 NUMBER COUNTE
Oo le 1 oT 1 4 1 phyll.

78

MMe wT ip oat et |

py (X ALL PARTICLES COUNTED)
MODE I-I-I

ex DB/KYD

100 °
50L” : 2 =|
} : : F Figure 20, Excess attenuation coeffi-
55 Sata : cient for mode 1-1-1 vs. concentration
1 Da ne nes > 500 NUMBER/ML for all organic particulate matter which

a to 1 L 1 =
Ds 100 NUMBER COUNTE

was counted.

CQ

p 215

2007
‘2 9,DB/KYD 5
@op( % SATURATION)
MODE 2-I-1 :
150 =
100} : 4
50- DR SoH ey cee : ; 5
° 40 60 80 100 120 °% SATURATION
Aietie L Lo 1 mL I S|
: fais
200[—
@5,DB/KYD Ayo4( GONYRULAX )
MODE. 2-I-I
150- |
took |
}
. a : U
ot : : —|
tie ca
50 100 150 NUMBER/ML
FN rT a
10 NUMBER COUNTED
4 a

380

Figure 21,

Total attenuation coeffi-

cient for mode 2-1-1 vs. per cent of
saturation with oxygen (no correction
made for cavity attenuation).

Figure 22,

cient for mode 2-1-1 vs. concentration

Total attenuation coeffi-

of Gonyaulax.

Aeis

200f
Deol PERIDINIUM )
yop 0B/KYD MODE 2-1-1
I50F
{
took
0 5 :
soir: ? . °
50 100 150 NUMBER /ML
+ + =n. + 1 + + + + tt +
5 10 20 NUMBER COUNTED
to 7 215 5
20a) -—<—— aioe ———
a, (BROKEN CELLS CONTAINING)
dy, DB/KYD tot CHLOROPHYLL
MODE 2-I-1
aoe =|
100k
50 ONG 07g O Sais : ial
100 200 300 NUMBER/ML
5 Ole Sapam ‘40. NUMBER pouNTeD
fo) eee | Se | 1 |

Figure 23. Total attenuation coeffi-

cient for mode 2-1-1 vs.

of Peridinium.

concentration

Figure 24, Total attenuation coeffi-

cient for mode 2-1-1 vs.

concentration

of broken cells which contained chloro-

phyll.

81

to. f215

ZOO
dege(E ALL PARTICLES COUNTED)
yg, DB/KYD MODE 2-1-1
150 |
Look a
5 0;- ° 3 ° ° =|
{
100 300 500 NUMBER /ML
; ; ; i O NUMBER COUNTE
a 20 40 60 80 TOON ED

82

Figure 25. Total attenuation coeffi-
cient for mode 2-1-1 vs. concentration
for all organic particulate matter
which was counted.

200

a eer |
4 5pDB/KYD
Ege a, _,(% SATURATION)
tot
MODE 1-2-1
150 =
100h . 4
50 , |
40 60 80 100 120 % SATURATION
) 1 n =
ae
a, DB/KYD
tot
49, (GONYAULAX )
MODE |-2-1
150- =
100 . a
q
al 4
50 100 190 NUMBER/ML
10 26 + + + + TED
6 i NUMBER COUN

Figure 27.
cient for mode 1-2-1 vs.
of Gonyaulax.

Figure 26, Total attenuation coeffi-
cient for mode 1-2-1 vs. per cent of
saturation with oxygen (no correction
made for cavity attenuation).

Total attenuation coeffi-
concentration

83

d4,,0B/KYD

dy, (PERIDINIUM )

MODE I-2-I

Figure 28. Total attenuation coeffi-
150 NUMBER/ML cient for mode 1-2-1 vs. concentration
NUMBER COUNTED of Peridinium. ;

f
200 | |
f

tor DB/KYD a (Gsoken CELLS CONTAINING
tot CHLOROPHYLL
MODE 1-2-1 P
150 a)
A
;
,
loo 0 ay
50 ;
Bio : ¥ Figure 29. Total attenuation coeffi-
% inne cient for mode 1-2-1 vs. concentration
100 200 300 NUMBER/ML of broken cells which contained chloro- —
nL, 2 Ole nat OAR NI NUBERI COUNTED, phyll.

34

200 ]

Stok BLK G yo AD ALL PARTICLES COUNTED)
MODE I-2-l

150 4
100} : al
50 5 P

? . . :

ts ies e Figure 30. Total attenuation coeffi-

Pe erie cient for mode 1-2-1 vs. concentration

200 490 sponumservm | fOr all organic particulate matter which
+
100 NUMBER COUNTED was counted,

85

150

86

% SATURATION,NUMBER COUNT, DEPTH 0.1 FT.

200
f ap,08/4Y0,
=--x—K
I50b ee.
| aS
mA

SUNSET
amas

Gy EXCESS ATTENUATION

% OXYGEN SATURATION

DEPTH OF TURBID LAYER SAMPLED

GONYAULAX COUNTED

PERIDINIUM

BROKEN CELLS CONTAINING
CHLOROPHYLL

O LALL PARTICLES COUNTED

PPOe+x

SUNRISE

== oa
12 HOURS

134%

OXYGEN SATURATION 4

NOTHING DONE TO SAMPLE
WALL WIPED

SAMPLE STIRRED =
(----),3 MINUTES EARLIER

n=O

oO

115%

'gO MINUTES
ww __-

Figure 31.

Serial sequences of attenua-

tion, oxygen content, and depth of turbid
layer taken during the night of 2 August

1960,

Figure 32.

Rapid serial sequence of

attenuation showing the effects of stir-
ring and other variables.

Effect of Stirring

Some observations can be made on the basis of figure
32. Even very gentle stirring always increases the excess
attenuation, if any is present at all. The sample was always
stirred very thoroughly when collected, due to the method
of collection by pumping from the sampling depth. Accord-
ingly, the largest attenuation was usually measured immed-
iately after sampling. Different degrees of stirring were
tried after the samples were collected, ranging from a
thorough stirring with the squeegee to a very gentle stirring
with a thermometer. Care was taken not to work any air
bubbles into the water when stirring with the thermometer,
and the organic material that sometimes settled down to the
bottom of the cavity was never disturbed by this action.
The attenuation would usually decrease with time, if the
sample was not disturbed in any way, but there was almost
always residual attenuation of 20 to 30 db/kyd, when the
sample had possessed appreciable attenuation initially.

Bubbles did sometimes appear on the sides of the
cavity, but wiping them off generally increased rather than
decreased the attenuation, presumably due to the unavoid-
able stirring of the sample. Ona few occasions bubbles
were observed floating close to the center of the cavity and
very large attenuations were measured under these circum-
stances. Only samples for which no air bubbles could be
seen (in the presence of a strong light beam) have been
included in the data collection. This limits the possible
bubbles to 0.1 mm or less, the largest observable diam-
eter being somewhat dependent upon the kind and amount of
suspended matter. Reverberation curves taken in rapid
succession give the same value for the reverberation time.
It may therefore be assumed that a single measurement of
attenuation does not change the acoustic properties of the
sample noticeably, but it does not assure that the sample
is unaffected by a sound field sustained over an appreciable
time.

87

388

Effect of Plankton

The attenuation as a function of the plankton content is
shown in figures 17 to 20 for the fundamental mode. Two
groups of organisms were dominant during the bloom, the
dinoflagellate Gonyaulax polyedra (fig. 17) and dinoflagel-
lates of the genus Peridinium (fig. 18), The attenuation in
figure 19 has been plotted as a function of the remaining
particulate matter containing chlorophyll. Finally, figure
20 gives the sum of the counts in the previous three figures.

The plankton samples had a volume of 1 ounce. The
count was performed by first shaking the sample well, and
then, using an eye dropper, filling a Sedgwick-Rafter
counting cell with a portion of the sample. The dimensions
of the chamber are 50 x 20 x 1 mm, and the volume contained
is therefore 1 cm® (or 1 ml). The count was accomplished
at 100 power with a microscope equipped with a Whipple
micrometer disk. All the particles within the horizontal
lines of the disk were counted during three complete passes
in the length direction of the chamber. The calibration
showed that the number obtained should be multiplied by
5.9 to give the number of particles per ml. It should be
noted that only a small fraction of the particulate matter
present was accounted for by the described procedure.

Influence of Sampling Methods

The rather poor correlation of attenuation and plankton
count might well be attributed to nonrepresentative sampling.
Plankton determinations for biological investigations are
often accomplished by towing a plankton net at a chosen
depth for a given time. The count can then be performed
as described above. However, the accuracy will be much
greater since all of the plankton from the large volume of
water swept out by the net will be concentrated in a small
sampling jar. This method was attempted during the pres-
ent investigation by letting the discharge from the pump

flow through a plankton net. At the same time the water
flow was recorded in liter/min. However, it was evident
that a large number of particles went through the bolting
silk. The finest netting available was No. 25 with an
aperture of 65U, and the average diameter of Gonyaulax
polyedra and Peridinium are 50u. Moreover, there was
no assurance that the plankton count in the pump water
would remain constant, since the plankton-rich layer was
known to move up and down due to the surface wave action.
A count accomplished by this means would therefore not
necessarily be representative of the water in the cavity.

Effect of Oxygen Content

Examination of figures 16, 21, and 26 shows that the
high attenuations are intimately connected with a high
content of oxygen. The highest concentration of phytoplank-
ton was usually found in a well defined layer which varied
between the surface and a depth of 20 feet. The oxygen
produced by photosynthesis of the phytoplankton can be at
saturation under a total pressure well above the atmospheric
pressure. It is therefore understandable that the water
samples can be supersaturated with oxygen when placed in
the cavity at atmospheric pressure. No determination of
the nitrogen content was attempted since there is no simple
chemical procedure available. A gasometric method must
be used and the equipment required was not available, nor
could it have been utilized due to the limited space available
on the tower.

Dissolved gas has very little effect on the sound propa-
gation in water, as already noted.°~’*” Gas in the form of
bubbles does, however, have a marked influence on the
sound propagation. Accordingly the high oxygen content
indicates that bubbles clearly must be considered in the
evaluation of the data.

Free bubbles have a short life in a small sample of a
liquid which contains no particulate matter.°°’*° Never-

89

90

theless, it has been suggested that bubbles can still be

present.** A bubble of equilibrium size must satisfy the
following equation:*°’*%
27
= —
[Din — 12 E,

where

D,, = Pressure due to dissolved gas and water vapor

= Hydrostatic pressure

XS}
}

= Surface tension

ss)
|

r = Radius of bubble

The last term caused by the surface tension can be
neglected when the radius of the bubble is large. The
partial pressure of the dissolved gases and the water vapor
must then balance the hydrostatic pressure. Equilibrium
bubbles are therefore only possible close to the surface.
However, these bubbles are unstable. A bubble slightly
larger than the equilibrium size will continue to grow since
the term due to the surface tensions gets smaller with
increasing diameter. These bubbles will soon rise to the
surface and disappear. A bubble smaller than equilibrium
size will be forced into solution due to the increasing pres-
sure from the surface tension. Bubbles have been utilized
in the past in the form of bubble screens,etc., but the
bubbles must be produced continually for as long as the
effect is desired. It was observed on a few occasions that
small bubbles were stabilized in the water by adhering to
particulate matter. This partly increased the effective
surface of the bubble and thereby slowed its rate of rise to
the surface and partly added mass to the bubble counter-
acting the buoyant force. Samples in which this occurred
showed a large attenuation, and the results are not included
in the data presented.

Bubble Nuclei

Pure water without any air bubbles has a very high
tensile strength.°° This is in stark contradiction to the
behavior of natural fresh or salt water. This discrepancy
is usually explained by the presence of gas nuclei in any
naturally occurring water mass. The above equation
determining the radius of an equilibrium bubble can be
considered as the criterion distinguishing bubbles and
nuclei. Bubbles smaller than the critical size must some-
how be stabilized, or they would soon go into solution.

The only agent capable of doing so seems to be particulate
matter such as dust. * Filtering or centrifuging are

capable of removing the nuclei. This fact can be considered
as substantiation of the above viewpoint. Gas nuclei can be
removed by applying high pressures to the sample, thereby
forcing the small air masses into solution.

The dividing line between nuclei and bubbles will ob-
viously change with the hydrostatic pressure. Nuclei close
to the critical size may grow into bubbles, if p, is reduced.
This is precisely what happens when the sample is pumped
up into the cavity.

The very high attenuations measured shortly after
sampling, or after the samples were stirred, can perhaps
be explained by the development of bubbles which were too
small to be seen by direct visual observation. The attenua-
tion remaining after the sample was left undisturbed and
also the attenuation measured for nonsaturated water is
difficult to explain as an effect of bubbles. It may be
remarked in passing that the method of wiping the cavity
walls seemed to be quite efficient in removing at least the
larger nuclei, since bubbles rarely developed on the sides
after a thorough wiping, even though the water was strongly
supersaturated.

“Ref. 36, p. 95-97

91

Losses Due to Bubbles

A large amount of work has been done on the problem
of the sound attenuation introduced by bubbles,*’127**»+%744
The most recent experimental evidence substantiates the
notion that the thermal, radiation, and viscous losses
adequately describe the attenuation of a bubble vibrating
close to its resonance frequency.* It is usually assumed
that the effect of bubbles much smaller than the resonance
size, corresponding to the measuring frequency, is very
small. However, the following discussion indicates that
this conclusion may not always be justified.

When the glass rod in the captured bubble experiment
was moved around close to the center of the cavity, it was
found that the attenuation caused by the bubble was sub-
stantially constant provided that the bubble was within a
distance of approximately 2 cm from the center. Now
suppose the bubble is divided into many smaller bubbles
with the same total volume as the single bubble and dis-
tributed within 2 cm from the center of the cavity. It must
be expected that the resonance phenomenon is basically
unchanged. The attenuation caused by the air volume,
however,: will have changed in a marked degree. The
reradiated sound will not contribute to the total loss by
cavity measurements, in accordance with the behavior of
the single bubble. The viscous losses will be the same or
perhaps somewhat larger, since the motion in the water is
essentially unchanged except in the immediate neighbor-
hood of the bubbles. The thermal losses will be consider-
ably smaller, since the air in the small bubbles will be
almost at the same temperature as the water throughout
a wave cycle. The thermal loss of a resonant bubble
arises if the disturbance of the gas is neither adiabatic
nor isothermal; in this case there must be a phase differ-
ence between the acoustic pressure and the particle velocity.

“Private communication from Dr. D. E. Andrews, NEL.

There are, however, at least two other sources of
loss experienced by a bubble: diffusion of gas and evapora-
tion of water vapor into and out of the bubble during the
acoustic cycle. It was not possible to find any investiga-
tion of these losses in the literature, presumably because
the main interest in the past has been centered on bubbles
close to resonant size. Moreover, the resonant bubbles
are adequately described by their viscous, thermal, and
radiation losses. However, for a given total volume of
gas phase within a sample, the surface area available for
diffusion and evaporation is inversely proportional to the
radii of the bubble. Accordingly one should expect a
greater effect due to diffusion and evaporation for the
nuclei. The theory of the growth of an air bubble ina
sound field by diffusion processes has been investigated
in a recent paper*° and provides a possible approach to
the problem of determining energy losses. Reference may
also be made to an experimental paper concerned with
ultrasonic cavitation traceable to the growth of air nuclei.*~°

Particulate Matter as a Source of Nuclei

The greatest part of the particulate matter in the
samples was undoubtedly of organic origin. The question
therefore arises whether organic material can produce
bubble nuclei and also act as the stabilizing center for a
nucleus.

The evidence found in the literature indicates that air
bubbles are almost never found in living cells.*” Bubbles
may form, however, if a cell is injured by pinching or
cutting and then exposed to reduced pressure. Bubbles
may form inside a cell even though the cell wall is not
broken by being pinched and they may form in a different
place than the pinched spot. This is explained by the
unavoidable twisting strains set up in the cell material
when manipulated. *”

There is still another possibility that not only will
explain the above experiments, but also the observations

93

94

made in connection with the present work. Plant cells
have, as a rule, a cell wall that completely encloses the
living protoplasm. The pressure in the protoplasm can
therefore be considerably higher than the pressure outside
of the cell. Oxygen is continually being produced when the
plant builds up organic material by photosynthesis. The
excess pressure in the protoplasm will increase the satura-
tion level required in the cell to make it possible fora
nucleus to grow into a bubble.

Both Gonyaulax polyedra and Peridinium have cell walls
in the shape of very elaborate testa that completely enclose
the living cell. Nuclei produced de novo by the photosyn-
thetic process will therefore not develop into bubbles under
normal circumstances. If the outside pressure is reduced
rapidly, such as occurs when the water is pumped up into
the cavity, it seems reasonable that bubbles may sometimes
develop and make the organism explode. This would explain
the two instances where strings of gelatinous material were
found floating in the surface of the cavity. Pinching a cell
will stretch the cell wall and thereby relieve the inside
pressure. Cutting the cell wall will of course have the
same effect. If a plant cell dies, it can act as center for
bubbles, and the dead cell must consequently contain nuclei.
The excess pressure in the cell is connected with the life
processes, and will presumably disappear when life ceases.
The cell wall will often be broken by death and expose the
cell content, and the effect of any nuclei in the protoplasm
will therefore be at a maximum.

A recent paper is concerned with the absorption of
ultrasonic acoustic waves in water containing algae in
suspension. The results given indicate that a very sub-
stantial volume viscosity could be attributed to the algae.
However, the measuring frequencies were so high (15-27
Mc/s) that no reliable deductions can be made with respect
to the present results.

The data presented in this investigation indicate that
a relationship may exist between broken cells and the
attenuation measured. A very clear-cut correlation could

not be expected, since nothing is known about the length

of time required for decaying plant material to stabilize
nuclei, nor about the possible production of new nuclei due
to the decay processes.

The observations presented here are in many ways
similar to the description given by Skudrzyk. * The explan-
ation presented in this reference is based on a continuous
production of bubbles, but the experiments were performed
at frequencies of the order of 100 kc/s. Bubbles close to
resonance will have such a small diameter that it is diffi-
cult to verify by visual inspection whether bubbles of this
size are present. A frequency of 100 kc/s corresponds to
about 0.065 mm diameter.

Serial Sequence of Measurements

One series of measurements was taken through the
night of 2-3 August 1960, in an attempt to identify variables
of importance to the sound attenuation other than the oxygen
content and the count of particulate matter (fig. 31). The
samples were taken in the layer of lowest light transmission
as indicated by the hydrophotometer. Changes in the differ-
ent counts reflects not only changes in the density of the
layer with highest turbidity, but also the influence of sur-
face waves and internal waves during the sampling period.
These changes must therefore be interpreted with some
caution.

The samples are supersaturated with oxygen at atmos-
pheric pressure, and the change in concentration through
ther michteis nmotweny oceat. he upper Ko-2 0) feet were
supersaturated through the whole series, and there was
no pronounced layer of maximum oxygen concentration.
The layer of maximum oxygen content did not coincide with
the layer of maximum turbidity.

*Ref. 4, p. 869-870

V5

96

The correlation between the sound attenuation and the
variables shown in figure 31 is very poor. The attenua-
tion goes down from its high daytime value to about 30
db/kyd at night, but this value is still significantly higher
than the attenuation of pure sea water, and corresponds to
the values obtained when a sample was left standing in the
cavity.

This behavior can be understood if it is postulated that
the plankton produces oxygen nuclei during the daytime
hours only. During the dark period the nuclei disappear
either because their stability in time is limited, or perhaps
because they are carried to the surface by convection cur-
rents where they develop into bubbles and escape.

Vi. SUMMARY AND CONCLUSIONS

PRIMARY RESULTS

Sea water without suspended particulate matter showed
no excess attenuation when compared with distilled water.
This is in agreement with earlier work performed at higher
frequencies, within the measuring accuracy of the equip-
ment utilized. Nearshore water with high oxygen content
and with suspended particulate matter showed excess
acoustic attenuation in the full range of the measuring
equipment, i,e., from 10 to 200 db/kyd. The largest atten-
uations were measured for water supersaturated with oxygen,
but excess attenuation was experienced even when the oxygen
content was only 65 per cent of the saturation value.

The comparison of excess attenuation with suspended
particulate matter (primarily plankton) and oxygen content
does not indicate a clear-cut correlation, but does suggest
the trend indicated above. Perhaps the most striking
feature of the measurements is the large range of attenua-
tions which can exist in samples whose gross properties
appear to be quite similar.

On the other hand the time sequence of attenuation
followed a definite pattern. Whenever any significant
amount of attenuation was measured, it would always
increase when the sample was stirred. The highest atten-
uations were consequently usually measured immediately
after sampling. The greater part of the excess attenuation,
however, would disappear when the sample was left undis-
turbed, but an excess attenuation of 20 to 30 db/kyd almost
always remained even after long periods of time (several
hours).

A bubble resonant at 5.2 kc/s will have a diameter of
approximately 1.2 mm and is easily visible. It is con-
cluded that the attenuation is not caused by bubbles close
to resonance, but may have been caused by much smaller

98

bubbles or nuclei. It seems likely that the large excess
attenuations following stirring are caused by very fine
bubbles, when the water is supersaturated, but the attenua-
tion remaining after periods without disturbance may be
caused by nuclei. These nuclei may either be stabilized

by dead particulate matter, or perhaps inside of living
phytoplankton. There is evidence that plankton, at least
under certain conditions, do contain nuclei.

The importance of the above considerations in the
evaluation of the transmission anomalies encountered in
the ocean remains to be seen. Vertical currents are
known to exist in nearshore areas, and the pressure
changes created may well result in bubble growth when a
water mass is brought close to the surface. It was ob-
served at two occasions that bubbles could be stabilized
by particulate matter, and there is reason to believe that
this phenomenon also takes place in nature. Bubbles
stabilized in this manner may again be carried down and
cause some of the anomalies observed.

SUGGESTIONS FOR FUTURE STUDIES

Little is known of the acoustic properties of water
containing a large number of air bubbles much smaller
than resonance size. Future work must therefore include
an investigation of small bubbles and nuclei, before a full
understanding of sea water measurements can be expected.
Special attention must be given to the influence of plankton,
since nuclei inside a living cell may have an especially
marked effect due to the surrounding organic matter. Only
when these problems have been answered satisfactorily is
it to be expected that the acoustic properties of plankton
itself can be ascertained.

Cavity methods are well suited for most phases of
this work. Single bubbles may be created in the center of
a cavity around a fine platinum wire by means of electroly-
sis. This method has been utilized in the past for

measurements on resonant bubbles. Hard-wall spherical
cavities may be particularly suited since the losses due to
the cavity are as small as possible and the influence of a
single very small bubble must be expected to be slight.

To determine the effect of nuclei contained in plankton
may require that pure cultures be raised in the laboratory
so that the conditions under which the plankton is growing
can be established precisely. On the other hand it may
be possible to show by much simpler means that plankton
at least sometimes contain nuclei by exposing samples of
sea water to a vacuum and examining the water micro-
scopically before and after applying the vacuum. In this
fashion it may be possible to show with certainty that a
pressure reduction at times can cause plankton cells to
explode and to conclude that they contain nuclei.

2g)

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4
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i

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105

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Commander Submarine For, PacFlt
Commander Submarine For, LantFlt
Commander Anti-Submarine Defense
For, PacFlt
Commander Training Command, PacFlt
Commander Submarine Development Group TWO
Commander Service For, PacFlt, Library
Commander Service For, LantFlt
Commander Key West Test and Eval. Det.
Naval Air Development Center, Library
Aeronautical Instrument Lab.
Naval Missile Center
Tech. Library Code 5320
Naval Ordnance Laboratory, Library (2)
Naval Ordnance Test Station, Pasadena
Annex Library
Naval Ordnance Test Station, China Lake
Tech. Director Code 753
Charleston Naval Shipyard
Portsmouth Naval Shipyard
Puget Sound Naval Shipyard
Naval Radiological Defense Laboratory
David Taylor Model Basin
Navy Mine Defense Lab., Library
Navy Training Device Center
Navy Underwater Sound Laboratory
Library (3)
ASW Tactical School, LantFlt
Naval Engineering Experiment Station
Library
Naval Research Laboratory
Library (2) Code 5120
Navy Underwater Sound Reference Lab.
Library
Air Development Squadron ONE (VX-1)
Beach Jumper Unit ONE
Beach Jumper Unit TWO
Fleet Sonar School
Fleet ASW School
Naval Underwater Ordnance Station
Library
Office of Naval Research, Pasadena
Naval Submarine Base, New London
Naval Medical Research Laboratory
Naval Personnel Research Field Activity
Washington, D. C.
Navy Hydrographic Office
Library (2)
Div. of Oceanography
Naval Postgraduate School, Library (2)
Navy Representative, Project Lincoln, MIT
Assistant SECNAV, R and D
Assistant Chief of Staff, G-2, US Army, IDB (3)
Chief of Engineers, US Army, ENGRD-MF
The Quartermaster General, US Army
R and D Div., CBR Liaison Officer
Redstone Scientific Information Center,
Redstone Arsenal
Army TRECOM, Research Reference Div.

Continental Army Command, ATDEV-8
Army Electronic Proving Ground
Tech. Library
Rome Air Development Center, RCOYL-2
Holloman Air Force Base, SRLTL
Beach Erosion Board, Corps of Engineers, US Army
Air Defense Command, ADOOA
Air University, Library AUL3T-5028
Strategic Air Command, Operations Analysis
Air Force Cambridge Laboratory CRREL-R
Headquarters, U.S. Coast Guard
Aerology & Oceanography Section
Marine Physical Laboratory, Univ. of Calif.
Scripps Institution of Oceanography, Univ.
of Calif.
National Research Council
Committee on Undersea Warfare (2)
U.S. Coast & Geodetic Survey
Director, Div. of Tide & Currents
U.S. Fish & Wildlife Service, Pacific
Oceanic Fishery Investigations,
Library, Honolulu
U.S. Fish & Wildlife Service, La Jolla
South Pacific Fishery Investigations
U.S. Weather Bureau
University of Alaska, Geophysical Institute
Brown University, Research Analysis Group
University of California at Los Angeles,
Engineering Dept.
Columbia University, Hudson Labs.
Harvard University, Director, Acoustics
Research Lab.
The Johns Hopkins University
Chesapeake Bay Institute, Library
University of Miami, Marine Laboratory
University of Michigan
Great Lakes Research Division
Research Institute
New York University
Meteorology & Oceanography Dept.
A & M College of Texas, Dept. of Oceanography
Pennsylvania State University, Ordnance
Research Laboratory
University of Texas
Defense Research Laboratory
Military Physics Laboratory
University of Washington
Department of Oceanography
Applied Physics Laboratory
Yale University
Bingham Oceanographic Labe
Lamont Geological Observatory
Woods Hole Oceanographic Institution
Laboratory of Oceanography (2)
U.S. Naval Academy
Civil Engineering Laboratory, L54
Office of Naval Research
Contract Administrator, S.E. Area
Chicago
Boston
New York
San Francisco
Allan Hancock Foundation
Arctic Research Laboratory
U.S. Geological Survey
U.S. Fish & Wildlife Service
Point Loma
Stanford
Washington, D.C. (2)
Woods Hole
Geophysics Research
Narragansett Marine Laboratory
Waterways Experiment Station
Navy Weather Research Facility
Cornell University, Dept of Conservation
Florida State University,
Oceanographic Institute
University of Hawaii, Marine Lab.
Oregon State College, Dept. of Oceanography
Rutgers University
AWS, Scott AFB, Illinois
Bureau of Comm. Fisheries, Bio. Lab.
Wash, D.C. Point Loma Sta.