NORDA REPORT 4

A NEW MODEL OF RESONANT ACOUSTIC
SCATTERING BY SWIMBLADDER — BEARING FISH

WHO]

DOCUMENT
COLLECTION

RICHARD H. LOVE

OCEAN ACOUSTICS DIVISION

NAVAL OCEANOGRAPHIC LABORATORY

AUGUST 1977

oe es Public Relea
Dis ca

| R i NAVAL OCEAN RESEARCH AND DEVELOPMENT ACTIVITY
! NSTL Station, Mississippi 39529

FOREWORD

Oceanic volume reverberation can adversely affect the performance of Navy sonars. Small swimbladder-
bearing fish are the primary cause of this reverberation at ship sonar frequencies. This report describes a new
acoustic model of a swimbladder-bearing fish. This model is an improvement over previous models and should be
of value in future volume reverberation studies.

This report was originally a dissertation submitted in partial fulfillment of the requirements for the Ph.D. degree
in acoustics at the Catholic University of America.

CHARLES G. DARRELL
Captain, USN
Commanding Officer
NORDA

oustic Scattering

sh

and Dev. Activity

RETURNED

A TS

a

WHO]

COLLECTION

EXECUTIVE SUMMARY

Anew model of a swimbladder-bearing fish has been developed in order to provide improved predictions of the
resonant frequency and acoustic cross section of such a fish. The model consists of a small spherical shell in
water, enclosing an air cavity which supports a surface tension. The shell is a viscous, heat-conducting Newtonian
fluid, with the physical properties of fish flesh. A comparison of the results obtained with the new model to
experimental data indicates that the new model constitutes a definite improvement over previous models. The new
model can predict the high values of damping and elevated resonant frequencies that previous models could not.
The model appears to be most accurate for fish in which tension in the swimbladder wall has a minor effect on
resonant scattering. This includes the fish which are of interest in studies of volume reverberation and therefore,

the new model should be of considerable value in such studies.

NN

wi

NIN

ACKNOWLEDGMENTS

It is my pleasure to acknowledge the guidance and advice provided by Dr. Thomas J. Eisler. My appreciation
also goes to Drs. Ronald New and John J. McCoy for their critical review of the manuscript. Special thanks go to Mr.
Robert S. Winokur, for his cooperation and support throughout the course of this work.

CONTENTS

Page

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Figure 1.
Figure 2.
Figure 3.
Figure 4.

Table |.
Table Il.

LIST OF ILLUSTRATIONS

Resonant frequency of a prolate spheroidal bubble .......... 0... 0. cee eee 5

Resonant frequency of the new model of a swimbladder-bearing fish......................... 24

Viscous and radiation damping factors for the new model of a swimbladder-bearing fish ......... 26

Thermal damping factors for the new model of a swimbladder-bearing fish..................... 2
LIST OF TABLES

Physical Properties aan. aa ect foci snc sme cteraie cecal cases eacnepeue CCIn ep etere cieeeesteetc rae een ate ernie seg 8

Experimental Data and Results of Comparisons to the New Model.....................0.2... 29

CHAPTER |

INTRODUCTION

Volume Reverberation

When sound is propagated in the ocean any inhomogeneities in the medium will scatter a portion of the
sound incident upon them. This scattered sound is termed volume reverberation. If volume reverberation levels
are high, the operation of an active sonar can be adversely affected. Specifically, high reverberation levels can
mask an echo reflected from a particular target of interest.

During World War II, researchers studying sonar echo ranging at the University of California Division of War
Research (UCDWR) discovered that volume reverberation levels were usually vertically stratified. Layers of
high reverberation, on the order of 100 m thick, were found, usually within the upper 1,000 m of the water column,
extending over large geographic areas of the Pacific Ocean [1,2]. Other researchers subsequently found such
layers in other oceans [3,4]. These layers came to be known as deep scattering layers (DSL). In addition, it was
found that these layers frequently rose to shallower depths around sunset and descended around sunrise. This
gave rise to the hypothesis that deep scattering layers were caused by biological organisms, which were known
to undergo diurnal vertical migrations [5].

The great difference between the acoustic impedances of air and sea water causes an air bubble to be a
much more effective scatterer than other objects of comparable size [6]. This fact led Marshall to examine the
possibility that small mid-water fish which contained air-filled swimbladders were the cause of DSL [7]. His study
strongly implicated such fish as major components of DSL. The primary scattering mechanism was considered
to be swimbladders which could resonate in the fundamental, or volume pulsation, mode when insonified at the
proper frequency. Subsequent research by Hersey and co-workers displayed the frequency dependence of
reverberation levels and provided further qualitative proof that resonant scattering by swimbladder-bearing fish
was the major cause of volume reverberation in the ocean [8-1 1]. The study of volume reverberation in the world
ocean has continued (for example, see reference 12) and it is now generally accepted that swimbladders of fish
are the predominant scattering mechanism in most geographic areas. However, it has not been until very
recently that quantitative comparisons between fish distribution data and acoustic volume reverberation data
have been made.

The fundamental volume reverberation parameter is the back-scattering coefficient of a unit volume of
ocean, M, which is the ratio of the scattered intensity at a unit distance from the unit volume, |,, to the
incident intensity, |;, [13],

p= VAL (I-1)

If it is assumed that the scattered signals add incoherently [13], then M can be defined in terms of the scatterers
as

(1-2)
j=1
where n is the number of scatterers in volume V and gq; is the ratio of scattered power to incident intensity of the
jth scatterer. o is called the acoustic cross section of the scaiterer.
Although M is the fundamental parameter, the quantity which is most often utilized in any discussion of
volume reverberation is the scattering strength per unit volume, S,, which is M expressed in decibels.
S,=10logM . (I-3)
S, is often simply called scattering strength [14].
In any acoustic measurement of volume reverberation, S, is obtained basically from the equation
S, = 10 log (I,/I,) . (1-4)

The determination of |, and |, is relatively straightforward and acoustic measurements of S, are fairly routine.
However, if a determination of S, is to be made from the distribution of scatterers, the equation employed is
n

S, = 10 log [aa x: I (I-5)
j=1

The difficulty in making quantitative comparisons between acoustic measurements of volume reverberation and
the distribution of swimbladder fishes lies in the determination of both n and g. The determination of the number
of fish and the species, size and swimbladder size of each is strictly a biological problem which will not be
considered here. However, the difficulty of this problem is not to be minimized. Given the size and swimbladder
size of an individual fish, calculation of o is also difficult, due to the complex structure of fish.

Simplified models have been developed in order to estimate o near resonance for an individual fish. These
models are based on the premise that only the swimbladder is a significant contributor to O near resonance.
Experimental evidence [15] shows, in fact, that this is so, whereas, at frequencies much higher than the
resonant frequency, the swimbladder and the body of the fish contribute about equally to o [16].

Experimental evidence indicates that the existing models have some shortcomings. After a review of the
models and the experimental data, it will be the purpose of this report to develop an improved model
of resonant scattering from an individual swimbladder-bearing fish in order to eliminate or at least decrease
the shortcomings of the existing models.

Existing Models

A swimbladder is essentially just an air bubble within the fish, so that the simplest acoustic model of a
swimbladder fish is an ideal spherical air bubble having the same volume as the swimbladder. The
frequency of the fundamental mode of resonance of a small ideal spherical air bubble in water was
determined by Minnaert [17] to be

W°a? = SY2Po_ (I-6)
Pow,
where W, is the circular frequency of resonance, a the equilibrium bubble radius, P, the ambient pressure, Y,
the ratio of specific heats of air, and p,,, the density of water. The acoustic cross section of a small ideal
spherical air bubble in water is [18]

4ma?
Ch re re Uti ROE | (1-7)

(ey Ss
oy Cre

where w is the insonifying frequency and c,, is the sound velocity in water. The limiting factor on these
equations is the size of the bubble, which is limited to values such that (wa/c,)<<1.

For a small real air bubble in water, the above equations for w, and o remain the same, except that the
term (w2a?/c,2) in equation |-7 is replaced by d? [18]. d is an unspecified damping constant which includes
the effects of heat conduction, surface tension, viscosity, and other processes.

Devin [19] studied the damping at resonance of real air bubbles in water. He found that the damping
constant at resonance, 5, was the sum of three damping processes: thermal damping, 5,,, viscous
damping, 5,,,, and radiation damping, 5,24,

5 = Sai + Sys + By (1-8)

5 is defined such that, at w = w,, d = 5. Devin determined the values of 5,,4, 5, aNd 5,,, for bubbles of the
size which are of present interest, to be

Wa
Orad a oa , (I-9)
= Any :
= 3 (Ya = 1) K, i i
ote Gee) a

where n,,, is the shear viscosity of water, k, is the thermal conductivity of air, A, is the density of air, and c,,

CHAPTER Il
A NEW MODEL

The Model

The purpose of this report is to improve the equations which are presently utilized to predict the
resonant frequency and acoustic cross section of a swimbladder fish, in order to facilitate the correlation of
acoustic and biological volume reverberation data. A new model for a swimbladder fish is proposed and the
appropriate equations developed for it. The model consists of a small spherical shell, enclosing an air cavity,
in water. The shell is chosen to be a viscous, heat-conducting Newtonian fluid, with the physical properties of
fish flesh, and the interface between the shell and the cavity supports a surface tension.

The shell is insonified by a harmonic plane compressional wave whose wavelength is large compared to
the shell diameter. For convenience, the center of the shell will be taken as the origin of the coordinate system
and the wave will travel in the positive direction along the z-axis. Spherical coordinates (r, 8, @) will be used
to describe the field, where r is the distance from the origin, 6 the polar angle and @ the azimuthal angle.
Since the problem is axisymmetric, 9/d@ = 0.

Acoustically, fish flesh may be considered to resemble soft rubber. In such a substance, shear waves are
much less important than transverse waves and the substance can be closely approximated by a viscous fluid.
Modelling fish flesh as a viscous liquid has advantages over modelling it as an elastic solid with a complex shear
modulus. Specifically, exact equations (Navier-Stokes) exist for the motion of viscous fluids, whereas a
phenomenological approach is required if a complex shear modulus is utilized.

The body of a fish which surrounds its swimbladder will cause an increase in stiffness over that of a free
bubble. It is assumed that most of that increased stiffness in concentrated in the swimbladder wall and may be
modelled by a surface tension at that interface. Unlike an elastic modulus, swimbladder tension may be under
the fish’s control and could account for otherwise unexpected variations in resonant frequency.

Limits and Physical Properties

Several limits will be placed on this model. These include limits on applicable depth, frequency, and size
ranges. The depth range of interest is from the surface to 1,000 m, which corresponds to an ambient
pressure of 10° to 108 dynes/cm?. The frequency range is 100 Hz to 40 kHz, which corresponds to a circular
frequency, w, of 2m x 10? to 8m x 10* rad/sec. The fish size range is 1 cm to 1 m. This roughly corresponds
to an inner shell radius, a, of 10-1 to 5 cm. (Appendix A contains a discussion of swimbladder volumes.) The
ratio of outer shell radius, b, to ais 2.5 b/a< 6 for small fish and 2.5 < b/a < 3.2 for large fish. Two further
limitations will be that 6 x 102cm/sec S$ was 2.4 x 104 cm/sec and that wb < 7 x 10* cm/sec.

In addition to swimbladder size, the surface tension at the swimbladder wall must be specified. For small
fish,

IA

102 dyne/cm < s < 10° dyne/cm
and for large fish,

102 dyne/cm < s < 109 dyne/cm.

The rationale for this choice of ranges is discussed in Appendix A.

Several other parameters must also be specified. These include the velocity of a compressional wave
(sound velocity), c; density, P,; specific heat at constant pressure, c,; ratio of specific heats, y; thermal
conductivity, k; shear viscosity, n,; and bulk viscosity n,; for air, sea water, and fish flesh. Subscripts a, w,
and f will be utilized to indicate the properties of air, sea water, and fish flesh, respectively. For convenience,
a viscosity parameter, &, is defined as

E = fn, + ny. (4)
The parameters are listed in Table 1, for a temperature of 10°C and, unless specified, a pressure of one

atmosphere (10° dyne/cm2). The properties of air were obtained from references 42 and 43 and those of
sea water from reference 44. The properties of fish flesh are discussed in Appendix A.

Table 1 — Physical Properties

Air Sea Water Fish Flesh
c, cm/sec 3.3 x 104 1.50 x 108 12550 1102
Po, gm/cm? 1.3 x 10-3 (at 1 atm) 1.026 1.050
1.3 x 10°' (at 100 atm)

Yy 1.40 1.01 1.01
Cc cal

>) Gm °C 0.24 0.93 0.89

’ om sec °C _ ze 5.5 x 10°° 1.34 x 10-3 1.32 x 10-3
N;, poise 1.8 x 10-4 1.4 x 10-2

&, poise 1 to 1.07

Preliminary analysis of the new model indicated that considerable simplification resulted, with virtually
no loss of accuracy, if several approximations were made concerning the physical properties. The first
approximations were that y,, = y; = 1. Secondly, since k,# k;>>k,, it was assumed that any heat generated
in the air is rapidly conducted away, so that the temperature in fish flesh and water is constant. Thirdly,
SINCE Ny>>N1s,,>>Ns,, it was assumed that n,, = N., = 0.

One further assumption which greatly simplifies the problem is made in this model. As discussed in
Chapter |, only the fundamental, or zeroth mode of oscillation will be considered.

Formulation of Equations

The first step in the determination of the resonant frequency and acoustic cross section of the new
model is the determination of the proper wave equations in water, fish flesh, and air. The wave equations
are determined from the basic equations of motion as given by Hunt [45], generally following the method
discussed by Epstein and Carhart [46]. The basic equations are the continuity equation,

>t + V: (pu) = 0 ; (Il-2)
the equation of conservation of momentum,
p af = —p(0°V)G — VP + EV(VU) — nV x (Vx) ; (II-3)
the equation of conservation of energy,
oT — | A> —W y.g4y-q _ = , \|-4

and the equation of state,

p = e(P,T) ; (II-S)
where p is density, t is time, U is a velocity vector, P is pressure, c, is specific heat at constant volume, T is
absolute temperature, B is the coefficient of thermal expansion, q is a heat conduction flux vector,

Ge Eo (II-6)
and @, is a viscous dissipation function.

Equations II-2 through II-5 are linearized by assuming that

T= © & oh, ed)

Py PometsPn) (II-8)

geil tee a, (I-9)
and

P(p,T) = Po(Po,To) + Pi » (II-10)

where the subscripts 0 and 1 refer to the average and perturbation values of the parameters, respectively.
The first-order equations are

oP. Lets O, (I-11)
oye EVV UN) = neVex (vy xu) | (Il-12)

(Cac

Poy a + By vt, SKN ETO (II-13)

since @, is of second order [45], and

Oo (rasa ea Gees cee (1-14)

It has been assumed that T,, = T,, = 0 and that y, = y, = 1. Hence, in fish flesh, equations II-11 through
ll-14 are

a so pSV! 1U,)) =" Ole (I-15)
Po SS = — VP, + EV(V-O,) — nV x (VXd)) , (ll-16)
and
0 Po
— pA EAU fe Il-17
on (aaa (11-17)

These equations are the same in water, with the exception that the viscosity is set to zero. The isothermal
compressibility, K,, is defined as

1 Are) (Il-18)
Kai esi (Ee) '
i Po aPo ij
and the adiabatic sound velocity, c, as
Y
Cae — z 2
PoK, (II-19)
Since y; = Y, = 1, equation II-17 is
Sp (11-20)
Pr; iat o?

and equation II-15 is

OP,
ot
According to Helmholtz’s theorem [47], the U, vector field can be represented in terms of a vector
potential A and a scalar potential Q such that

ts Po Cie Wan Ut) a Ole: (11-21)

U, VxA-VQ,

(Il-22)
with VA) Oe, (I-23)
so that V0, <= — V2", (I-24)
and Vxi, = —-VWA. (II-25)
Thus, equations II-21 and IIl-16 become
P
p,c2V2Q = aa (I-26)
and
Po; tv x A) — nV x (V2A) = — VP, — EV(V2Q) + Po, v9) : (Il-27)
If the curl of equation I-27 is taken,
Vx VP. = Vx VO =] V <V(V20) —0- (II-28)
Thus,
Do, 9 x A) — nV x (V2A) = 0, (\I-29)
or
nN V2A — py, 38 =). (I-30)
Therefore,
UP, + EV (V2) ~ Py (VO) = 0. (1-31)
Then substituting equation II-26 into I-31 and taking the divergence yields:
3 ( oP, ) Une
2 2 eye eae = Il-32
V?P, + ance aT Ge at OF ( )

Equations II-30 and II-32 are the wave equations in the fish flesh. In water, where the viscosity is zero, the
wave equation is

1 92P

VPs a oer aE = 0. (I-33)

Cre

The harmonic time dependence is now introduced, such that

P, = P,e™ , (I-34)

T, = Toe , (II-35)
and

A=A,e-* , (II-36)

Then equations II-30, II-32, and II-33 are, after eliminating the numerical subscripts,

10

NsV2A sF iWP,A = 0 , (II-37)

(ie ioe SS ae nD 279 , (11-38)
PoC? Cy
and
W2
V2P seers = (0) , (II-39)

The coefficient of thermal expansion, B, is defined as
pas (ea, (II-40)

so that, in air, equation Il-14 is

P
Pi, = = nes Po,Baly . (II-41)

In addition, equations II-11 through II-13 for air are, after substitution for U:

Sat 7 Gees =) (I-42)
0 a (I-43)
Poa ay x A 02 Dp (VY) VP, ’
and
oT, (Cy, “i Cy.) 2
PosPva eta” Poa ae ae ae OQ = TWF, = ©. (I-44)

Taking the curl of equation II-43 and utilizing equation II-28 indicates that

VxA=0, (I-45)
so that

VP, — Po, a4 (VO) =0 (I-46)

Then, substitution of equation II-41 into Il-42 yields:

zs YaP

Vox Tee ial | (II-47)

Hence, taking the divergence of equation II-46 and substitution of equation I-47 yields:
2 a*

VP; — Pose [sa cr ~ BT: |e: (II-48)

In addition, substitution of equation II-47 into I-44 yields:
oT hiGa = GA) oe a
Poalpa maith aa Bee 2 aa i K,V°T, =O); (Il 49)

Introduction of the harmonic time dependence into equations |I-48 and II-49 yields, after eliminating the
numerical subscripts:

wep + 2Y¥ep — w2p.,8,7 = 0 (II-50)
a

and
vet + sek) — JOVaG = op = Og - (I-51)

Equation II-50 can be written in terms of P alone by taking the Laplacian and substituting equations II-51 and
Il-50 for V?T and T, respectively, into the resulting equation. The result of this procedure is

2 ; THe
vV4p | — ait —— | yep + /° PoaTpa P =u (II-52)

Equation II-52 is reduced by considering it as a quadratic equation in V2. Then

(V2 = ikise) (V2 Fk) Po 0. | (II-53)
where
= WV. iWPo,Cp, + iWPo,Cp [ = w?K,7 Ya" = 2iWK.(Y2 a 2) |
2(K;,52,)° Gr aE Ka inet K, 3 1 Dreccicte Po,Ca cm
(II-54)

The second and third terms under the radical are much smaller than 1 for the ranges of parameters selected for
this study, so that the radical can be approximated as

eae (\I-55)
(2) 1 7
Thus,
Doe ae IWPo Con w?(Y, =a 1) a iw?K,Y4" Ss
crn ecient es
and
a? 1@siKaVes
2 = SEE eV
agi cz * Ape CSC (I-57)

An examination of the relative magnitudes of the terms in equations II-56 and II-57 shows that

k. 2 w~ !WPoaCpa (I-58)
la wo mK sen
or
ie WPo,Cp4 ) I-59
ky Ce) — (I-59)
and
vee eos
Cpe | (I-60)

Since k,, and k,, are never equal, the general solution of equation 11-53 is

P=W+h , aie)

where wW, and w, satisfy the equations

(V2 + k,2)W, = 0 (II-62)

and

(V2 + k,,?) Wo OW: (I-63)

Equations I-37 through II-39 can be written in a form similar to equations II-62 and II-63. Thus

(V2 +ky2)A = 0 , (II-64)
(V2 +k,2)P = 0 , (I-65)
and
(V2 +k,2)P = 0, (II-66)
where
iwp
ke? = ie (I-67)
Ww? iwe \-1
ae eo ae Ss \l-68
kp? CG? ( 1 PoC; ) ’ ( )
and
2
kee = = (II-69)

Comparisons of the equations for air and water indicate that w, represents a compressional wave. Hence, W,
represents a thermal wave.

Equation II-64 can be transformed into a scalar equation. It can be shown that a vector T exists such that
[48]

= Ver (Il-70)
and
Ae Vian (I-71)

Since the problem is axisymmetric, 0/3 @ = u, = 0, which implies that [,, = 0. Thus

Vx F=F(0) + 6(0) +> [era - SE |

30 (II-72)
so that
A = PA, > (II-73)
where T, 6, and @ are unit vectors. Therefore
Ge A ]
ZA = bay Ne ape SL EEG) I-74
ESS) [v A» r sin?@ ( )

13

and equation II-64 can be written as
VAG = oe a k3,2 Ag =0. (I-75)
The plane wave impinging on the shell will be represented by
Py, = Aekow , (II-76)

where P,, is the perturbation pressure of the wave in water, and A is the pressure amplitude. P,,, can be
expanded in spherical waves as

= AD (224 1)i Pi okKowt) P, (cos 8) , (II-77)
2=0

where & is the mode number, j, (k,,,r) is a spherical Bessel function and P, (cos 6) is a Legendre polynomial.
The incidence of the wave upon the shell gives rise to five additional waves, which are represented by the
solutions of equations II-62, II-63, II-65, II-66, and II-75. These equations can be solved by standard
separation of variables techniques. The solutions are:

= 2 Dagig (k,,") P 9 (cos 6) , (II-78)
W, = > B.gig(ks,F) Pe(cos 6) , (II-79)
2=0
ai anaes (Kar) + Ey on o(kar)] Pp (cos) , (II-80)
Phe = = By ghe(ko,r) Po(cos 6) ) (II-81)
2=0
and 0
Aor = DS [Figi olka) + GioMo(kat)] P’ g (os 8) , (I-82)

Q=I
where B, D, E, F, and G represent the amplitudes of the waves, n Kz,f) is a spherical Neumann function,

and h,(k,,") is a spherical Hankel function of the first kind. P,,, indicates the scattered compressional wave in
water, so that

P, = Py, + Pu. - (1I-83)

One of the assumptions of this model is that only the fundamental mode contributes to the scattering. Thus
only the &£ = 0 mode is considered. Examination of equation II-82 shows that for & = 0,

A, = 0. (11-84)

The remaining equations are now written for & = 0:

Py = Ajo(Koyt) + Bwho(Koyr) (II-85)
P, = Bilo(kar) + Epno(kar) , (II-86)
Wi = Dajo(k,,") . (I-87)

and
Wo = Bajo(Ko,") ;

where the numerical subscripts have been eliminated from the coefficients for convenience.

The scattering cross section of the shell is
Oo = ©, / fF ’

where @®, is the average scattered power and |, the incident intensity.

1
®, 7 rahe Uws qv,

where v is the area of an element of a sphere [49].
vVv=nv,
dv = 1rdx ;

where n is the normal to the area and x is the solid angle. Also,

ioe
i = 2 Pe Uy;
so that it is necessary to determine U,,, and u,, in terms of P,,, and P,,,, respectively [49].
In the fundamental mode u = — VQ, so that utilizing equation II-31 for water,

00
Pw — Pow ag = 9

Introducing the harmonic time dependence yields:

O.. = == (Pes
Mw WPoy
so that
one OPV.
TWs WPo, or
and
Uy — — Salli ORs
l WPo, 02

Thus, from equations II-76 and II-97,
A2

iF = >
Z2Pouce
From equation II-81,

Pws = ByNo(Ko,,F) 5

where
eikowr

No(koyF) an ik r
2w

15

(I-88)

(II-89)

(II-90)

(II-91)

(II-92)

(II-93)

(II-94)

(II-95)

(II-96)

(II-97)

(II-98)

(II-99)

(II-100)

so that for r— oo,
ho’ (Koyf) = (11-101)

Thus, as r+,

: oe Bera i
are Uwe Wo, Koy (II-102)
and
= e0B.Ba (Il-103)
WPoyKoy,
Therefore,
AnIB,, 2
oO = aw ; ba
K,2At (II-104)

and in order to determine o, it is necessary to solve equations II-85 through II-88 for (B,,/A).

Boundary Conditions

Equations II-85 through II-88 contain five unknown coefficients. Thus, five boundary conditions are
required for the solution of this set of equations. These conditions are the continuity of normal velocity, u,, and
normal stress, T,,, at both interfaces and continuity of temperature at the inner interface. The first step in
solving for B,,/A, therefore, is to obtain u,, T,, and T in terms of P,,, P;, W,; and Wo.

Equation II-50 gives T, in terms of P,. Substitution of equations I-58 and II-60 through II-63 into equation
Il-50 leads to

2 { [va — Posferoe Jus + (va — 1) He (1-105)

Po,Ca oa

u,,, is given in terms of P,, by equation II-96. Following the same procedure to obtain u,,, equation I-31 may
be written as

P, + EV2Q, — p, Be ai (II-106)

Then substitution of equation II-26 into II-106 yields, with introduction of the harmonic time dependence:
i iwé
%=->-[1-~e]e -
WPo, PoC (II-107)

Similarly, from equation II-46 and II-61:

i
ES Dp., (Y + Wo) . (Il-108)
Thus,
= = Ae Hi OG eRe
Ot Oy [ Pc?) ar (1-109)
and
= i OW, dW.
y= —-— pee eens -
a Po, at + oe | (II-110)

The normal stress in fish flesh is [45]

Qu
Tort = = P, oP Sette . hs
eG aaa (II-111)
so that utilization of equation II-109 yields:
ig = 02P, :
Tan Spe ; (II-112)
"WP, PoC? Or
Also,
ta oP, (II-113)
and
a ey (II-114)

At the air fish-flesh interface the surface tension must be included in the stress equation. For a bubble in
water [50],
2
Py = Pou te (\I-115)
where P ,, and P,,,, are the total pressures inside and outside the bubble and R is the radius of the bubble at any
instant. If g represents the small changes in the radius of the bubble and a is the average radius, such that

R = at+g, (II-116)

where g<<a, then

Qi hone Coe _
aia ai Tae) rer g/a) , (II-117)
so that
2s
Pin = Rong as (1 3 g/a) . (II-118)

Since P,,and P ,,,, are total pressures, equation II-118 can be linearized utilizing equation II-10. Thus, the first
order equation is
Peay) = Pinch soe (I-119)

Due to the harmonic time dependence,

UAMOR tas 11-120
Web o O9) ( )
Thus
:
Poy = Pm = =a - (\l-121)

The boundary conditions are:

BC 1 Ue Ul ate —ib), (II-122)

BC 2 Tee area bir (II-123)

BC 3 u, = U, at r=a, (1-124)

2iu,,S
BC 4 Ty = Trra = wae at r=a, (II-125)
BC 5 fe 00)" vat’ Gr—a (II-126)

Equations |I-122 through II-126 have been labeled BC 1 through BC 5 for later convenience. Substitution of
equations II-105, II-109, II-110, II-96, II-112, II-113, Il-114, and [I-85 through II-88 into equations II-122
through II-126 yields:

;
BC 1, (_) [ Akewio’ Koy) + Bakeyo’ (Koy) |
1 ors - ,
= (1) (1 - ES +) [Baio hab) + Eiken’ kab) | (127)
BC 2, Ajo(k2b) + Byho(K2,,0) = Brjo(Kab) + E,no(Ka,b)
ing 77 ” .
+ (BS-) (1 ~ SEE) [Btetie' hb) + Eikatn" kab) | (1-128)
1 IW - y ,
BC 3, (ie) (1 - oe ) [ Bikaio’ (kaya) + Esky’ (koa) |
Ea
( Paz )[ Dak, oho’ (Ki,4) + Bakoaio’ (Koa) | ; (II-129)
BC 4, Bio(kaa) + Eino(kea) + (22) (1-3) [Bike 2ie" (keya) + Exke?no"(kea) |
Poy Po Ci*

. 2 a ais ;
= Dajo(k;,8) + Bajo(k2,a) + ( —— ) [ Dak, ao (Ki,€) + Bakoalo (k,,a) | » (II-130)

(l-131)

iPoaCpaCa® -
and BC 5, ( Ya ebb rea) Dajo(Ki,8) + (Ya — 1) Bajo(kK2,a) = 0

CHAPTER Ill
SOLUTION

The first step in the solution of equations II-127 through II-131 for(B,, /A) is to solve II-131 for D, and
substitute the result into equations II-129 and II-130. For conciseness the boundary conditions will now be

written in a tabular form:

BC By B, E,

1 $i; Si. Sis
2 S>, Soo Sos
3 0 S32 S33
4 0 S42 Sag

The Sis are taken such that

$B, + SB, + S,E, + $,B, = aA
The S,’s and as are as follows:

Si = — (2) ho’ (head)

Ow
Ww? ee
S12 = ( 9o,02Ko, ) jo (K2,b) ,
W2
S,, = son ) No! (Ke,b
ie ( PorCi2Ks, 0 ( = )

tye | (PE) tect)

Ww

S2; = S- No(K2,,b) ,

2
32 (Geecten Jo’ (Kaa),

w?2 i
S33 = IGescres)) No’ (Kaa) ,

Sa = ( 1 ) (Wa) Ky Jo! (K1,) jo (Ko,€)
Poa i 24y 07
1 (Ye | jo(K;,€)

Seo = jolkaa) + (ZPS-) jo"(kaa)

S43 = Mo(kz,a) + ( mer) No"(Ka,a) ,

19

nnoo
wo
.S

A
a,
a,
0
0
(III-1)
(III-2)
(III-3)
(II-4)
(II-5)
(III-6)
(III-7)
(IIl-8)
(III-9)
(IIl-10)
(I-11)
(I-12)
(I-13)
(IIl-14)

_ a~ 1)Jo(K2,4) 2s Ky alo’ (ks ,€)
S = (Y : O\R25 1ajo la )
a 1, C.-C... [ xs ( Wa? Po, ) ( Jjo(K; 8) |

(ve Z RK )

+ jolkega) + (

where equation II-68 has been used for k,,2.
The solution of the boundary conditions for B,, is

28K, ae
wa) Jo’ (Koa) (Ill-15)

By _ a,U + a,W_ (Ill-16)
A $,,U0+S;,W
where
U = S44(S22S33 — S23S32) + S54(So3S42 — So2Sa3) (IIl-17)
and
W = Sa4(Si3S32 al $12S33) =f S34(Si2Sa3 = $1384) : (IlI-18)

This solution is easily arrived at after several pages of substitutions.

Equation III-16 could be solved numerically with a computer by choosing values for the various physical
parameters and calculating the S,'s and a,'s. However, this will not be done because the purpose of this
report is to obtain simplified equations for w, and o which can be solved without resorting to use of a
computer. The simplified solution is based on the ranges of the physical properties and the limits of the
variables given in Chapter Il. Itis obtained by assuming that | k,,a | is large and that k,,b, k,,b, and k,,a are
small and accepting any errors which are less than ten percent. Order of magnitude comparisons are then
made between various terms and any term which is always less than ten percent of another term is
neglected. This simplification process is shown in Appendix B. The result of this process is:

Sy = (1 + A+i0 | ae

3PoCy =) te 2s
Seer |G)! (eae

i CwPo 3P0,Ca? 2s 11& =
+i (who 2 Po.Cat = eos
( wap, ) [ (wats, ) ( 30.0,78) os ( 319,2C,2a2 ) | | ’ (III-19)

where
AS ae a _ 1) (III-20)
Po,2C;4.a8 9Po,,Cw?
and
= (Zoe) | (Pose, 1) - (Pate YC
A = = Spyera a3 [ —— 2 1) (once sls 4 )\G) | : (IIl-21)

Equation III-19 can now be substituted into II-104 to yield o. However, before doing this, equation III-19
will be manipulated so that the final results can be easily compared to those of other researchers. Thus, the
resonant frequency is determined by setting the imaginary part of the denominator of equation III-19 equal
to zero and the real terms of the denominator represent the damping [18]. Therefore, letting

20

CE) (1— ae) eB)
W?a? Do, 3Po,Caa 3Po,C;2a?

and solving for w?a? yields the resonant frequency of the new model:

3P,Ca? ( ees )
dytigy wide ctor 30,6228
Wp-ae = (; 118
ui 3Po,2C;2a? )
Now [42],

Po,Ca? a YaPo,

and from equation II-115,

2s 2s
Po. = Po, tf eae Row ara

Therefore, substitution of equations III-24 and III-25 into III-23 yields:

W 2a? = Fr 182
(1 + ap,20287)
3Po,2C;2a2

The damping factor, H, will be defined by

w Po, wa 26
SW es
1) PoCw Po, Wa?

+e (= yi (1+ 2s Ny.

wa 2Po,Cpa Po, W2a3

Thus, following Devin [19],
1

1 1 1

a= +an +a e=d,

H Hrad Hvis Ain
where

— Do PoSw

Had @2P,a ,

Hvis Tz a ’
and

Hy = gto (2Prstms)" (1 428)”
28 3(Ve—a) WK, Po, Was

As before, at w = w,, H = Q. Thus, substitution of equations III-26 and III-27 into III-19 yields:

ei

(Il1-22)

(IIl-23)

(IN1-24)

(IN1-25)

(IN1-26)

(IN1-27)

(IN1-28)

(II1-29)

(Il1-30)

(IN-31)

A ( Wo \ , i (Fe, = ( 1+ Uns ) es
WH ) ay ) 3Po,C,2a2
Then, substitution of equation III-32 into II-104 yields the scattering cross section of the new model:

Pow
A4na2'( 1+ A)? + 2
7 ame (BE) ARM (\lI-33)

(BR) + SE - 10 ge
(3 2 + Ww2 1 ee 3Po,2C,2a2

where A, A, ®,, and H are given in equations III-20, III-21, II|-26, and III-27.

22

CHAPTER IV
DISCUSSION

Results

The equations developed for w,, 0, and H in Chapter III for the new model can be considered to be
accurate within ten percent only for the parameter ranges given in Chapter II. The ranges of &, for which
exact values are not known, and s, which may be varied, were extended beyond the limits of present
available data. However, the largest values of a, w, and P,, used in this model are not beyond the range of
possible values. These limitations were imposed by the assumption that k,,a is small. This required that, for
the error caused by using just the first term of the spherical Bessel function expansions to be less than ten
percent,

wa < 2.4 x 10* cm/sec.

The limitation of k,,a being small is a factor in all the previous models, but in most cases it is not mentioned
or, if it is mentioned, no limiting value is given.
The range of & utilized in the solution of the new model was
1 poise < &€ < 104 poise.
However, a more likely range of &, as shown in Appendix A, is
50 poise < & < 2 x 103 poise.
This narrower range will be utilized in the remainder of the discussion because it results in a simplification of
equations III-26 and III-34 without, in all probability, affecting the results. Equations III-26 and III-34 simplify

to P 2s

3Y
@peae = et 3y, — 1 IV-4
0 Do, Da | Y ) (IV-1)
a ane (Bs)
PSs Ream CREA Sela UR Er Py Cae (IV-2)

W,2 W,2 2
min cae |) ||
respectively, when the upper limit of & is reduced to 2 x 10° poise. A and A are negligible compared to 1 for
&< 2 x 103 poise. A and A were the only remaining terms which contained b. Thus, for the a/b range given
in Chapter II, b is unimportant for § $2 x 10° poise.

Figure 2 gives w,a as a function of depth and surface tension, calculated from equation IV-1. s has
essentially no effect on w,a for s/a < 10° dyne/cm? at any depth and for s/a < 10° dyne/cm? for depths
below 100 m. For s/a values around 5 x 10’ dyne/cm?, pressure has essentially no effect on wa for depths
less than 100 m and at a depth of 1000 m, w,a is increased by only 50 percent over its value near the
surface.

At resonance,

Ana2Q2 Pow \*
Oo = 4mla awe iS
( Pos ) (IV-3)
and
= PorCw
OF Poy, @oa ’ (IV-4)
— Wo Po? hy
Qi tae (IV-5)
and
ja Wa ( a a ( 2s —1
On 3(¥, — 1) Ka As ses) (IV-6)

Thus, 0 is proportional to Q? at resonance.

23

HSI4 ONINVAE-YSGGVIENIMS V SO TAGOW MAN AHL SO AONSNDAYS LNVNOSSY =e AYNOIS

$49jaW — H1d30
¢Ol 201 Ol l

2W9/auAp ul D/s

,Ol

99S/Wd —D°Mm

p jOl XS = 07S 2

ee eee l a

24

Figure 3 shows Q,,, and Q,,, as functions of w,a. Q,,, increases with increasing W,a and decreases with

increasing &/a. Values of

2 x 102 poise/cm < &/a < 10° poise/cm
give values of Q,,, which are comparable to values obtained for Q in near-surface swimbladder resonance
experiments [15, 33, 36-38]. Q,,, decreases with increasing w,a and, depending upon the value of &/a, may
be significant for all w,a considered.

Figure 4 shows Q,, as a function of w,a, a, s/a, and depth D. Any individual numbered curve indicates
that Q,, increases gradually with increasing s/a. The set of numbered curves indicates that Q,, increases
with increasing a and D. The lettered curves are for the case where the effect of surface tension is
insignificant, and also indicate that Q,, increases with increasing a and D. A comparison of figures 3 and 4
shows that Q,, is significant only at depths above 100 m and then only if €/a < 10° poise/cm.

At off-resonance frequencies

ty = (Sy Gn 2 (Iv-7)
lis Fi Qiis ’ (IV-8)
and 1 2s
+ a
V2 p W,2a3
Hn = (S2)" | 2 Ja, wv)
' Po, W2. as

Thus, at frequencies above resonance, H,,, may be the dominant damping term, even for relatively high
values of E/a. Conversely, at frequencies below resonance, H,,, may be the least important damping term.
The variation of H,, with (w,/w) is complicated by the presence of the surface tension term. !t can be shown,
utilizing equation IV-1, that

Wee eee Fe yaa vane)
Po, Wo? a2
so that, for the purpose of establishing a trend,
w Ya 25 -1 i
His (22) (1+ tn) On (IV-11)
Hence, at frequencies above resonance and at frequencies below resonance for which (o,w°a® >>(28/ a),
(0) Va
ns (=) Qn (IV-12)
At frequencies below resonance for which (2s/a) >> Po,W*a?,
Wo \ "2 / Po, 2a i
Hin ~ (3) (a— ) Qi = (IV-13)

Thus, H,, becomes relatively less important than H,,, aS frequency increases and conversely.

Comparison to Free Bubble

The validity of the results developed for the new model must be determined. This will be done by first
checking that the equations developed for the new model approach the equations for a free bubble in water
as a limiting case and then comparing values calculated using the new model with experimental values.

The equations obtained in Chapter III for the resonant frequency and scattering cross section of the new
model can be readily compared to the equations given for a free bubble. Equations I-6 and |-7 give w, and o
for an ideal bubble, neglecting surface tension. Equation II|-26 reduces exactly to equation I-6 when & = s =
O. If thermal losses are also neglected, equation III-33 reduces to equation I-7 when € = s = 0, except for
factors of (P,/Po,). If surface tension is included, w, for an ideal bubble is given by equation I-12 with
1);,, = 0. Equation Ill-26 reduces exactly to equation I-12 when & = n,,, = 0. Thus, when viscous and thermal
dissipation effects are eliminated from the new model, the results are equivalent to results obtained for an
ideal bubble.

When the viscosity of water is considered for an air bubble in water, w, is given by equation I-12. A
comparison of equation II|-26 with I-12 shows that the form of the viscosity factor for the new model differs
from that given for a bubble. However, as was noted earlier, equation |-12 is only valid for small viscosities,
so that this difference is not surprising [25]. Both equations I-12 and III-26 indicate that w) decreases as the
viscosity increases, which should be the case for a viscously damped system [51].

25

102

° Qvis, €/a =10
2
102
Q
5 & Qvis, €/a = 102
o
Eg 2
&
10
5

Qvis ’ &/a = 105
2 . .
€/a in poise/cm

2 5 104 2 )

Wad - cm/sec

10° 10°

FIGURE 3. VISCOUS AND RADIATION DAMPING FACTORS FOR THE NEW MODEL OF A
SWIMBLADDER-BEARING FISH

26

108

10
9
8
5 7
6
5
4
2 3
2
102
5
=
Go
2
1: D= a=0.1 1025
10 2: D= a=0.1
3: D= a=0.3 alOmr
v3 1D a=0.3
5 5: D= a=1.0 1025
6: D= a=0.1
7: D= a=1.0 2: 028
8: D= a=3.0
9: D= a=0.3 Dinm
2 10: D= a=3.0 ain cm
1s D a=1.0 s/a in dyne/cm*
2 2) 2 5
108 104 108

Wa - cm/sec

FIGURE 4. THERMAL DAMPING FACTORS FOR THE NEW MODEL OF A SWIMBLADDER-
BEARING FISH

27

The damping in the new model can also be compared to that for a real bubble in water as determined by
Devin [19]. For ease in comparison with Devin’s results, equations IV-4, IV-5, and IV-6 will be rewritten as:

Poy oa
OF eee IV-14
‘ Po; Cw ( )
On = ae IV-15
vis Wo Po,? ( )

and

B(Ve = 1) Ka \ 2s IV-16
Om = Wo”a (cones) (1 ? aura ) ; \

Equation IV-14 is equivalent to equation I-9, except for the factor (P,/>,). Devin has considered only the
shear viscosity, so that equation Il-1 appears to indicate that equations IV-15 and I-10 differ by a factor of
24. However, nN, and n, are related to the dilatational viscosity ny, by [45]

Mm = Ta + MN, - (IV-17)

Therefore, if ny is neglected, equation II-1 indicates that § = 2n,. Hence, equations IV-15 and I-10 are
equivalent. Equations IV-16 and I-11 are also equivalent, except for the surface tension factor in equation
IV-16. Devin considered surface tension, but since the surface tension of an air bubble in water is only 74
dynes/cm, its effect is insignificant for the frequencies and bubble sizes of present interest. Hence, the form
of the damping factor at resonance for the new model is essentially equivalent to that of a real bubble in
water.

Comparison to Experimental Data

The previous comparisons compared the spherical model to spherical bubbles. In order to compare the
model to experimental measurements made on swimbladder-bearing fish, it is obvious from figure 1 that the
effect of swimbladder shape should be included. This is done by letting

Wo, = GW (IV-18)

where W,, is the theoretical resonant frequency for the spherical model, ¢ is given by equation I-25, and Wy,
is the value which is utilized in the comparisons to the experimental data.

There are five sets of experimental data to which calculations based on the new model can be
compared. These are contained in references 15, 33, 36, 37, and 38. In each of these, the measured
resonant frequency, W,,,, and the calculated Q are either given directly or can be obtained from a curve. The
data in reference 36, which was collected a decade before any of the other data, have variations which
require some interpretation to obtain a,, and Q. This required interpretation makes the w,,, and Q values
obtained from reference 36 less reliable than those obtained from other sources.

Comparisons of the data to the model require that values for a and € be known. In some cases, a and €
were measured and the values given. In the other cases, it was possible to determine a indirectiy from other
data in the report. When values of € were not given, average values were obtained from other sources.
Measurements of a and € were all made at atmospheric pressure. There is some question as to what effect
increasing depth has on swimbladder volume [e.g. 11, 12]; that is, whether the fish retains a constant
swimbladder volume or a constant swimbladder mass (Boyle’s Law) or some other, intermediate process.
(McCartney and Stubbs [87] assumed an intermediate process, due to the effect of tension in the
swimbladder wall.) It is probable that different species react to changing depth in different ways and that
different experimental methods can affect the way a fish would normally react. Hence, there are
uncertainties in the actual values of a at depth. Since the swimbladder is attached to various other parts of
the fish, there is no reason to expect a change in swimbladder volume to cause a uniform change in its
linear dimensions. Thus, the uncertainty in a at depth produces a smaller uncertainty in €. Various depth
variations in a were examined when comparing the data to the model. However, since the individual
researcher is in the best position to interpret his own data, the final comparisons utilized the depth variations
chosen by those researchers.

The new model requires values of & and s. Since the exact value, or range of values, of & is presently
unknown and s is quite likely under the control of the individual fish, direct comparisons of the model and

28

experimental data are not possible. However, indirect comparisons are possible. The first step in a
comparison is to determine the experimental values of w,,, Q, a, and €. Then W,, is set equal to wW,,, and s
calculated utilizing equations IV-1 and IV-18 and figure 1. If surface tension has a negligible effect on w,,,
can be assumed to be equal to 10* dyne/cm, based on the data given in Appendix A. Q,,, and Q,, are then
calculated from equations IV-4 and IV-6 and Q,,, is determined from the equation:
1 1 1 1

Q 4 Qraa i Qvis bi @iy f (IV-19)
Finally, § is calculated from equation IV-5. The results of this procedure are given in Table II, where L is the
fish length, D, a, @,,,, and Q are experimental data, and s and & are calculated by equating the model to the
data.

The new model can be indirectly compared to the experimental data by examining the values of s and &
given in Table li. The values shown in Table I! are within the limits chosen in Appendix A to give reasonable
ranges for s and €. This is a necessary condition for the model to be valid, but it is by no means a sufficient
condition. In order that the model can be used with some degree of confidence to predict resonant
scattering from swimbladder-bearing fish, the variations in s and & shown in Table II must also be explained.

The five sets of data can be separated into two groups, based upon the magnitude of tension in the
swimbladder wall. Swimbladder tension had little or no effect on the measurements made by Coate, Batzler and
Pickwell, and Sundnes and Sand, where s < 10° dyne/cm. Swimbladder tension appears to have had a significant
effect on the measurements by McCartney and stubbs and Sand and Hawkins, where

2 x 10° dyne/cm < s < 4 x 10° dyne/cm.

Table Il — Experimental Data and Results of Comparisons to the New Model

Source Fish L D a Wo, Q s eS
cm m cm rad/sec dyne/cm poise
Coate [36] Crappie 20 46 1.10 2400 4.6 104 300
Batzler and Goldfish U 6 0.51 5650 Si 6 x10 130
Pickwell [15] Goldfish 6 6 0.44 6600 3.8 5 x104 160
; Anchovy 11 6 0.40 8000 4.5 7 x104 130
McCartney and Coalfish 30 10 Wee 4150 ileal 1.7x 10° 2600
Stubbs [37] 20 1.08 5200 (1.7 1.9x 108 1750
30 1.03 6000 2.5 1.7x 10° 1200
Pollack 35 30 Uses} 4800 1.4 3.6 x 10° 2200
40 1.09 5800 16} 6.1 10° 2600
Ling 50 30 1.60 3150 225 3} S102 1500
Cod 35 30 leo2 3500 2.0 1.6x 108 2000
Sand and Cod 16 11 0.69 10700 1.0 3.8 x 10° 2500
Hawkins [33] 25 0.58 g000. «1.3 2.8105 1000
30 0.55 8800 1.6 2.1x105 830
35 0.53 9900 2.2 3.3x 105 610
40 Ori 10900 2s) 3.8x 105 540
45 0.50 12000 3:2 5.1 10° 430
50 0.48 12800 $)45) 4.7x* 10° 370
Sundnes and Charr 40 2 0.95 3000 5x2 10¢ 240
Sand [38] (Averages 6 0.87 4000 45 104 300
for 5 10 0.80 4900 5.0 10¢ 280
fish) 15 0.74 5800 6.1 108 220

29

Goldfish, anchovies, and charr are physostomes, that is, their swimbladders have an opening into their
stomachs. Hence, based on consideration of equation II-115, it is to be expected that the tension in
physostome swimbladders would be relatively low. The data show that this is so. The other fish are
physoclists, that is, their swimbladders are completely closed. These fish may well obtain some benefit, either
hydrostatic or otherwise, by maintaining a tension in the swimbladder wall. The data indicate that cod, ling,
pollack, and coalfish maintained relatively high values of s. Physiologically, it should be possible for a
physoclist to vary the tension in the swimbladder wall from very taut to flaccid. It appears that the crappie
swimbladder was in a flaccid condition.

Since physoclists probably can vary the tension in their swimbladders, any shock to their systems, such as
a rapid increase in pressure, might cause the swimbladder tension to change dramatically. This seems to be
the situation for the cod examined by Sand and Hawkins. This fish had been allowed to become adapted to a
depth of 11 m for at least 48 hours. At this depth, its measured resonant frequency was several times higher
than that of a bubble of the same presumed size and shape. This difference was attributed to a high
swimbladder tension. Rapidly shifting this fish to a depth of 25 m significantly decreased the measured resonant
frequency to a value only about 10 percent greater than that expected for a bubble of the same presumed size and
shape. This ratio remained essentially constant as the fish was shifted in 5 m increments to 50 m. The
decrease in w,,, as the fish was shifted from 11 m to 25 m is readily explained by the new model by a
sudden decrease in s, which seems quite feasible from a physiological standpoint.

Thus, the variations in the values of s calculated using the new model can be explained by the
differences between physostomes and physoclists and by the ability of physoclists to vary swimbladder
tension. Hence, the new model is an improvement over the models of Andreeva and Lebedeva. By using a
variable swimbladder tension, rather than a fixed shear modulus, to model tissue stiffness, the new model
can predict resonant frequencies significantly higher than those expected for a free bubble, whereas the
other models cannot.

As a practical matter, a parameter that can be randomly varied over several orders of magnitude is not
very useful in a predictive model. However, the variations in tension indicated in Table II were obtained from
fish which were subjected to other than natural conditions, such as rapid changes in depth. It is quite
possible that, under more natural conditions, values of swimbladder tension would be much more uniform.
Thus, it could very well be that further experiments, in which the experimental conditions more closely
approximate natural conditions, could provide much information about swimbladder tension. This would
enhance the value of the new model as a predictor of resonance in large physoclists.

Although the variations in s can be adequately explained by variations in fish physiology, the uncertain
variation of a with depth definitely causes some degree of variation ins. In order to calculate s, the difference
between w,,,and W, for a bubble of the same assumed size and shape is required. Thus, the accuracy of the
estimate of a greatly affects the accuracy of the calculated value of s.

Sundnes and Sand measured the resonant frequencies of charr in order to determine their swimbladder
volumes. In the case of these physostomes, this was a valid procedure. However, in the case of large
physoclists, where swimbladder tension affects w,,,, this procedure would be invalid. Hence, experiments
which might be designated to acoustically examine swimbladder tension require an accurate and
independent measurement of swimbladder volume at depth.

The values of € given in Table II are also grouped on the basis of the magnitude of swimbladder tension.
For the cases where

s < 105 dyne/cm,

the values of — range from 130 to 300 poise, with the values being quite consistent for a particular set of data.
For the cases where
s > 2 x 105 dyne/cm,

the values of E range from 370 to 2,600 poise, with wide variations within a particular set of data. Based on the
consistent results obtained, and the reasonableness of their values, it appears that the new model can be
used to predict the damping for the first group. However, the high values and variability of § in the second
group must be examined more closely before any conclusion can be reached.

30

There are essentially two sources of the variations in the calculated values of € between the two groups
and within the group where s> 2 x 10° dyne/cm. One source is the experimental data and the other is the
model. The most likely, and probably greatest, source of error in the experimental data is the uncertainty in
swimbladder volume. Since the calculated values of & are proportional to a2, an error in a can cause a large
error in —. However, as mentioned earlier, the experimental data were examined assuming various
swimbladder volume-depth relationships. None of these relationships produced consistent results or values
as low as those obtained for the first group, although in some cases the variability was reduced. For example,
if aconstant a=0.69 cm is assumed for the cod examined by Sand and Hawkins, & varies from 1,500 to 900
poise as the depth increases from 25 to 50 m. This variation is smaller than that for the swimbladder-depth
relationship which was utilized, but the values are still much higher than those of the first group and are by no
means consistent. It is possible, though, that the cod-like fish, which have a well-developed muscle system,
have an intrinsically higher tissue viscosity than fish such as goldfish or anchovies.

At the beginning of this chapter, it was assumed that

— < 2 x 10° poise
and the equations were simplified accordingly. It might be suspected that this could be a source of error in the
cases where & > 2 x 10% poise. However, the error caused by simplifying equations IIl-26 and III-34 to
equations IV-1 and IV-2 is less than ten percent for values as high as & = 6 x 10° poise. Thus, the simplified
equations are not a significant source of error.

The other primary source of error is the model itself. Since it does not seem to be possible to attribute all
the variations in & to the errors in the data, it appears that the remaining variations are due to inadequacies
in the model.

An examination of all the data in Table II indicates that € generally increases with s. There is no indication
that this is so for the group of data for which s < 10°dyne/cm. However, there is definite correlation between s
and &— when the two groups of data are compared. Also, despite the wide variations of € within the group of
data for which s > 2 x 10° dyne/cm, there is a rough indication that € increases with s for this group of data.
Thus, it appears that increasing tension in the swimbladder wall has little or no effect on & for s < 10° dyne/cm,
but that it does cause E to increase for s > 2 x 10° dyne/cm. The apparent increase in viscosity with increasing
tension in the swimbladder wall for s > 2 x 105dyne/cm would not be expected to occur in a Newtonian
fluid. This indicates that modelling a fish as a Newtonian fluid may not be appropriate for s > 2 x 10°
dyne/cm and that some type of non-Newtonian model, such as a dilatant or viscoelastic fluid, may be more
appropriate. However, the present model is appropriate for s <10° dyne/cm.

31

32

CHAPTER V
CONCLUSIONS

The new model of a swimbladder-bearing fish as a viscous shell enclosing an air cavity with surface
tension at the inner interface was developed because older models, of which Andreeva’s model [26] is mosi
widely employed, do not accurately predict the results obtained in experiments on the characteristics of
swimbladder resonance. The principal difference between the older models and the data is that the values
of Q predicted by the models are always higher than those obtained experimentally. A second difference is
that the older models can not account for the high resonant frequencies obtained for some large
physoclisits. The new model sought to correct these differences by explicitly including the viscosity of fish
tissue and by including a tension in the swimbladder wall.

For convenience, the equations developed for the new model will be restated here:

3YaPo 2s
W 2a2 = —=— + SV ata) es 4
0 Da, Da (3y ) (V-1)
2
4nae(5)
Pe eM NO TS (V-2)
Wo? ( Wo? 2
ei (e+) |
weH2 w?
H = Wo Po;Cw (V-3)
rad @2o,a ’

ye (V-4)

Menaea 20o2Cp, \” ( 2s ) =i
ian Siva) ( WK, +) as Po Wass ve
Comparable equations for an air bubble in water are available at only w = W), where H = Q. The equations
for o and Q obtained for the new model are essentially the same as those for an air bubble in water, the only
significant difference being in the value of the viscosities of fish flesh and water. However, the equations for
Wp for the new model and an air bubble in water for which viscosity and surface tension are included are not
the same. If the viscosity of fish flesh were used in the equation obtained for an air bubble, w, could be zero
or imaginary. However, for the new model, the effect of viscosity on w, was shown to be small enough to be
neglected.

Equations V-1 and V-2 are valid for an upper limit for § = 6 x 10° poise. If € > 6 x 10° poise, then the
outer shell radius, b, appears in the o-equation. This implies that this is a boundary layer-type problem.
Thus, if the fish flesh-water interface is outside the boundary layer, the magnitude of b is immaterial.

All of the models are spherical in nature. However, the swimbladders of some fish are sufficiently
elongated that their shapes can have a significant effect on w). Thus, equation V-1 should be modified to
include this effect. Hence:

and

Wo2a2 = C[—*2 ow + = (3y, — 1) | : (V-6)
f

Although G was determined for a bubble in water, its use here is quite reasonable, especially when the
similarity between the equations for a bubble and the new model are considered.

Several other conclusions can be reached for the new model. One is that, for the ratios of outside to
inside shell diameters considered, the actual value of the outside diameter has no effect on the results.
Another is that thermal losses are not very significant and, in most cases, can be neglected. A third is that,
for low values of H, (W./wH)? can be comparable to [(w,2/w?) — 1]? at off-resonant frequencies. Hence,
considering the experimental values of Q obtained, it is apparent that H, rather than Q, should be used to
calculate o.

A comparison of the new model with available experimental data indicates that the new model
constitutes a definite improvement over previous models. The new model can predict low values of Q and
elevated values of wy which the previous models could not. In addition, the new model can be used to
obtain the magnitude of damping at any frequency, whereas many previous models only produced the
value at resonance.

33

The comparison of the new model with experimental data indicates that the model is most accurate for
fish in which s < 105 dyne/cm. Table II indicates that this includes physostomes up to at least 40 cm in
length. In addition, Appendix A indicates that small physoclists, 10 cm or less in length, are also included
in this group. Most of the fish found in deep scattering layers are smaller than 10 cm and none have the well
developed musculature of fish such as cod. Thus, it appears that the new model will be of considerable
value in studies of volume reverberation. Table II indicates that a value of & of approximately 200 poise is
appropriate for these studies.

The variations in s and & required to match the values calculated using the new model to the
experimental data for large physoclists indicate that the new model is not completely adequate for these
fish. Even so, the new model is still better than previous models and, as such, is of some value in studies of
resonance of large physoclists. Table II indicates that values of s = 10° dyne/cm and 10 poise < — $2 10
poise should give reasonable results for these studies.

The new model introduces two new parameters, & and s. Few measurements of these parameters exist
and those that do vary widely. The accuracy of the new model could be determined with much more
confidence if better information on & and s were available. In addition, the variations of swimbladder volume
with depth for different fish species are required for all models. Hence, further experiments to determine s.
€, and swimbladder size and shape versus depth are recommended. This recommendation is made with
the realization that the complexities involved in these experiments will be significant.

34

25.

26.

REFERENCES

C.F. Eyring, R.J. Christensen, and R.W. Raitt, “Reverberation in the Sea,” J. Acoust. Soc. Am. 20, 462-475
(1948).

R.W. Raitt, “Sound Scatterers in the Sea,” J. Marine Res. 7, 393-409 (1948).

R.S. Dietz, “Deep Scattering Layer in the Pacific and Antarctic Oceans,” J. Marine Res. 7, 430-442 (1948).
J.B. Hersey and H.B. Moore, “Progress Report on Scattering Layer Observations in the Atlantic Ocean,” Am.
Geophys. Union Trans. 29, 341-354 (1948).

M.W. Johnson, “Sound as a Tool in Marine Ecology, from Data on Biological Noises and the Deep Scattering
Layer,” J. Marine Res. 7, 443-458 (1948).

V.M. Albers, Underwater Acoustics Handbook-!I (Pennsylvania State University, University Park, PA, 1965)
p. 47.

N.B. Marshall, “Bathypelagic Fishes as Sound Scatterers in the Ocean,” J. Marine Res. 10, 1-17 (1951).
J.B. Hersey, H.R. Johnson, and L.C. Davis, “Recent Findings about the Deep Scattering Layer,” J. Marine
Res. 11, 1-9 (1952).

J.B. Hersey and R.H. Backus, “New Evidence that Migrating Gas Bubbles, Probably the Swimbladders of
Fish, are Largely Responsible for Scattering Layers on the Continental Rise South of New England,”
Deep-Sea Res. 1, 190-191 (L) (1954).

H.R. Johnson, R.H. Backus, J.B. Hersey, and D.M. Owen, “Suspended Echo-Sounder and Camera Studies
of Midwater Sound Scatterers,” Deep-Sea Res. 3, 266-272 (1956).

J.B. Hersey, R.H. Backus and J. Hellwig, “Sound-Scattering Spectra of Deep Scattering Layers in the
Western North Atlantic Ocean,” Deep-Sea Res. 8, 196-210 (1962).

G.B. Farquhar, Ed., Proceedings of an International Symposium on Biological Sound Scattering in the
Ocean, Rep. 005 (Maury Center for Ocean Sci., Washington, DC, 1970).

S. Machlup and J.B. Hersey, “Analysis of Sound-Scattering Observations from Non-Uniform Distributions of
Scaiterers in the Ocean,” Deep-Sea Res. 3, 1-22 (1955).

R.J. Urick, Principles of Underwater Sound for Engineers (McGraw-Hill, New York, 1967), p. 188.

W.E. Batzler and G.V. Pickwell, “Resonant Acoustic Scattering from Gas-Bladder Fishes,” in Proceedings
of an International Symposium on Biological Sound Scattering in the Ocean, Rep. 005, G.B. Farquhar, Ed.
(Maury Center for Ocean Sci., Washington, DC, 1970), pp 168-179.

F.R.H. Jones and G. Pearce, “Acoustic Reflection Experiments with Perch (Perca fluviatilis Linn.) to
Determine the Proportion of the Echo Returned by the Swimbladder,” J. Exp. Biol. 35, 437-450 (1958).

M. Minnaert, “On Musical Air-Bubbles and the Sounds of Running Water,” Phil. Mag. 16, 235-248 (1933).
“Physics of Sound in the Sea, Part IV,” Nat. Defense Res. Comm. Div. 6 Sum. Tech. Rep. 8, Chap 28,
pp 460-467 (1946).

C. Devin, Jr., “Survey of Thermal, Radiation, and Viscous Damping of Pulsating Air Bubbles in Water,” J.
Acoust. Soc. Am. 31, 1654-1667 (1959).

A.|. Eller, “Damping Constants of Pulsating Bubbles,” J. Acoust. Soc. Am 47, 1469-1470(L) (1970).
W.M. Fairbank, Jr., “Damping Constants for Nonresonant Bubbles,” J. Acoust. Soc. Am. 58, 746(L) (1975).

H.B. Briggs, J.B. Johnson, and W.P. Mason, “Properties of Liquids at High Sound Pressure,” J. Acoust. Soc.
Am. 19, 664-677 (1947).

G. Houghton, “Theory of Bubble Pulsation and Cavitation,” J. Acoust. Soc. Am. 35, 1387-1393 (1963).

A. Shima, “The Natural Frequency of a Bubble Oscillating in a Viscous Compressible Liquid,” ASME J. Basic
Eng. 92, 555-562 (1970).

D.Y. Hsieh, “Variational Method and Nonlinear Oscillation of Bubbles,” J. Acoust. Soc. Am. 58, 977-982
(1975).

|.B. Andreeva, “Scattering of Sound by Air Bladders of Fish in Deep Sound-Scattering Ocean Layers,” Sov.
Phys. Acoust. 10, 17-20 (1964).

35

Cle

28.

Ae)
30.
31.

32.

33.
34.

35.

36.

37.

38.

39.

40.

41.

42.
43.
44.

45.
46.
47.
48.
49.

50.

51.

L.P. Lebedeva, “Measurements of the Dynamic Complex Shear Modulus of Animal Tissues,” Sov. Phys.
Acoust. 11, 163-165 (1965).

E. Meyer, K. Brendel, and K. Tamm, “Pulsation Oscillations of Cavities in Rubber,” J. Acoust. Soc. Am. 30,
1116-1124 (1958).

L.P. Lebedeva, “Sound Scattering by Fish,” J. Ichthyology 12, 144-149 (1972).

M. Strasberg, “Gas Bubbles as Sources of Sound in Liquids,” J. Acoust. Soc. Am. 28, 20-26 (1956).

R.L. Capen, “Swimbladder Morphology of Some Mesopelagic Fishes in Relation to Sound Scattering,”
Naval Electronics Lab. Report 1447, San Diego (March 1967).

R.H. Gibbs, Jr., R.H. Goodyear, R.C. Kleckner, C.F.E. Roper, M.J. Sweeney, B.J. Zahuranec, and W.L.
Pugh, “Mediterranean Biological Studies, Final Report,” Smithsonian Inst. Washington, DC (July 1972).
O. Sand and A.D. Hawkins, “Acoustic Properties of the Cod Swimbladder,” J. Exp. Biol. 58, 797-820 (1973).
M. Strasberg, “The Pulsation Frequency of Nonspherical Gas Bubbles in Liquids.” J. Acoust. Soc. Am.
25, 536-537 (1953).

D.E. Weston, “Sound Propagation in the Presence of Bladder Fish,” in Underwater Acoustics, V.M. Albers,
Ed. (Plenum, New York, 1967), Vol. 2, Chap. 5, pp. 55-88.

M.M. Coate, “Effect of a Single Fish on Low Frequency Sound Propagation,” Naval Ordnance Lab. Report
4514, White Oak, MD (April 1957).

B.S. McCartney and A.R. Stubbs, “Measurements of the Target Strength of Fish in Dorsal Aspect, Including
Swimbladder Resonance,” in Proceedings of an International Symposium on Biological Sound Scattering
in the Ocean, Rep. 005, G.B. Farquhar, Ed. (Maury Center for Ocean Sci., Washington, DC, 1970), pp. 180-
211.

G. Sundnes and O. Sand, “Studies of a Physostome Swimbladder by Resonance Frequency Analyses,” J.
Cons. Int. Explor. Mer. 36, 176-182 (1975).

W.N. Tavolga, “Sonic Characteristics and Mechanisms in Marine Fishes,” in Marine Bio-Acoustics, W.N.
Tavolga, Ed. (Pergamon Press, New York, 1964) pp. 195-211.

A.N. Popper, “The Response of the Swimbladder of the Goldfish (Carassius auratus) to Acoustic Stimuli,” J.
Exp. Biol. 60, 295-304 (1974).

R.H. Love, “Predictions of Volume Scattering Strengths from Biological Trawl Data,” J. Acoust. Soc. Am.
57, 300-306 (1975).

L.E. Kinsler and A.R. Frey, Fundamentals of Acoustics (Wiley, New York, 1962), 2nd Ed.

M. Jakob and G.A. Hawkins, Elements of Heat Transfer (Wiley, New York, 1957), 3rd Ed., p. 10.

H.U. Sverdrup, M.W. Johnson, and R.H. Fleming, The Oceans: Their Physics, Chemistry, and General
Biology (Prentice-Hall, Englewood Cliffs, N.J., 1942), Chap. 3, pp. 47-97.

F.V. Hunt, “Propagation of Sound in Fluids,” in American Institute of Physics Handbook, D.E. Gray, Ed.
(McGraw-Hill, New York, 1963), 2nd Ed., Sect. 3c, pp. 3-28 to 3-59.

P.S. Epstein and R.R. Carhart, “The Absorption of Sound in Suspensions and Emulsions. |. Water Fog in
Air,” J. Acoust. Soc. Am. 25, 553-565 (1953).

P.M. Morse and H. Feshbach, Methods of Theoretical Physics, (McGraw-Hill, New York, 1953), Part 1,
Chap. 1, pp. 1-118.

O.D. Kellogg, Foundations of Potential Theory, (Dover, New York, 1953), p. 156.

P.M. Morse and K.U. Ingard, Theoretical Acoustics, (McGraw-Hill, New York, 1968), Chap. 8, pp. 400-466.
F.W. Sears and M.W. Zemansky, University Physics, (Addison-Wesley, Reading, MA., 1955), 2nd Ed.,
Chap. 13, pp. 224-236.

W.T. Thomson, Vibration Theory and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1965), Chap. 2, pp.
36-50.

36

APPENDIX A
PHYSICAL PROPERTIES OF FISH

Several of the physical properties of fish flesh which must be known for this study have not been
accurately measured. In this appendix, the available data will be discussed and, where necessary,
appropriate approximations determined. Properties which must be specified include compressional wave
velocity (or sound velocity), c,; density, Qo,; specific heat at constant pressure, c,,; ratio of specific heats, y;;
thermal conductivity, K; shear viscosity, ne: and bulk viscosity, Ne In addition, the ratio of swimbladder
volume to total fish volume and the surface tension, s, at the air-fish flesh interface must be determined.

Experimental evidence indicates that swimbladder volumes of small mid-water fishes range from about
0.5 to 5 percent of the total fish volume [32]. For larger, near surface marine fishes, swimbladder volumes
are about 4 to 5 percent of the total fish volume [A1]. For small fish, the ratio of outer radius, b, to inner
radius, a, of the fish flesh shell is chosen to be 2.5 Sb/a $6, which corresponds to swimbladder percentage
volumes of about 6 to 0.5 percent. For large fish, b/a is chosen to be 2.5 <b/a $3.2, which corresponds to
swimbladder percentage volumes of about 6 to 3 percent. The fish size range of interest is about 1 cm to 1m,
which roughly corresponds to 0.1cm <a S$ 5cm (82, A2, A3].

The acoustic properties of fish flesh have been measured by several researchers [31, A1, A4-A6].
Experimental values of density range from about 1.02 to 1.09 gm/cm’, with an average of about 1.05
gm/cm%. Experimental sound velocities range from about 1.50 x 105 to 1.60 x 105 cm/sec, with an average
of 1.55 x 105 cm/sec. The average values are used in this report.

Measurements on the thermal properties of fish flesh are not available. However, data on the thermal
conductivity and specific heat at constant pressure of human and dog tissue do exist [A-7]. These data
provide sufficiently close approximations to the required values. Thus, for fish tissue

a Cal
C,, = 0.89 gm°C
and
|
Kalo 20X 10-24 _.
u cm sec °C
These values are quite close to those for sea water of 35 parts per thousand salinity, which are [45]
cal
Cow = 0.93 =—5
and i ginie
Ky = 1.34 x 10-9 __.
cm sec °C
Also for sea water [45],
Vw 2 1.01.
Thus, it will be assumed that, since other thermal properties of flesh and sea water are very similar,
y, ~ 1.01.

Both the shear viscosity, n,,, and the bulk viscosity, n,,, are required for the present model. However, for
convenience, a viscosity parameter, &, will be defined as

& = 4/3 ng + Ny - (A-1)
The ratio, n,/n,, for animal tissue is similar to that for water, which is approximately 3 [A8, A9]. Thus,
& RAS Ney ¥ 1.4 Noy - (A-2)

Only one set of data on the viscosity of animal tissue is available. However, other data exist from which
tissue viscosity can be determined indirectly. These data are measurements of absorption, complex shear
modulus, and cell viscosity of animal tissue. In all cases it will be assumed that the viscosity of all animal
flesh is approximately equal.

The direct measurements of viscosity were performed on mammalian tissue [A10]. n.,, was determined
by four different methods. The results ranged from 100 to 420 poise, with an average value of 175 poise.
Thus the values for & range from 430 to 1800 poise, with an average value of 760 poise.

37

Viscosity can be determined from measurements of absorption of sound by utilizing the equation [A111]:
2P,c3a

g = =P (A-3)
where a is the absorption and w the circular frequency. Many measurements of absorption in tissue have
been made at high frequencies. However, tissue exhibits relaxation phenomena at high frequencies so that
viscosities at frequencies above relaxation cannot be directly related to those below and can be used only
as lower limits. It has been found that for muscle, n, relaxes near 400 kHz and n, relaxes near 40 kHz
[A12]. Only one set of absorption measurements has been conducted below the Megahertz range. These
measurements were made at 300 to 350 kHz [A13]. Utilizing equation A-3, & was calculated from these data
to range from 30 to 220 poise, with an average value of 130 poise. However, n, has already relaxed at the
measurement frequencies, so that its contribution to the absorption is unknown. Thus, equation A-3
provides a lower limit to €&. An upper limit can be determined for & if it is assumed that the absorption is due
solely to n,. Then, at lower frequencies, & would range from 130 to 950 poise, with an average value of 580
poise. Thus, from the absorption measurements:

30 poise < &€ < 950 poise.

Viscosity can also be estimated from measurements of complex shear modulus by utilizing the equation
[AQ]:

_
AE ger (A-4)

where u, is the imaginary part of the complex shear modulus. One set of measurements of the complex
shear modulus of fish tissue has been made from about 2 to 14 kHz [27]. Utilizing equations A-4 and A-1, &
was calculated from these data to range from about 4 to 90 poise, with an average value of 28 poise.

The viscosity of animal tissue can also be estimated from the viscosity of animal cells if it is assumed
that the tissue viscosity is equivalent to the cell viscosity. Separate measurements have been made on the
viscosities of both cell protoplasm and membrane. Thus, to estimate the viscosity of the complete cell, a
geometric average of the membrane and protoplasmic viscosities is calculated based on the proportional
thicknesses of membrane and protoplasm. The equation used to calculate cell viscosity is

ae ce Sp Gm lig (A-5)

where €,, & and €,, are the viscosity parameters of the cell, protoplasm and membrane, respectively, and a
and b are the inner and outer membrane radii. Cell radii range from 2 x 10-4 to 15 x 10-4 cm and cell
membranes are 75 x 10-8 to 10-6 cm thick [A14]. Thus

5 x 10-4 < 2-8 <5 x 10-3,
and a/b 1. Measurements have been made on n, of protoplasm and n, of membranes. For protoplasm, n,

was found to range from 4 x 10-2 to3 x 10-1 poise [A15]. For membranes, n, was found to range from 2.7
x 10’ to 2.7 x 108 poise [A16]. Therefore, utilizing equations A-2 and A-5,

60 poise < € < 1,600 poise.
This estimate is probably subject to the greatest error of the three indirect estimates of viscosity due to all
the assumptions required.
Summarizing the ranges of & determined by the various methods in their probable order of accuracy:

direct measurement,

430 poise < & S 1,800 poise;
absorption,

30 poise < € < 950 poise;
complex shear modulus,

4 poise < & < 90 poise;
cell viscosity,

60 poise < & < 1,600 poise.

38

The total range of data is
4 poise < € < 1,800 poise.
Thus, for this study, limits of
1 poise < & < 10% poise
will be used, with a more likely range being
50 poise < & < 2 x 10° poise.
The swimbladder wall of a fish is a membrane which is capable of supporting tension. In the present
model, the swimbladder wall has zero thickness, so that the tension in the membrane is effectively a surface

tension. Measurements of the internal swimbladder pressure can be used to calculate surface tension
since

AP = (2s/a), (5)
where AP is the difference between the internal swimbladder pressure and the ambient pressure [50].
Several researchers have measured internal swimbladder pressures, but since it is probable that a fish can
control the tension in the swimbladder wall, only measurements on live, unanesthetized, fish are
considered. Excess internal pressures from 2 x 104 to 6 x 10° dynes/cm? have been measured in fishes
which were 2 to 20 cm long [A17]. These results correspond to surface tensions of 6 x 10° to 7 x 10*
dyne/cm for fish with swimbladder radii of about 0.1 to 1.0 cm. The majority of these fish were less than 10
cm long with swimbladder radii less than 0.5 cm.

Surface tension is probably a function of fish size, so that measurements on larger fish are needed for
the present study. No measurements of excess pressure of larger, unanesthetized fish are available.
However, another means can be used to estimate the upper limits of surface tension in larger fishes. As
discussed in the text, Sand and Hawkins have attributed high experimental resonant frequencies to
swimbladder tension [33]. If this is true, then an upper limit of surface tension can be calculated by
assuming that the fish is a free bubble with surface tension and utilizing equation I-12, neglecting viscosity,
and equation I-25 to account for spheriodal swimbladder shapes. Although this method does not
necessarily give accurate estimates of surface tension, it does produce values which can be used as upper
limits. These limits are useful because it is the possible range of surface tension that is required. Surface
tensions calculated from resonance measurements range from about 10° to 10° dyne/cm for swimbladder
radii from 1 to 2.5 cm [33,37].

The surface tension of an air bubble in water is 74 dyne/cm [50]. Hence, the range of surface tension for
small swimbladders (as 0.1 cm) is chosen to be

10? dyne/cm <s <= 10® dyne/cm.
For larger swimbladders (a& 5 cm) the range is chosen to be

102 dyne/em £s ¥ 109 dyne/cm.

39

Al.

A2.

A3.

A4.

AS.

A6.

A7.

A8.
AY.

A10.

Al}.

Al2.

A13.

A14.

A15.

Ai6.

A17.

APPENDIX A REFERENCES

F.R.H. Jones and N.B. Marshall, “The Structure and Functions of the Teleostean Swimbladder,” Biol. Rev.
28, 16-83 (1963).

R.W.G. Haslett, “Measurement of the Dimensions of Fish to Facilitate Calculations of Echo-Strength in
Acoustic Fish Detection,” J. Cons. Int. Explor. Mer 27, 261-269 (1962).

R.H. Love, “Maximum Side-Aspect Target Strength of an Individual Fish,” J. Acoust. Soc. Am. 46, 746-752
(1969).

E.V. Shishkova, “Investigation of the Acoustic Properties of the Bodies of Fish,” Tr. Vses, Naughn.-issled.
Inst. Morsk. Rybn. Khoz. Okeanogr. 36, 259-269 (1958). [English Transl.: Associated Technical Services,
Inc., East Orange, N.J. (1960).]

M. Freese and D. Makow, “High-Frequency Ultrasonic Properties of Freshwater Fish Tissue,” J. Acoust.
Soc. Am. 44, 1282-1289 (1968).

R.W.G. Haslett, “The Back-Scattering of Acoustic Waves in Water by an Obstacle II: Determination of the
Reflectivities of Solids Using Small Specimens,” Proc. Phys. Soc. 79, 559-571 (1962).

T.E. Cooper and G.J. Trezek, “A Probe Technique for Determining the Thermal Conductivity of Tissue,” J.
Heat Transfer, Trans. ASME 94, Series C, 133-140 (1972).

W.D. O’Brien, Jr., Personal communication, April 1974.

T.A. Litovitz and C.M. Davis, “Structural and Shear Relaxation in Liquids,” in Physical Acoustics, W.P.
Mason, Ed. (Academic Press, New York, 1965) Vol. 2A, Chap. 5, pp. 281-349.

H.E. von Gierke, H.L. Oestreicher, E.K. Franke, H.O. Parrack and W.W. von Wittern, “Physics of Vibrations
in Living Tissues,” J. Appl. Physiol. 4, 886-900 (1952).

J. Lamb, “Thermal Relaxation in Liquids,” in Physical Acoustics, W.P. Mason, Ed. (Academic Press, New
York, 1965) Vol. 2A, Chap. 4, pp. 203-280.

F. Dunn, “Temperature and Amplitude Dependence of Acoustic Absorption in Tissue,” J. Acoust. Soc. Am.
34, 1545-1547 (1962).

D.E. Goldman and T.F. Hueter, “Tabular Data of the Velocity and Absorption of High-Frequency Sound in
Mammalian Tissues,” J. Acoust. Soc. Am. 28, 35-37 (1956).

E.D.P. DeRobertis, W.W. Nowinski, and F.A. Saez, Cell Biology, (W.B. Saunders, Philadelphia, 1970), 5th
Ed!; p: 18; p. 152.

L.V. Heilbrunn, An Outline of General Physiology, (W.B. Saunders, Philadelphia, 1955), 3rd Ed., Chap. 8,
pp.69-88.

A. Katchalsky, O. Kedem, C. Klibansky, and A. DeVries, “Rheological Considerations of the Haemolysing
Red Blood Cell,” in Flow Properties of Blood and Other Biological Systems, A.L. Copley and G. Stainsby,
Eds. (Pergamon Press, New York, 1960).

J.H. Gee, K. Machniak, and S.M. Chalanchuk, “Adjustment of Buoyancy and Excess Internal Pressure of
Swimbladder Gases in Some North American Freshwater Fishes,” J. Fish. Res. Board Can. 31, 1139-1141
(1974).

40

APPENDIX B
SIMPLIFIED EXPRESSION FOR (Bw/A)

Simplification of (B,,/A) as given in equation III-16 is based on the ranges of physical properties and the
limits of the variables given in Chapter II.

The first step in the simplification of equation III-16 is the simplification of the spherical Bessel,
Neumann, and Hankel functions in the S,'s. In the k,,a terms:

, sin(k,,a)

jo(ki.€), = (aca (B-1)
and

—, _ cos(k,.a) sin(k,.a)

Jo'(K1,8) = sea a ae (B-2)
Now, if

sin(k,,a) = sin[(1+i) xa] , (B-3)
then

sin(k,,a) = sin(xa)cosh(xa) — i sinh(xa)cos(xa) (B-4)
Similarly,

cos(k,,a) = cos(xa)cosh(xa) — i sin(xa)sinh(xa) . (B-5)
An examination of the parameters involved shows that xa > 10, so that,

cosh(xa) % sinh(xa) 2 = (B-6)
and

sin(k,,a) = i cos(k,,a) . (B-7)
Thus,

Hes) =e Ee)
and ie

ae any 1

Jo'(Ki,8) = —Jo(K:.8) ft ea | (B-9)

One of the assumptions in the model is that the shell is small compared to the wavelength of the incident
compressional wave. This means that k,,,b, k,,b, and k,,a are small. The definition of “small” will now be
determined by examining the expansions for spherical functions of small argument:

jo(z) = (1-2+....), Bio
jo'(2) = -2 (0-34...) a
jo"(2) = — (1 -e.. ) (B-12)
no(z) = -2(1 -3 4... ) (B-13)
no'(2) =se (1+ -....), coe
noi(z)= — 2 (1 to), Bee

41

hoz) = (1-2+...)-£(1-3+... i (B-16)
ho'(2) = -> (1-24... js5(1+3-... ). (B-17)

” 1 322 2i z4
het(z)=-4 (1-324...) -8 (+2-... ). (B-18)

In order to limit errors in these functions to under ten percent, it must be assumed that

eae nen (B-19)
Oa Ona (B-20)

and
Keseae 6 10e! (B-21)

This implies that in fish flesh and water

wb < 7 x 104 cm/sec (B-22)
and in air

wa < 2.4 x 104 cm/sec . (B-23)

Then with these limitations and the acceptance of at most a ten percent error, equations B-10 through B-18
can be written as:

ete (B-24)
jo'(2) = -, (B-25)
Jo (Z) = -4, (B-26)
no(z) = -—>, (B-27)
no'(z) = > . (B-28)
no"(z) = -£ (B-29)
ho(z) = 1 -+, (B-30)
ho'(z) = -Z+4 . (B-31)
ho"(Z) = _ -4 (B-32)

It is necessary to keep both the largest real and largest imaginary term in the spherical Hankel function
expansions, regardless of their comparative magnitudes.

Equations B-24 through B-32 will now be utilized to simplify equations IIl-2 through III-15. In addition,
equations II-59, II-60, II-68, and II-69, will be substituted for k,., kz,, k.,, and k,,,. These substitutions yield,
after neglecting any term which causes an error of less than ten percent, the following expressions for the
S,'s:

F ,
S., w2b ick

Bpo,Gu? Pog DBE a

42

Pe (B-34)
Sa aa

3Po,C/?
a YSU SS eeey sae
w2b
rhs ee (B-36)
iu (B-37)
Sa = = + Ob
iwe ;
ara ae (B-38)
fs 3&2 2iE B-39
S23 =e (ais) [1 iy Po262b2 * Po, Wb2 | :
(B-40)
a> > 1 ’
Ss w2a au
ae 3PoC
¢, 3iEw F
: an (B-42)
Soo = (tesa ) ( 2 PoC? )
aie Yo (B-43)
S34 ~ 3p,,c2 7 | seh (ome 3) Ma )(2p,,6 3)
ee iwe (B-44)
22 3Po,C;
C; 3&? 25 | :
S43 = - (ar) } lpetctay Po, Wa? ; wie
2s 2s(va—1) ] (_@K._)"
Se Wp tarrcea Aoeacccee ll Seay)
alata) (fan CH 26, zi) [ Wrioee = ba ann Obs . le oot

43

Substitution of equations B-33, B-36, B-37 and B-40 into III-16 yields:

- (5) veiw

By 3Poy Cw?

AG a ; 7 , (B-47)

A W2b ~— icy u— —1-21]w

SPoyCw2 — Poy Wb? wb

or

Be =i ;

A Y-zZ ti
where

v= (a=. uw (B-49)

3 Poy Cw?

and

z= (4 )E (ste) e-w ] (B-50)

The remainder of the simplification process for equation III-16 will be conducted in several steps. In each
step, the order of magnitude of the various terms will be compared and terms which cause an error of less than
ten percent will be neglected. This procedure will be done in such a way that no term is neglected which could
become important due to a later subtraction. It should be mentioned that most of the terms which are
neglected result in errors of much less than ten percent and in those cases where the errors approach ten
percent, that error is approached only at the limiting values of the terms involved. In order that the
simplification process can be checked by the interested reader, the results of each step will be given, rather
than just the final result.

The next step is to simplify the terms in the parentheses in equations IIl-17 and III-18. The results are

) ; (B-51)

_ Cy ig 4a3
$22833 — So3S32 = i=) = (ha) ( 11+ 3

S23S42 7 S22S43 =

(Gas) sos) sop ien) et) omen) lig

(B-52)
a wb _ 2) _ | (__w?&b_) (4
$3832 _ $12833 a (aera ) (1 =) ! ease ) ( b3 ) "
(B-53)
and
CS ee Oa oe _ wb ae _@&*b
42943 13942 = (Gamer ) (ferscaas )
- Cy iE Abs (B-54)
( Po, Wb? ) (once) Gh Tas )

The next apparent step would be io simplify the expressions for U and W. This, however, can lead to errors
when the subtractions indicated for Y and Z are made. Hence, the next step is to simplify Y and (Y-Z). Before
writing the simplified expressions for Y and (Y-Z), several intermediate steps will be given in order to assist the
interested reader in checking the final expressions.

44

Substitution of equations B-43, B-46, B-51, and B-52 into III-17 yields:
C; 2s 2sc,(Y,— 1)
= — (—+_) (1 - ———— )+
" ( p,008?) ( 3Po,Ca°a ) Po; Poala “Bo, c.20Fa | (enaon Cy, .
wesc] [+ GE) Gem)” | (01-44)
6 Po;2C;P0,Ca2Cp, a? wa? Po, WK, b3
C; wa ek 11wW& ass
(5,,0,7a) ( 3 )(1 Bh Speer ( i a)
a7 emai we ave | WK, —) Picea ia
+ a- 0 (get—)'(1-8) - [se] (ae) (er)
a a WK, VC mace
3Po)2C;2a? 2P0,Cp, bs )
3 ( 2s 4a3
+ = — ee
W(eayccee ) a) (11 "TE )
ah &s(Ya— 1) Oia Niaitfay) oe
Tere ( Spee) ( bs )
4 Pete [1+ = )( Cpa )"
PoaCa? Po;Cp, iPorcs205.c,2a2 wa2 20 ,WK,
‘ (or 2& ) ( -=) 2 WK, \% es
an (saa ) ( 3P,a i be J War!) ae! (1 b )
[eae] ( WK, )*( as )
—— = | ———
Po, a? 2PoaCpa b°
11€2(y, — 1) ae
+ —_—_— -
3Po,2C;2a2 eran | ores Cpa is @ bs ) e522)
Substitution of equations B-43, B-46, B-53, and B-54 into IIl-18 yields:
rat wb _ a = “4 2s 2s(Y,—1) WK, Ye
ae (Sp.28 ) (1 o ) | ( 3Po,Ca7a ) * bpp,0.208 ( 2Po2Cpa )
_3w?EK, (V2 — 1) dS] [1 ( ) ( Coa Ni ]
2precs2pnceen 20o,C;2C,, wa2 229, WKa
w w2b w?E&2b ca
oe ——— ah | ee
( o.02) ( Po Cy ) ( 3Po,3C,3a2 ) ( 3P,b2 )

wb(Y.— “1 WK, )

3PoCa 2PoaCpa

w&b(y,— 1) ain tye | ( WK, y
Po Pc sas Sorat (Gere =) Po, wb? 2P0.Cpa

45

- beter GE) 01+) |
ee ea
3&s(Ya— 1) i= "

Po,Ca? PoC? a? 2Po2Cpa

Va

+e (GSS) [1+ (Ger) (tr) ] |
aBa) | Gost) (0S) ERS)

ste) ee ee

6 Po,2C;b2 2052p. PoPc3as 20o.Co.

_ pclya—1) (ses ) B-56
“Po, Wb? = 2o,Cp, | ; {
Substitution of equations B-55 and B-56 into B-49 yields:

__ Pogla2we;b pees wc;b aie
(Po.Ca2)Y¥ = eae) (1 TRE, * (5...) Gs )

+

(gnceee ro 2s re ay _ wb in wea
3Po,2C,a? ) ( oe) ( a) (ara ) ( zp,08)

+

11&? wb , ay (_&o*b % c(¥2— 1) WK, ig
(arp nc.t. ea ) ( bs ) (ap,sasa ) [ Po, b? ] race )

V2

2c,sb(y,—1) WK, Ye a w2c,b(Y, — 1) WK, el
RT (orace) [ SPowCw2a | (Ga) ( D)

a

=) ( WK, Ye ay web(y, — 1) wk, \”
3Po?C,a4 soe) ( bs ) SPo,Ca \ Gexes,

= Ye 4a3
opened] [1 > (BE) (ete) ])
18 Powlw? Po)? CC, a? wa? 2 Po, WK, bs

46

2&c.C (Ya — 1) ( OK, ) (1- 110& )
peste ene ee
W*Po,,P,a°D* 2P0.Cpa BPC?

+i(e)[ (S82) (- mez) -']

11&°¢y © [SvG(¥a — 1) | WK, \” 2s
iis ( 9Po,Po,°Ca°*b* ) es WPp,,ab? laZe ) (1 . Po,w°a? )

2&c,,C,(Ya— 1) ( WK, Ne 4 — 1108
WP oy Poj°D 2Po.Cpa ( BPo,C\" ) (B-61)
Then a comparison of all terms in equation B-61 yields:
( Po;Po,Ca2 )z ( 2Ec, ) Cw Po(Y ma) (ie ) 2s
ote z= - Ger) -E SIG) (1 + se)
Cc; 3WPpo,,a2b2 apReab= auch BA Po, W2as

Cw Poy (Pu Pole) (1 - 2s )-1]
= Uae Spee) [ Po, W2 ae 3Po,Ca7a
; WAZo, 5 POH Wa = 1) WK, ve 2s
ee 9 Poy Po, Ci2a2b? )+ ; WP,,ab? (caren ) (' a Po, W?as ) * (B-62)

Examination of equations B-60 and B-62 shows that all the viscothermal terms are negligible.
Subtraction of equation B-62 from B-60 and a subsequent comparison of terms yields:

Po; PoaCa? , fa ( wa ZEGH
( Ci )(v Z) z a) ieara sae)
3Po,C KAS 2s
Pow @)?a2 ee Ce Coa )" (1 ss Fate)

; Gn 3P,Ca? = 2s i 2 lathes )
+i Gone) [( w? a? )(3 CALY, pe Gree ] eee)

Equation II|-32 indicates that | By, leis required to determine o. This implies, from equation B-48, that | Y k
is required. If

Ve ig ae ING (B-64)

’

then

lyl2 =Y2 + Y2, (B-65)

where Y, and Y, are the real and imaginary parts of Y, respectively. Thus, in light of equation B-65, itis possible
to simplify equation B-60 by comparing magnitudes of real and imaginary terms. This comparison yields:

(Fe )y i = ) [ aN ey I (B-66)

51

where

Nee Pascua hin) (8-67)
PoP C4 a8 9PowCw?
and
_ 2wé&b3 PoCe (Por? aa) (=) 8-68
a ( 3Po,C2.a? ) [ ( PowCw? ') a x 4 b3 | : ( )

This set of equations does not give Y,, and therefore B,, to the desired accuracy of ten percent, but it is
sufficiently accurate in the final answer for | Be lz

Thus, the simplification process of equation III-16 yields:

By id 1 2ECy
3PoCw(Ya— 1] Ka ) ( 2s
i [ Po,, 0? a? (s5.-85 wi Po, Was )
Weis 3P,Cq? eS meeak= ] I
+i (—Stt —— aa lee == ,
(car ) [ (a7) ( TRC) : (cers, (B-69)

where A and 2d are given by equations B-67 and B-68.

52

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READ INSTRUCTIONS
REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM
1. REPORT NUMBER 2. GOVT ACCESSION NO,| 3. RECIPIENT'S CATALOG NUMBER
NORDA REPORT #4

4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED

FINAL

6. PERFORMING ORG. REPORT NUMBER

8. CONTRACT OR GRANT NUMBER(a)

A NEW MODEL OF RESONANT ACOUSTIC SCATTERING
BY SWIMBLADDER-BEARING FISH

» AUTHOR(s)

RICHARD H. LOVE

10. PROGRAM ELEM
AREA & WORK U

NT, PROJECT, TASK
IT NUMBERS

. PERFORMING ORGANIZATION NAME AND ADDRESS 5

NAVAL OCEAN RESEARCH AND DEVELOPMENT ACTIVITY
NSTL STATION, MISSISSIPPI 39529

62759N/2552/301

12. REPORT DATE |
AUGUST 1977
sdoe bi 20 anusatone 10

15. SECURITY CLASS. (of this report)

- CONTROLLING OFFICE NAME AND ADDRESS

NAVAL OCEAN RESEARCH AND DEVELOPMENT ACTIVITY
NSTL STATION, MISSISSIPPI 39529

. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office)

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. DISTRIBUTION STATEMENT (of this Report)

APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report)

. SUPPLEMENTARY NOTES

- KEY WORDS (Continue on reverse side if necessary and identity by block number)

VOLUME SCATTERING, BIOLOGICAL SCATTERING, ACOUSTIC MODELS,
BIOACOUSTIC MODELS, REVERBERATION

20. ABSTRACT (Continue on reverse side if necessary and identify by block number)
The primary ‘cause of oceanic volume reverberation is resonant scattering by
the swimbladders of small fish. A new model of a swimbladder-bearing fish has |
been developed in order to provide improved predictions of the resonant
frequency and acoustic cross section of such a fish. Development of a new
model was undertaken because comparisons of the predictions of previous models
with experimental data show these models to be inadequate. Primarily, the |

experimental data indicate that damping in fish tissue is consistently greater |

DD , tee 1473 EDITION OF 1 NOV 65 IS OBSOLETE UNCLASSIFIED

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20.

than that predicted by previous models. In addition, in several instances,
resonant frequencies have been measured which are significantly higher than
can be accounted for by these models.

The new model consists of a small, spherical shell, enclosing an air cavity,
in water. The shell is a viscous, heat-conducting Newtonian fluid, with the
physical properties of fish flesh. The interface between the shell and the
cavity supports a surface tension. The shell is insonified by a harmonic
plane compressional wave, whose wave-length is large compared to the shell
diameter.

The solution to the problem consists of obtaining the amplitude of the
scattered compressional wave. This was accomplished by first determining
the appropriate wave equations from the linearized equations of motion.
Eigenfuction solutions to these equations were then obtained. However,
theoretical evidence indicates that only the fundamental mode is an important
contributor to volume reverberation, so that higher modes were neglected.
Application of the boundary conditions then led to the required solution for
the amplitude of the scattered compressional wave. Due to the complexity of
this solution, order of magnitude analyses of the various terms were con-
ducted in order to obtain a simplified expression. Equations for both the
resonant frequency and acoustic cross section were then determined from this
simplified expression.

A comparison of the results of the new model with experimental data indicates
that the new model constitutes a definite improvement over previous models.
The new model can predict the high values of damping and elevated resonant

frequencies that previous models could not. The comparison indicates that
the model is most accurate for fish in which tension in the swimbladder wall
has a minor effect on resonant scattering. This includes the fish which are
of interest in studies of volume reverberation and therefore, the new model
will be of considerable value in such studies.

a
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