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A technique for predicting relaxational effects on finite-amplitude wave propagation in liquid media
J. Sound Vib. (1970) 13 (2),201-209
A TECHNIQUE FOR PREDICTING RELAXATIONAL EFFECTS ON FINITE-AMPLITUDE WAVE PROPAGATION IN LIQUID MEDIA B. B. CARY AND F. H. FENLON Applied Physics Laboratory. Electronics Division 0/ General Dynamics, Rochester, New York, U.S.A. (Received 21 July 1969)
A generalized Burgers' equation is derived which governs the propagation of finiteamplitude impulses in relaxing liquids. If the attenuation dependence on frequency is known, then, as is done in filter theory, a time domain representation can be found in operator form. As an illustrative example, the attenuation operator is constructed for sea water. The resulting Burgers' equation is then compared with a previously published variant of Burgers' equation which had been adopted for describing salt water relaxation effects on impulsive waveforms. It is concluded that the technique introduced here for incorporating relaxational effectsis valid for impulsesin generalwhereasthe earliermethod is restricted to waveforms whose chief spectral components satisfy the criterion WT < 1 where T is the relaxation time. 1. INTRODUCTION
Burgers' equation governs the propagation of spreading finite-amplitude waves in lossy media which are non-dispersive. Cary [1] has previously presented a numerical method for solving this equation. In the present paper, we extend Burgers' equation to include relaxational effects. The technique for incorporating relaxation is applied to salt water as a specific example. Prior to presenting our derivation of an extended Burgers' equation, we shall briefly mention some previous work . Polyakova [2, 3, 4] has developed a theoretical expression for the absorption coefficient in a relaxing medium as a function of frequency. She then made use of successive approximations [3] to study the influence of a relaxation mechanism on the growth of the second harmonic for plane waves and made predictions concerning its changing phase relative to the fundamental. Finally, Polyakova [4] derived anon-linear wave equation including dispersion but ignoring viscous losses. She then proceeded to solve the problem of a steady, plane shock wave with relaxation effects from a system of equations in Eulerian co-ordinates. Qualitative steady-state shock profiles were derived for various values of the dispersion. Soluyan [5] has applied Polyakova's [4] equations to the case of plane progressive sine waves and calculated the sawtooth wave shape for the limiting cases WT ~ 1 and WT ~ 1, where T is the relaxation time and w the angular frequency. In these studies.Jt was assumed that the relaxation process could be represented by the simple firstorder chemical kinetic process
(1) Here go is the equilibrium value of any appropriate progress variable that monitors the degree of completion of the chemical reaction or structural adjustment. For example, in a diatomic gas undergoing rotational relaxation, the specific heat at constant volume is a parameter which is sensitive to the degree of rotational excitation. In a salt solution, g can 201
202
B. B. CARY AND F. H. FENLON
be taken as the product of the degree of dissociation of the salt by its molar concentration. Equation (1) is in the form used by Polyakova [2, 4] and is equally applicable to structural relaxations. We shall discuss its application to aqueous salt solutions. Mazo [6] has written down the hydrodynamic equations of continuity, momentum and energy. He has coupled these with a more general rate expression than equation (1). He linearizes his equations and shows that, as is commonly assumed, it is the volume change associated with a chemical reaction and not the small amount of heat released or absorbed which primarily affects acoustic propagation. He demonstrates the validi ty of this assumption for an aqueous solution of MgS0 4 • This implies that it is justifiable to ignore heat transfer effects in considering the dissociation of MgS0 4 in salt water. The molar content of MgS0 4 is on the average 0·02. It is the dissociation of MgS0 4 that causes the anomalously high absorption of sea water between 10 and 104 kHz. A molar concentration of 0·014 of MgS0 4 in water yields the same absorption curve with frequency as that determined empirically for sea water [7, 8]. In regard to experimental work, Romanenko [9] has qualitatively verified Polyakova's [3] predictions of phase shift between second harmonic and fundamental. Ryan [10] has measured plane wave second harmonic growth in an MnS04 solution for a case in which wT=4·54. It is a characteristic of all relaxation processes that an acoustic signal with a frequency high enough to yield WT > 1 travels with a faster phase velocity than a signal satisfying WT < 1. The relaxation does not have time to equilibrate in each half cycle of signal propagation when W'T> 1. Consequently, a high-frequency signal suffers no increased absorption owing to the presence of the relaxation. By contrast, at WT < 1, the relaxation has time to attain thermochemical equilibrium in each half cycle. In view of the foregoing remarks, for sinusoidal waveforms, we can state that the principal influence of a relaxation process upon a propagating impulse will be to selectively increase the rate of attenuation of the lower frequency components of its spectrum. In addition, the higher frequency components will travel faster than the lower ones changing spectral phase relations. Both of these effects are cumulative with distance. Now our generalized Burgers' equation is only good to second-order inertial terms and to first-order viscous terms. Therefore, we assume that the contributions to total absorption from shear viscosity and a relaxation process are purely additive, since any coupling between them is of third order. It is of interest to note that shear viscosity itself can be viewed as a very rapid structural relaxation occurring in a shorter time than most chemical processes. Marsh [11] has presented a generalized variant of Burgers' equation in order to include the influence of salt water dispersion on the waveforms received from sofar bombs. Marsh [II] did not present a detailed derivation of his equation. The equation that we derive here is not the same, and we shall discuss the differences between them. 2. DERIVATION OF THE GOVERNING NON-LINEAR WAVE EQUATION IN A RELAXING MEDIUM
In weak shock waves with Mach numbers € less than 0·5 or sound pressure levels less than 200 dB relative to 1 ftbar, the entropy changes across the wave can be ignored, since they are of third order in E. Consequently, an appropriate equation of state suitable for relaxing fluids can be derived by making a Taylor series expansion of the excess pressure over static in terms of the deviations of the density and the relaxation parameter from their equilibrium values. On this basis Polyakova [4] derived the following equation of state:
(1 + dtd) P/ T
=
(2 2 .d), (eP P) ( d) /2 Co + coo' T dt P +"2: 8 2 go 1 + T dt P . .i,
p
(2)
FINITE-AMPLITUDE WAVE PROPAGATION
203
The relaxation parameter g does not explicitly appear in equation (2). The partial derivative coefficient of the second-order term in the density has an implied dependence upon ~o (the product of the molar concentration of the salt with its equilibrium degree of dissociation). If the second-order term is neglected , then the dependence of the dispersion upon salt concentration is accounted for through the appearance of c~ and c~. It should be emphasized that the relaxation time T for the salt dissociation is strongly dependent on the temperature and virtually independent of concentration for solutions with molar concentrations similar to that of salt water. Glotov [7] has derived an explicit expression for T as a function of temperature and concentration for MgS0 4 solutions. The weak dependence of T on the concentration validates the use of equation (1) which implies that T is independent of initial molar concentration. Fox [12] has presented data which shows that the dispersion in aqueous MgS0 4 solutions is directly proportional to the molar concentration of salt. The measurements were conducted over the range of 0·1-0·6 M. In addition, Tamm [13] presents data which shows that the excess absorption in aqueous MgS0 4 solutions is directly proportional to the molar concentration for concentrations up to 1·0 M. In order to derive the wave equation, we invoke the conservation equations of continuity and momentum: (3) Pt + pu, + P, v = 0,
(4) Since we are primarily interested in progressive waves, we shall transform from x, t into retarded co-ordinates z, T; thus, z = p.r,
r T=t--
co'
and
a
a
I a coaT'
-=p.~---
or GZ a a at= er Thus, making use of the fact that
for progressive waves and using the small signal relation pi
V
Po
Co
--=in second-order terms, we see that equations (3) and (4) become
(1 -~) p~ PVT + pv (P-V% -
=
(~) (1 +
:J
VT -
P-Po v"
LVT) = - (p- :z - :0 DT)P +
(27] + 7)')
(5)
(p. ~ - :0 DTy o.
(6)
204
B. B. CARY AND F. H. FENLON
In order to eliminatep, if we apply the operator (1 + rD T ) to equation (6), invoke the relation p'/PO = v/co in second-order terms, then with the aid of equations (2) and (5), we obtain
c~
( 1 + C6 •T D T
(3.)
) ( JLv z - ca VV T
=
(1 + 'T DT )
17 +17' (22po Cb )
VTT
+ (m.r) 2co VTT,
(7)
where
(a
2p Po ) • ,8=1+ 2 2Co 0P2 go The f3 term is a measure of the non-linearity of the fluid. Equation (7), which is in Eulerian co-ordinates has a form similar to that of Polyakova's [4] wave equation derived in Lagrangian co-ordinates. Equation (7) differs in that it includes a viscous loss term on the right. In the linear lossless approximation, equation (7) can be reduced to that given by Lamb [14]. It will now be desirable to transform our spatial co-ordinate z to Blackstock's [15] stretched co-ordinate f in order to make our wave equation applicable to cylindrical and spherical geometries. A new dimensionless retarded time y = wTis also introduced. Our new dependent variable W is given by
where q = 0 for plane waves, q = 1/2 for cylindrical waves, and q = 1 for spherical waves. In terms of the new variables, equation (7) becomes
(,)q{( 1+'T--D co)y G
c~ Co ) (Wf-WWy) = ( l+2"'r-Dy Co Xo '0
-I
x;
where c~ -
m.rco} Wyy, +-2t:1
(8)
t'IEX c
ca
m=--2-' Co
is the Gol'dberg number which is a measure of the relative importance of non-linear to dissipative effects and is defined for sinusoidal waves as
(J
G == 21T(3E etA
For an impulse propagating in a non-relaxing medium, G is defined by G = f3lEx c A, 2)/(;VIX)
where A is defined as (41T and x; is a length which has to be chosen. If the impulse has a time duration of L1t, then Xc can be defined as
Co Llt Xc =
b1T '
where b is an integer and Co is the low-frequency sound speed which is appropriate providing
'T/Llt < 1. In a relaxing medium et is not simply proportional to the frequency squared. In fact, a relaxing medium acts like a high pass filter. Therefore, A becomes a function of frequency itself; hence, the above approach loses its utility and it is better to replace IX by an operator containing time derivatives. We shall discuss the construction of such an attenuation operator for the specific case of salt water in the next section. The last term on the right-hand side of equation (8) will usually be negligibly small and will be dropped hereafter.
FINITE-AMPLITUDE WAVE PROPAGATION
205
3. THE ATTENUATION OPERATOR FOR SEA WATER The attenuation of sea water for frequencies of 1 kHz to 1 MHz can be expressed in a fashion similar to that of Marsh [11] as follows:
(9) As stated earlier the MgS0 4 relaxation causes the attenuation IX given by equation (9) to have the frequency characteristic of an inefficient high pass filter. Inefficient in the sense that the attenuation is insufficient to block frequencies for which WT < 1. If equation (9) represents the filter characteristic of the medium in the frequency domain, then we are seeking an appropriate representation of the filter characteristic in the time domain . The time domain representation is an operator containing time derivatives and can be constructed intuitively by replacing w 2 in equation (9) by a second-order derivative with respect to the retarded time y, i.e. The signs are chosen to make the operator yield the correct result wlien W = siny. The chosen operator is
D;.
ex =
(g2 +
1_
T~;C'~~:~) D;) (~:) D;. 2
(10)
Now we can complete the derivation of a generalized Burgers' equation which includes relaxation effects. In equation (8), we replace the term WyyjG by (xiiW)/({3€). Then substituting for ii from equation (10) and multiplying both sides of equation (8) by [1 - or x (co/xc) Dy], we obtain
The dependence on molar concentration of the salt appears on the left through the presence: of c~jcij in the dispersion operator. On the right the dependence upon salt concentration is given through 0 2 and g2 which are directly proportional to salt concentration [13]. Equation (1I) is in a form suitable for application of the Banta method as developed by Cary [1]. The Banta method is a superior technique for solving equation (11). Bellman [16J has discussed the advantages to be gained by using the differential equation itself as a recursion relation to avoid limitations in step size usually imposed by the two-dimensional grid required by a finite difference method. To apply the technique one simply defines a new dependent variable W* where
W*(j,y) =
[1- ~~'T(~:r n;] W.
(12)
A new w* is obtained for each increment LJ1 Then equation (12) must be integrated to find W before taking another step LJ1 Owing to the extra integration that must be performed at each stage, the calculation proceeds more slowly than in the non-relaxing case [1]. Typical values of a and g for sea water taken from Glotov [7] are a2
= 0·9 x 102 nepers/m,
m=7 x lO-S, g = 1'2 X lO- I S sec- nepers/m,
f3 = 3'8, Co
=
1500 mjsec.
206
B. B. CARY AND F. H. FENLON
Reference [14] treats a wide variety of relaxation phenomena. Many processes possess attenuation expressions which do not differ radically from equation (9). Therefore, it should be possible to construct an appropriate operator ii for a variety of cases. It was stated in the Introduction that Marsh (l] had adopted a somewhat different wave equation than that presented here to study impulse propagation in the sea. If we convert Marsh's [11] equation to our nomenclature and change his dependent variable from pressure to velocity, we obtain (13)
where
(3er
(]'=-.
Xc
Equation (13) is applicable to spherically spreading waves. The left-hand side of (13) can be reduced to the left-hand side of (11) if we change variables from 0 tojas follows (for spherical waves): o f= ooln00
Therefore, equation (13) becomes
(::a + D
y)
(WI
WWy) = efIUO[a~:co
-
w"y
+ ~ Wm
] .
(14)
The operator [(Xc/TCO) + D y] on the left side of equation (14) incorporates the effects of dispersion as we know from Lamb's linear lossless dispersion equation [14]. This is similar to the left side of equation (8) providing c;/c~ == 1. For sea water m = (c~ - (5)/c5 == 7 x 10-5 making the left sides of equations (8) and (14) equivalent. The principal difference between equations (8) and (14) is in how the right side was formed. We note that the right-hand side of (14) was constructed in such a way that as one takes the limit at either high or low frequencies the equation reduces to the form
Wf where G
=
efl u o
-
WWy =G Wm
(3€x c A. Consequently, we can rewrite Marsh's [11] equation (14) as
+ Dy) (WI (3:... TCo
1
WWy) = el 1uo [ x C o Wyy + G W m Co T
L
] ,
(15)
II
where the high frequency Gol'dberg number is
with
(16)
FINITE-AMPLITUDE WAVE PROPAGATION
207
Therefore, An is given by the high frequency asymptotic limit for IX from equation (9). It is important to note that we have neglected the heat transfer term in the classical small signal thermoviscous attenuation coefficient, i.e.,
47T 2
O:c
[
,
= 2poCo ,\2 21) +?'] +
?'](y -
Pr
1)] .
(17)
The Prandtl number is defined as
Pr= ?']Cp
K'
where Cp is the specific heat at constant pressure and k the heat transfer coefficient. In sea water with a salinity of 30 parts/thousand at 18°C, y = 1·0207 and Pr = 5·6. The dilatational viscosity?']' of water is about 2?'] [17, p. 321]. Consequently, the ratio of the viscous to the heat transfer losses in equation (17) is 21) + 7]' _ • 3 ?,](y-l)/Pr -106 x 10 . This ratio agrees quite favorably with a value of 1·32 x 103 derived from measurements taken in water [17, p. 281]. Similarly GL=fJEXcAL , where by comparing equations (14) and (15) we find
fJExc fJ A GL = 2--r-2 = "xc L' a
T
Co
and therefore, (18) For a sinusoidal wave, (19) We see that A L given by equation (18) is the direct result of substituting the asymptotic limit for IX at low frequency (W7" < 1), from equation (9) into (19). Consequently, while Marsh's [11] equation (15) is adequate for WT ~ 1, it does not accurately represent the behavior of the absorption in the vicinity of WT == 1. Marsh applied his equation to sofar bomb waveforms at ranges upwards of 1·5 km. The major spectral components of these waveforms satisfied the inequality WT < 1; hence, Marsh's [11] equation was adequate. Equation (11) is not subject to such a limitation and is applicable to impulses which have a spectrum whose components have values for W7" ranging from less than unity to greater than unity. 4. LIMITATIONS OF THE MODEL
The Polyakova [4] equation of state (2) was derived by adopting a general phenomenological approach to the relaxing medium. Hunt [18] has criticized this type of approach calling for a two-fluid theory which would directly couple the chemical and transport processes. Meixner [19] and Mazo [6] have presented more rigorous formulations along the lines suggested by Hunt [18]. Mazo's [6] treatment is preferable to that of Meixner [19] if one is interested in the chemical aspects of the relaxation process. Mazo [6] has shown that the thermoviscous and chemical absorptions and dispersions are additive to first order. Moreover, Mazo [6] showed that the effects of diffusion currents are negligible in liquids but not in gases.
208
B. B. CARY AND F. H. FENLON
Equations (8) and (11) are only valid up to a peak particle velocity Mach number of about 0·05 which corresponds to a sound pressure level of 181 dB relative to I ,ubar. For such weak shocks it is felt that the assumption of addi tive coupling between chemical and thermoviscous absorption is still permissible. REFERENCES 1. B. CARY 1968 J. acoust. Soc. Am. 43, 1364. Prediction of finite-amplitude waveform distortion with dissipation and spreading loss. 2. A. POLYAKOVA 1959 Soviet Phys. Aeoust, 7, 85. Thermodynamic theory of absorption of finiteamplitude sound in relaxing media. 3. A. POLYAKOVA 1960 Soviet Phys. Acoust: 6, 356. Propagation of finite-amplitude waves in relaxing media. 4. A. POLYAKOVA, S. SOLUYAN and R. KHoKHLov 1962 Soviet Phys, Acoust. 8,78. Propagation of finite-disturbances in a relaxing medium. 5. S. SOLUYAN and R. KHOKHLOV 1962 Soviet Phys, Aeoust. 8,170. Finite amplitude acoustic waves in a relaxing medium. 6. R. MAZO 1958 J. chem. Phys. 28, 1223. Absorption and dispersion of sound in chemically reacting fluids. 7. V. GLOTOV 1964 Soviet Phys. Acoust. 10, 33. Calculation of the relaxation time for the degree of dissociation of magnesium sulfate in fresh and sea water as a function of temperature. 8. A. WEISSLER and V. DEL GROSSO 1951 J. acoust, Soc. Am. 23, 219. The velocity of sound in sea water. 9. B. ROMANBNKO 1960 Soviet Phys. Acoust, 6, 375. Distortion of a finite-amplitude waveform propagated in a relaxing medium. 10. R. RYAN, C. ATTANASIO and R. BEYER 1965 J. acoust, Soc. Am. 37, 602. Propagation of finiteamplitude ultrasound in a relaxing liquid. 1L H. MARSH, R. MELLEN and W. KONRAD 1965 J. acoust, Soc. Am. 326. Anomalous absorption of pressure waves from explosions in sea water. 12. F. Fox and T. MARION 1953 J. acoust, Soc. Am. 25, 661. Ultrasonic dispersion in water solutions of magnesium sulfate. 13. S. FLUGGE (Ed.) 1961 Handbuch der Physik Band XI Akustik, Berlin: Springer-Verlag. pp.
237-238. 14. J. LAMB 1965 In Physical Acoustics 2A, Properties of Gases, Liquids and Solutions (Ed. by W. Mason). New York: Academic Press. p, 218. 15. D. BLACKSTOCK 1964 J. acoust, Soc. Am. 36,217. On plane, spherical, and cylindrical sound waves of finite amplitude in lossless fluids. 16. R. BELLMAN Adaptive Control Processes: A Guided Tour. Princeton N.Y.: Princeton University Press, pp, 96-97. 17. S. FLUGGE (Ed.) 1961 Handbuch der Physik Band XI Akustik. Berlin: Springer-Verlag. 18. F. HUNT1955 J. acoust, Soc. Am. 27, 1019. Notes on exact equations governing the propagation of sound in fluids. 19. J. MEIXNER 1952 Acustica 2, 101. Allgemeine Theorie der Schallabsorption in Gasen und Flussigkeiten unter Berucksichtigung der transporter Scheinlungen. APPENDIX LIST OF SYMBOLS
w T
g .t p p
o Co
angular frequency relaxation time a thermodynamical variable which monitors the relaxation process time pressure density subscript denoting thermochemical equilibrium superscript denoting deviation of quantity from its equilibrium value small signal speed at thermochemical equilibrium
FINITE-AMPLITUDE WAVE PROPAGATION
frozen small signal speed c' -c z
Ceo
m=~
c~
particle velocity and peak particle velocity at source range and source radius shear viscosity dilatational viscosity
v, Vo
r, ro 'YJ 'YJ' Z
=p,r used in retarded co-ordinates to emphasize smallness of spatial derivatives retarded time = t - rico liquid non-linearity parameter
fL
T f3
W=~
rovo q "" 0 for plane wave; = t cylindrical wave; = 1 for spherical wave E Mach number IX attenuation coefficient for relaxing medium IXc small signal thermoviscous attenuation coefficient ,\ wavelength
J t ,wh ere Xc = Co b7T
A' d . 0 f impu ' I se an db'IS any integer , at rs uration
Gol'dberg number = f3Ex c A for an impulse
G
=
27TfJE l' ' 'd IX'\ lor SIOUSOt
a = fJEXcZ
f
= 0 for plane waves =
2VO'o(V~ - VO'o} for cylindrical waves
= 0'0
In (0'10'0) for spherical waves
47TZ
A "",\zlX frequency constants in salt water absorption expression y=wT v
G,
g
a
D,=-
at
2
47T h ere A L = ~,w 1\
(XL
is tthe asymptotic ' I'imit. 0 f eex: lor f is
W7'
Z
' I'tmit, 0 f asymptotic
AH =
47T h ere IXn IS is th ~,w t e 1\
W·
=
(1 + C;7'~D,) W Co
C
Z
8=~Tw
c5
<1
Xc
l' lor
WT
>I
209
A TECHNIQUE FOR PREDICTING RELAXATIONAL EFFECTS ON FINITE-AMPLITUDE WAVE PROPAGATION IN LIQUID MEDIA B. B. CARY AND F. H. FENLON Applied Physics Laboratory. Electronics Division 0/ General Dynamics, Rochester, New York, U.S.A. (Received 21 July 1969)
A generalized Burgers' equation is derived which governs the propagation of finiteamplitude impulses in relaxing liquids. If the attenuation dependence on frequency is known, then, as is done in filter theory, a time domain representation can be found in operator form. As an illustrative example, the attenuation operator is constructed for sea water. The resulting Burgers' equation is then compared with a previously published variant of Burgers' equation which had been adopted for describing salt water relaxation effects on impulsive waveforms. It is concluded that the technique introduced here for incorporating relaxational effectsis valid for impulsesin generalwhereasthe earliermethod is restricted to waveforms whose chief spectral components satisfy the criterion WT < 1 where T is the relaxation time. 1. INTRODUCTION
Burgers' equation governs the propagation of spreading finite-amplitude waves in lossy media which are non-dispersive. Cary [1] has previously presented a numerical method for solving this equation. In the present paper, we extend Burgers' equation to include relaxational effects. The technique for incorporating relaxation is applied to salt water as a specific example. Prior to presenting our derivation of an extended Burgers' equation, we shall briefly mention some previous work . Polyakova [2, 3, 4] has developed a theoretical expression for the absorption coefficient in a relaxing medium as a function of frequency. She then made use of successive approximations [3] to study the influence of a relaxation mechanism on the growth of the second harmonic for plane waves and made predictions concerning its changing phase relative to the fundamental. Finally, Polyakova [4] derived anon-linear wave equation including dispersion but ignoring viscous losses. She then proceeded to solve the problem of a steady, plane shock wave with relaxation effects from a system of equations in Eulerian co-ordinates. Qualitative steady-state shock profiles were derived for various values of the dispersion. Soluyan [5] has applied Polyakova's [4] equations to the case of plane progressive sine waves and calculated the sawtooth wave shape for the limiting cases WT ~ 1 and WT ~ 1, where T is the relaxation time and w the angular frequency. In these studies.Jt was assumed that the relaxation process could be represented by the simple firstorder chemical kinetic process
(1) Here go is the equilibrium value of any appropriate progress variable that monitors the degree of completion of the chemical reaction or structural adjustment. For example, in a diatomic gas undergoing rotational relaxation, the specific heat at constant volume is a parameter which is sensitive to the degree of rotational excitation. In a salt solution, g can 201
202
B. B. CARY AND F. H. FENLON
be taken as the product of the degree of dissociation of the salt by its molar concentration. Equation (1) is in the form used by Polyakova [2, 4] and is equally applicable to structural relaxations. We shall discuss its application to aqueous salt solutions. Mazo [6] has written down the hydrodynamic equations of continuity, momentum and energy. He has coupled these with a more general rate expression than equation (1). He linearizes his equations and shows that, as is commonly assumed, it is the volume change associated with a chemical reaction and not the small amount of heat released or absorbed which primarily affects acoustic propagation. He demonstrates the validi ty of this assumption for an aqueous solution of MgS0 4 • This implies that it is justifiable to ignore heat transfer effects in considering the dissociation of MgS0 4 in salt water. The molar content of MgS0 4 is on the average 0·02. It is the dissociation of MgS0 4 that causes the anomalously high absorption of sea water between 10 and 104 kHz. A molar concentration of 0·014 of MgS0 4 in water yields the same absorption curve with frequency as that determined empirically for sea water [7, 8]. In regard to experimental work, Romanenko [9] has qualitatively verified Polyakova's [3] predictions of phase shift between second harmonic and fundamental. Ryan [10] has measured plane wave second harmonic growth in an MnS04 solution for a case in which wT=4·54. It is a characteristic of all relaxation processes that an acoustic signal with a frequency high enough to yield WT > 1 travels with a faster phase velocity than a signal satisfying WT < 1. The relaxation does not have time to equilibrate in each half cycle of signal propagation when W'T> 1. Consequently, a high-frequency signal suffers no increased absorption owing to the presence of the relaxation. By contrast, at WT < 1, the relaxation has time to attain thermochemical equilibrium in each half cycle. In view of the foregoing remarks, for sinusoidal waveforms, we can state that the principal influence of a relaxation process upon a propagating impulse will be to selectively increase the rate of attenuation of the lower frequency components of its spectrum. In addition, the higher frequency components will travel faster than the lower ones changing spectral phase relations. Both of these effects are cumulative with distance. Now our generalized Burgers' equation is only good to second-order inertial terms and to first-order viscous terms. Therefore, we assume that the contributions to total absorption from shear viscosity and a relaxation process are purely additive, since any coupling between them is of third order. It is of interest to note that shear viscosity itself can be viewed as a very rapid structural relaxation occurring in a shorter time than most chemical processes. Marsh [11] has presented a generalized variant of Burgers' equation in order to include the influence of salt water dispersion on the waveforms received from sofar bombs. Marsh [II] did not present a detailed derivation of his equation. The equation that we derive here is not the same, and we shall discuss the differences between them. 2. DERIVATION OF THE GOVERNING NON-LINEAR WAVE EQUATION IN A RELAXING MEDIUM
In weak shock waves with Mach numbers € less than 0·5 or sound pressure levels less than 200 dB relative to 1 ftbar, the entropy changes across the wave can be ignored, since they are of third order in E. Consequently, an appropriate equation of state suitable for relaxing fluids can be derived by making a Taylor series expansion of the excess pressure over static in terms of the deviations of the density and the relaxation parameter from their equilibrium values. On this basis Polyakova [4] derived the following equation of state:
(1 + dtd) P/ T
=
(2 2 .d), (eP P) ( d) /2 Co + coo' T dt P +"2: 8 2 go 1 + T dt P . .i,
p
(2)
FINITE-AMPLITUDE WAVE PROPAGATION
203
The relaxation parameter g does not explicitly appear in equation (2). The partial derivative coefficient of the second-order term in the density has an implied dependence upon ~o (the product of the molar concentration of the salt with its equilibrium degree of dissociation). If the second-order term is neglected , then the dependence of the dispersion upon salt concentration is accounted for through the appearance of c~ and c~. It should be emphasized that the relaxation time T for the salt dissociation is strongly dependent on the temperature and virtually independent of concentration for solutions with molar concentrations similar to that of salt water. Glotov [7] has derived an explicit expression for T as a function of temperature and concentration for MgS0 4 solutions. The weak dependence of T on the concentration validates the use of equation (1) which implies that T is independent of initial molar concentration. Fox [12] has presented data which shows that the dispersion in aqueous MgS0 4 solutions is directly proportional to the molar concentration of salt. The measurements were conducted over the range of 0·1-0·6 M. In addition, Tamm [13] presents data which shows that the excess absorption in aqueous MgS0 4 solutions is directly proportional to the molar concentration for concentrations up to 1·0 M. In order to derive the wave equation, we invoke the conservation equations of continuity and momentum: (3) Pt + pu, + P, v = 0,
(4) Since we are primarily interested in progressive waves, we shall transform from x, t into retarded co-ordinates z, T; thus, z = p.r,
r T=t--
co'
and
a
a
I a coaT'
-=p.~---
or GZ a a at= er Thus, making use of the fact that
for progressive waves and using the small signal relation pi
V
Po
Co
--=in second-order terms, we see that equations (3) and (4) become
(1 -~) p~ PVT + pv (P-V% -
=
(~) (1 +
:J
VT -
P-Po v"
LVT) = - (p- :z - :0 DT)P +
(27] + 7)')
(5)
(p. ~ - :0 DTy o.
(6)
204
B. B. CARY AND F. H. FENLON
In order to eliminatep, if we apply the operator (1 + rD T ) to equation (6), invoke the relation p'/PO = v/co in second-order terms, then with the aid of equations (2) and (5), we obtain
c~
( 1 + C6 •T D T
(3.)
) ( JLv z - ca VV T
=
(1 + 'T DT )
17 +17' (22po Cb )
VTT
+ (m.r) 2co VTT,
(7)
where
(a
2p Po ) • ,8=1+ 2 2Co 0P2 go The f3 term is a measure of the non-linearity of the fluid. Equation (7), which is in Eulerian co-ordinates has a form similar to that of Polyakova's [4] wave equation derived in Lagrangian co-ordinates. Equation (7) differs in that it includes a viscous loss term on the right. In the linear lossless approximation, equation (7) can be reduced to that given by Lamb [14]. It will now be desirable to transform our spatial co-ordinate z to Blackstock's [15] stretched co-ordinate f in order to make our wave equation applicable to cylindrical and spherical geometries. A new dimensionless retarded time y = wTis also introduced. Our new dependent variable W is given by
where q = 0 for plane waves, q = 1/2 for cylindrical waves, and q = 1 for spherical waves. In terms of the new variables, equation (7) becomes
(,)q{( 1+'T--D co)y G
c~ Co ) (Wf-WWy) = ( l+2"'r-Dy Co Xo '0
-I
x;
where c~ -
m.rco} Wyy, +-2t:1
(8)
t'IEX c
ca
m=--2-' Co
is the Gol'dberg number which is a measure of the relative importance of non-linear to dissipative effects and is defined for sinusoidal waves as
(J
G == 21T(3E etA
For an impulse propagating in a non-relaxing medium, G is defined by G = f3lEx c A, 2)/(;VIX)
where A is defined as (41T and x; is a length which has to be chosen. If the impulse has a time duration of L1t, then Xc can be defined as
Co Llt Xc =
b1T '
where b is an integer and Co is the low-frequency sound speed which is appropriate providing
'T/Llt < 1. In a relaxing medium et is not simply proportional to the frequency squared. In fact, a relaxing medium acts like a high pass filter. Therefore, A becomes a function of frequency itself; hence, the above approach loses its utility and it is better to replace IX by an operator containing time derivatives. We shall discuss the construction of such an attenuation operator for the specific case of salt water in the next section. The last term on the right-hand side of equation (8) will usually be negligibly small and will be dropped hereafter.
FINITE-AMPLITUDE WAVE PROPAGATION
205
3. THE ATTENUATION OPERATOR FOR SEA WATER The attenuation of sea water for frequencies of 1 kHz to 1 MHz can be expressed in a fashion similar to that of Marsh [11] as follows:
(9) As stated earlier the MgS0 4 relaxation causes the attenuation IX given by equation (9) to have the frequency characteristic of an inefficient high pass filter. Inefficient in the sense that the attenuation is insufficient to block frequencies for which WT < 1. If equation (9) represents the filter characteristic of the medium in the frequency domain, then we are seeking an appropriate representation of the filter characteristic in the time domain . The time domain representation is an operator containing time derivatives and can be constructed intuitively by replacing w 2 in equation (9) by a second-order derivative with respect to the retarded time y, i.e. The signs are chosen to make the operator yield the correct result wlien W = siny. The chosen operator is
D;.
ex =
(g2 +
1_
T~;C'~~:~) D;) (~:) D;. 2
(10)
Now we can complete the derivation of a generalized Burgers' equation which includes relaxation effects. In equation (8), we replace the term WyyjG by (xiiW)/({3€). Then substituting for ii from equation (10) and multiplying both sides of equation (8) by [1 - or x (co/xc) Dy], we obtain
The dependence on molar concentration of the salt appears on the left through the presence: of c~jcij in the dispersion operator. On the right the dependence upon salt concentration is given through 0 2 and g2 which are directly proportional to salt concentration [13]. Equation (1I) is in a form suitable for application of the Banta method as developed by Cary [1]. The Banta method is a superior technique for solving equation (11). Bellman [16J has discussed the advantages to be gained by using the differential equation itself as a recursion relation to avoid limitations in step size usually imposed by the two-dimensional grid required by a finite difference method. To apply the technique one simply defines a new dependent variable W* where
W*(j,y) =
[1- ~~'T(~:r n;] W.
(12)
A new w* is obtained for each increment LJ1 Then equation (12) must be integrated to find W before taking another step LJ1 Owing to the extra integration that must be performed at each stage, the calculation proceeds more slowly than in the non-relaxing case [1]. Typical values of a and g for sea water taken from Glotov [7] are a2
= 0·9 x 102 nepers/m,
m=7 x lO-S, g = 1'2 X lO- I S sec- nepers/m,
f3 = 3'8, Co
=
1500 mjsec.
206
B. B. CARY AND F. H. FENLON
Reference [14] treats a wide variety of relaxation phenomena. Many processes possess attenuation expressions which do not differ radically from equation (9). Therefore, it should be possible to construct an appropriate operator ii for a variety of cases. It was stated in the Introduction that Marsh (l] had adopted a somewhat different wave equation than that presented here to study impulse propagation in the sea. If we convert Marsh's [11] equation to our nomenclature and change his dependent variable from pressure to velocity, we obtain (13)
where
(3er
(]'=-.
Xc
Equation (13) is applicable to spherically spreading waves. The left-hand side of (13) can be reduced to the left-hand side of (11) if we change variables from 0 tojas follows (for spherical waves): o f= ooln00
Therefore, equation (13) becomes
(::a + D
y)
(WI
WWy) = efIUO[a~:co
-
w"y
+ ~ Wm
] .
(14)
The operator [(Xc/TCO) + D y] on the left side of equation (14) incorporates the effects of dispersion as we know from Lamb's linear lossless dispersion equation [14]. This is similar to the left side of equation (8) providing c;/c~ == 1. For sea water m = (c~ - (5)/c5 == 7 x 10-5 making the left sides of equations (8) and (14) equivalent. The principal difference between equations (8) and (14) is in how the right side was formed. We note that the right-hand side of (14) was constructed in such a way that as one takes the limit at either high or low frequencies the equation reduces to the form
Wf where G
=
efl u o
-
WWy =G Wm
(3€x c A. Consequently, we can rewrite Marsh's [11] equation (14) as
+ Dy) (WI (3:... TCo
1
WWy) = el 1uo [ x C o Wyy + G W m Co T
L
] ,
(15)
II
where the high frequency Gol'dberg number is
with
(16)
FINITE-AMPLITUDE WAVE PROPAGATION
207
Therefore, An is given by the high frequency asymptotic limit for IX from equation (9). It is important to note that we have neglected the heat transfer term in the classical small signal thermoviscous attenuation coefficient, i.e.,
47T 2
O:c
[
,
= 2poCo ,\2 21) +?'] +
?'](y -
Pr
1)] .
(17)
The Prandtl number is defined as
Pr= ?']Cp
K'
where Cp is the specific heat at constant pressure and k the heat transfer coefficient. In sea water with a salinity of 30 parts/thousand at 18°C, y = 1·0207 and Pr = 5·6. The dilatational viscosity?']' of water is about 2?'] [17, p. 321]. Consequently, the ratio of the viscous to the heat transfer losses in equation (17) is 21) + 7]' _ • 3 ?,](y-l)/Pr -106 x 10 . This ratio agrees quite favorably with a value of 1·32 x 103 derived from measurements taken in water [17, p. 281]. Similarly GL=fJEXcAL , where by comparing equations (14) and (15) we find
fJExc fJ A GL = 2--r-2 = "xc L' a
T
Co
and therefore, (18) For a sinusoidal wave, (19) We see that A L given by equation (18) is the direct result of substituting the asymptotic limit for IX at low frequency (W7" < 1), from equation (9) into (19). Consequently, while Marsh's [11] equation (15) is adequate for WT ~ 1, it does not accurately represent the behavior of the absorption in the vicinity of WT == 1. Marsh applied his equation to sofar bomb waveforms at ranges upwards of 1·5 km. The major spectral components of these waveforms satisfied the inequality WT < 1; hence, Marsh's [11] equation was adequate. Equation (11) is not subject to such a limitation and is applicable to impulses which have a spectrum whose components have values for W7" ranging from less than unity to greater than unity. 4. LIMITATIONS OF THE MODEL
The Polyakova [4] equation of state (2) was derived by adopting a general phenomenological approach to the relaxing medium. Hunt [18] has criticized this type of approach calling for a two-fluid theory which would directly couple the chemical and transport processes. Meixner [19] and Mazo [6] have presented more rigorous formulations along the lines suggested by Hunt [18]. Mazo's [6] treatment is preferable to that of Meixner [19] if one is interested in the chemical aspects of the relaxation process. Mazo [6] has shown that the thermoviscous and chemical absorptions and dispersions are additive to first order. Moreover, Mazo [6] showed that the effects of diffusion currents are negligible in liquids but not in gases.
208
B. B. CARY AND F. H. FENLON
Equations (8) and (11) are only valid up to a peak particle velocity Mach number of about 0·05 which corresponds to a sound pressure level of 181 dB relative to I ,ubar. For such weak shocks it is felt that the assumption of addi tive coupling between chemical and thermoviscous absorption is still permissible. REFERENCES 1. B. CARY 1968 J. acoust. Soc. Am. 43, 1364. Prediction of finite-amplitude waveform distortion with dissipation and spreading loss. 2. A. POLYAKOVA 1959 Soviet Phys. Aeoust, 7, 85. Thermodynamic theory of absorption of finiteamplitude sound in relaxing media. 3. A. POLYAKOVA 1960 Soviet Phys. Acoust: 6, 356. Propagation of finite-amplitude waves in relaxing media. 4. A. POLYAKOVA, S. SOLUYAN and R. KHoKHLov 1962 Soviet Phys, Acoust. 8,78. Propagation of finite-disturbances in a relaxing medium. 5. S. SOLUYAN and R. KHOKHLOV 1962 Soviet Phys, Aeoust. 8,170. Finite amplitude acoustic waves in a relaxing medium. 6. R. MAZO 1958 J. chem. Phys. 28, 1223. Absorption and dispersion of sound in chemically reacting fluids. 7. V. GLOTOV 1964 Soviet Phys. Acoust. 10, 33. Calculation of the relaxation time for the degree of dissociation of magnesium sulfate in fresh and sea water as a function of temperature. 8. A. WEISSLER and V. DEL GROSSO 1951 J. acoust, Soc. Am. 23, 219. The velocity of sound in sea water. 9. B. ROMANBNKO 1960 Soviet Phys. Acoust, 6, 375. Distortion of a finite-amplitude waveform propagated in a relaxing medium. 10. R. RYAN, C. ATTANASIO and R. BEYER 1965 J. acoust, Soc. Am. 37, 602. Propagation of finiteamplitude ultrasound in a relaxing liquid. 1L H. MARSH, R. MELLEN and W. KONRAD 1965 J. acoust, Soc. Am. 326. Anomalous absorption of pressure waves from explosions in sea water. 12. F. Fox and T. MARION 1953 J. acoust, Soc. Am. 25, 661. Ultrasonic dispersion in water solutions of magnesium sulfate. 13. S. FLUGGE (Ed.) 1961 Handbuch der Physik Band XI Akustik, Berlin: Springer-Verlag. pp.
237-238. 14. J. LAMB 1965 In Physical Acoustics 2A, Properties of Gases, Liquids and Solutions (Ed. by W. Mason). New York: Academic Press. p, 218. 15. D. BLACKSTOCK 1964 J. acoust, Soc. Am. 36,217. On plane, spherical, and cylindrical sound waves of finite amplitude in lossless fluids. 16. R. BELLMAN Adaptive Control Processes: A Guided Tour. Princeton N.Y.: Princeton University Press, pp, 96-97. 17. S. FLUGGE (Ed.) 1961 Handbuch der Physik Band XI Akustik. Berlin: Springer-Verlag. 18. F. HUNT1955 J. acoust, Soc. Am. 27, 1019. Notes on exact equations governing the propagation of sound in fluids. 19. J. MEIXNER 1952 Acustica 2, 101. Allgemeine Theorie der Schallabsorption in Gasen und Flussigkeiten unter Berucksichtigung der transporter Scheinlungen. APPENDIX LIST OF SYMBOLS
w T
g .t p p
o Co
angular frequency relaxation time a thermodynamical variable which monitors the relaxation process time pressure density subscript denoting thermochemical equilibrium superscript denoting deviation of quantity from its equilibrium value small signal speed at thermochemical equilibrium
FINITE-AMPLITUDE WAVE PROPAGATION
frozen small signal speed c' -c z
Ceo
m=~
c~
particle velocity and peak particle velocity at source range and source radius shear viscosity dilatational viscosity
v, Vo
r, ro 'YJ 'YJ' Z
=p,r used in retarded co-ordinates to emphasize smallness of spatial derivatives retarded time = t - rico liquid non-linearity parameter
fL
T f3
W=~
rovo q "" 0 for plane wave; = t cylindrical wave; = 1 for spherical wave E Mach number IX attenuation coefficient for relaxing medium IXc small signal thermoviscous attenuation coefficient ,\ wavelength
J t ,wh ere Xc = Co b7T
A' d . 0 f impu ' I se an db'IS any integer , at rs uration
Gol'dberg number = f3Ex c A for an impulse
G
=
27TfJE l' ' 'd IX'\ lor SIOUSOt
a = fJEXcZ
f
= 0 for plane waves =
2VO'o(V~ - VO'o} for cylindrical waves
= 0'0
In (0'10'0) for spherical waves
47TZ
A "",\zlX frequency constants in salt water absorption expression y=wT v
G,
g
a
D,=-
at
2
47T h ere A L = ~,w 1\
(XL
is tthe asymptotic ' I'imit. 0 f eex: lor f is
W7'
Z
' I'tmit, 0 f asymptotic
AH =
47T h ere IXn IS is th ~,w t e 1\
W·
=
(1 + C;7'~D,) W Co
C
Z
8=~Tw
c5
<1
Xc
l' lor
WT
>I
209