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P. G Hansen.

Measurement of attenuation of low-frequency sound (5-8 kc/s) in small samples of sea water online

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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 Q of
the order of 2 000 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



/o

A/



where / is the resonance frequency for a given mode of
oscillation and A/ 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



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.



12



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 SCAB*
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.



14



TRANSDUCERS




SOFT SOLDERED
l^-SPOT WELDED



SECTION A-A



SOAB RUBBER BLOCKS



Figure 1. Schematic of preliminary watertight cavity.



TRANSDUCERS




GLASS
"PROBE



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


No.


1


17


11


14


No.


2


12


8


10


No.


3


8*


5*


7


No.


4


6


4


5


No.


5


4^


2^

^4


3*



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



15



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 consisted of four pieces of angle aluminum. The line
of support was approximately ^ 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 Q 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



16



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 F

RUBBER SOCK - ^assgrngsassssfegssi ,
BRASS



STAINLESS
STEEL BAND




COAXIAL CABLE



EPOXY COAT
PLEXIGLAS




10-32
FIBER



4-40 BRASS TUBE




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



17



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.



Mi r-


1 II




MAIN DIAL
INCREMENT DIAL



////////////////////////////////////



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



19



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 5 to 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 / in c/s and the re-
verberation time t in seconds is expressed by t A/ = 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 A/ 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.



20





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




Figure 8. Data- collecting equipment.



24



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. 39 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.



25



ACOUSTICAL WAVE EQUATION FOR VISCOUS
FLUIDS



Fundamental Equations*



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



p It + p ("* v > u - ^ v 2 s - (u +x ) v (vS) + vp = P i? (i)



where p is fluid density, v is the velocity vector, P the
pressure, F the body force per unit mass, \i and X 1 the
first and second viscosity coefficients and 7 is the general
(three-dimensional) gradient operator. The velocity and
density must also satisfy the continuity equation



^+7 • (pO) = (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=f(p) (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,



26



Assumptions



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

(a) The changes in P, v, 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 v is regarded as small compared with
the speed of sound for the fluid occupying the chamber.

(b) The motion is regarded as irrotational,



V x v =



(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



~ - u x v 3 v - (\x 1 +\ ) v (vS) + Vp = (la)



~+ P V-5 = (2a)



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



27



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



= v (4)



zt



Accordingly (2a) can be expressed as



— (p+P o V.s) =



(p-p ) = -p V-s = -p (5)

where is a convenient notation for the dilation V • s.
Equation (3) implies that for small changes



AP = ^Ap = ^ (6)

dp ° p n



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



■K ® (71



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



28



The Stress Tensor



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



Hh = K®hh (8)



where 6 - h is the unit tensor (which has unit value for I = k
and vanishes for i 4- h). 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



t. 7 = |j 2?.. + \ ® 5,, (9)

yk a TsK o vk



Here 2L ^ is the strain (or deformation) tensor






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



*The stress tensor t^ as employed here is such that tensile
stresses t , t 23 , t 33 are positive or pressures negative.
The conventional indicial notation is employed, where i, k
can take on values 1,2, 3 independently.



29



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



ZD. k .(a

t =n D..+X ® 6.^ +n -TJ^+X f^.. (11)

vk o vh o i,K i M i d£ l«



which retains the symmetry property mentioned earlier.
The Navier-Stokes equation for a viscous fluid as given by
(1) employs (11) with \x = and -X ® 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 in a closed system. Eckart* has shown that this will
be assured under all conditions of deformation, if and only
if



u > and JL > -f |i, (12)



It may be remarked that the traditional stipulation regarding
the second viscosity coefficient (namely X 1 = -f [i ) 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 X exceeds the lower limit imposed by (12).

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



30



The Acoustic Wave Equation

In view of the identity

V (v-u) = v 2 v + v X (7Xu) (13)

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



Po It- (2^+^)v 3 v + Vp = (14)

at



Moreover, if (4) and (7) are employed then the latter rela-


2 4 5 6 7

Online LibraryP. G HansenMeasurement of attenuation of low-frequency sound (5-8 kc/s) in small samples of sea water → online text (page 2 of 7)