dispersion relationships in sediments and sea water

Dispersion relationships in sediments and sea water C. W. Horton Sr.

Applied Research Laboratories and Department of Physics, The University of Texas at Austin, Austin, Texas 78712 (Received 24 October 1973)

The Kramers-Kronig dispersion relationship is used to discuss the interdependence between velocity and attenuation in sediments and sea water. It is shown that if the attenuation is proportional to the n th power of the frequency, there is no velocity dispersion if n = 1 or 2 but that dispersion exists for nonintegral values of n. The theoretical relation agrees with the experimental measurements of L. Hampton for a sediment with n- 1.37. The relaxation process of sea water is considered and it is shown that the velocity dispersion is slightly greater than the precision of velocity measurements.

Subject Classification: 30.20.

INTRODUCTION

It is possible to show on the basis of fundamental ar- guments concerning causality that the complex index of refraction of a wave satisfies an integral known as the Kramers-Kronig dispersion relationship. The practical importance of this relationship ig that the real part of the index of refraction can be predicted if the complex part is known and vice versa. This relationship has been applied extensively in network theory, electro- magnetic theory, and elementary particle theory but it has not been applied to any significant extent in acoustics. The reader is referred to papers by van Kampen • and Toll z for a fundamental demonstration of the theorem.

Goldhaber 3 has given a brief history of the subject and a clear discussion of the physical ideas involved. Ginz- berg 4 has discussed the applicability of the dispersion relationship to acoustical problems.

Although the dispersion relationship can be expressed in different forms depending on the behavior of the complex index of refraction at infinity, a form that is suf- ficiently general for the problems discussed in this paper is the following. (This is Goldhaber's Eq. I-1.30.) The complex index of refraction n(co) satisfies the integral relationship

n( • ) - n( •o) = ( • - •o) p n( o.,')ao., ' ,ri .• (•' - o.,)(o.,' - •o)' (1) provided lim•.. {n(co)/co{ = 0 in the upper half of the complex co plane. P means the Cauehy principle value of the integral. The parameter coo is a real constant which may be zero or i ,nfinity in special eases. In order that a real signal remain real after passing through the medium, n(co) is subjected to a condition, namely,

n(-co)=n*(+co), (2)

where the asterisk means complex conjugate. A second requirement on the complex index of refraction is that it have no poles in the upper half of the complex co plane. This is a cons•equence of a causality argument.

When the index of refraction is separated into real and imaginary parts , ns(co), n z (co), and Eq. 2 is utilized, the real part of Eq. 1 may be written in the form

2(•- •) j' • n• (•') •' a•' ns(co) = ns(co0) + P (co,z (3) - • o (•'•'- •") - •) '

Two special cases of this integral will be needed in the

following. If co0--ø% Eq. 3 becomes

2 f0•n• (•') •' a• ' n•(•) =n•(oo) + - P _ •. '

while if coot 0, Eq. 3 becomes

(4)

2co a ns(co)=ns(O)+ ' P co,(co,z coz). (5) 7t 0 --

Equation 4 is slightly modified form of Goldhaber's Eq. I-1.33 and Eq. 5 is identical with his Eq. I-1.15.

I. APPLICATION TO SEDIMENTS

If a plane pressure wave of the formp 0 exp [ik(co)x - icot] propagates through an acoustic medium; the real part of k(co), say b(co), is called the dispersive part of the wavenumber k and is related to the phase velocity vph (co) by the relationship

b(•,) = •/v,,(•,) . (6)

The imaginary part of k(co), say a(co), is called the ab- sorptive part of the wavenumber k. a(co) is the attenuation of the wave in nepers per unit distance and is the physical quantity measured in many experiments. The major purpose of the present article will be to use experimental values of a(co) to predict the associated phase velocity.

If a reference velocity c o is introduced, the complex index of refraction is related to the wavenumber k by the definition

n( co) : Co k(co)/co. (7)

The real and complex parts of this equation are related directly to physically observable quantities. Thus for the real part,

ns(co) = c o b( co)/co = Co/v,,h(co), (8)

while for the imaginary part,

n •r ( co ) : c o a ( co ) / co . ( O )

Experimental studies such as those of Hampton s and Hamilton, ø for example, often show that the experimental values of at[enuation can be expressed quite well in the form

a(co) : L'• (co/%.) ", (10)

547 J. Acoust. Soc. Am., Vol. 55, No. 3, March 1974 Copyright ̧ 1974 by the Acoustical Society of America 547

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548 C.W. Horton Sr.: Dispersion in sediments and sea water 548

where L is a length, cot is a reference frequency, and n is a positive number that need not be an integer. Specific values of n may be quoted. Hamilton ø finds n = 1 for ocean sediments, and n = 1 is widely cited for seismic waves in clastic. sediments. Hampton 5 finds n= 1.37 for one of the sediments he constructed in the

laboratory. The viscous loss in water yields n= 2.

When Eqs. 9 and 10 are combined, one has

nz ( co) = Co co"4/Lco• . (11)

The choice between Eqs. 3, 4, and 5 depends on the value of n.

Case A, 0 < n < 1

If 0<n< 1 , Eq. 4 is suitable. Fortunately, the re- sulting integral has been tabulated by Erd•lyi, 7 and one finds

nR(co) =nR(oo) + (co/Lcor)(co/cor)"ttan(•rn/2) . (12)

In this case it is natural to take Co as the phase velocity at infinite frequency so that nR(øo) = 1, and th• phase velocity at any other frequency is given by

1/V•h(CO) = 1/Co+tan(•n/2)(1/Lcor)(co/cor) "'•. (13)

Case B, n = 1

The important case n = 1 requires the use of Eq. 3. When Eq. 11 with n = 1 is substituted into Eq. 3, one has

f o ø co r d co t - c 0 p _ = 0) + - '

It is possible to show that the integral va•shes so that n•(•) • n•(•0) , and there is no dispersion for n = 1. This conclusion is in •reement with experiment• measurements.

Case C, I < n < 3 If the exponent n fails in the range 1 < n < 3, Eq. 5 may

be utilized to give

ns(co) =ns(O)+ tan[(n - 2)•/2](Co/Lco,)(co/cor) "'• . (15)

If n= 2 as in pure water, the second term on the right vanishes and v,•(co) is a constant independent of the frequency in agreement with experiment. If 2 < n < 3, the phase velocity decreases montonically to zero as co in- creases. This appears to be a satisfactory behavior. However, if 1 < n< 2, tan[(n - 2)•/2] is negative and the phase velocity becomes infinite at a finite frequency. This situation is intolerable yet Hampton's experimental data show that n= 1.37 arises. The difficulty of an infinite phase velocity can be avoided if it be assumed that there are two loss mechanisms with different power laws so that

a( co) = g•L'•( co/co,)" + g2L'•( co/co,)", (16)

with 1 < m < 2 and 2< n< 3. There is a corresponding sum of terms in Eq. 15 to allow for the two loss mechanisms, but it is possible to adjust the weighting factors g• and g•. so that the term corresponding to 1 < m < 2 is dominant over the frequency range of interest.

As an example of the application of these formulas, Eq. 15 is applied to the empirical attenuation function

a(f)= (f /fr) • .a7 dB/ft, (17) that Hampton • used to fit his measurements on a black sediment of •õ% volume concentration. The reference

frequency fr is 12 kHz. The associated phase velocity given by Eq. 1õ is

V•h(f)=Co[1 -- O. Oll(f/fr)•'a*] '•. (18) The Kramers-Kronig formula will not predict c o so that an adjustable constant is available. Equation 18 is plot- ted in Fig. 1 together wit h Hampton's experimental data for this sediment. Equation 18 yields an infinite phase velocity, but only at 2.2 GHz where other loss mech- artisms will have become important.

Igarashi • has measured attenuation and phase velocity in marine sediments in the Santa Barbara shelf off the coast of California. He fitted his attenuation mea-

surements (made at 15, 30, and 60 kHz)with Eq. 10 and found values of n ranging from 1.13 to 1.26. His measured values of phase velocity increased with frequency as predicted by the present analysis.

II. OBSERVATIONS ON THE COMPLEX INDEX OF

REFRACTION

In the examples discussed in the last section, the integral could be evaluated in a closed form. In the case of experimental data this may not be possible and a nu- merical integration may be necessary. An alternative procedure is to write down analytic functions for the complex index of refraction which satisfy the conditions of physical realizability and causality. One is assured from the mathematical derivation of Eq. 1 that the real and imaginary parts of this analytic function satisfy the Kramers-Kronig dispersion relationship. Thus, for example, all of the cases treated in the last section can be deduced from the complex function (n >0)

n(co) = n(co0) + [tan (•n/2) + i](co/cor)"'•. (19)

This function satisfies Eq. 2, has no poles in the upper half of the co plane, and behaves as a polynomial at infinity.

looj 0.98[ - 0.96 •_• n

• 0.94•

J - BLACK SEDIMENT 35% CONCENTRATION

0.92 I I I

2 5 10 20 50 100 200

FREQUENCY - kHz

FIG. 1. The dependence on frequency of the phase velocity in a sediment. Dashed line: experimental values of Hampton. • Solid line: theoretical prediction based on experimentally ob- served attenuation.

J. Acoust. Soc. Am., Vol. 55, No. 3, March 1974


549 C.W. Horton Sr.: Dispersion in sediments and sea water 549

III. DISPERSION IN SEA WATER

The approach of the last section is synthetic, but it leads to an easy formulation of the problem of attenuation associated with relaxation processes. Schulkin and Marsh 9 have fitted experimental observations of attenuation in sea water with an empirical formula (a pressure correction is omitted)

a(f )=[SAfvf•'/(f•+f•')]+(Bf•'/fv) Np/m, (20)

where S= salinity in parts per thousand, A =2.34x 10 'ø (when f is in kHz), B = 3.38x 10 'ø (when f is in kI-Iz), and fv= relaxation frequency near 150 kI-Iz at ambient tem- peratures. The first term is connected with the absorption produced by MgSO4 and the second term is associated with the viscous losses inpure water. Converting Eq. 20 to an index of refraction relative to a reference velocity Co, and introducing angular frequencies, one has

,

nt(øø)=M(øø/øør)+ l+(•o/•or) •' ' (21) where

M = (c0/2•r) B x 10 -a, (22)

N = (Co/2rr)SA x 10 -a ' (23)

The extra factors of 10 'a in Eqs. 22 and 23 are introduced because the empirical numbers in Eq. 20 are based in frequency expressed in kilohertz while Eq. 9 requires f in Hertz. It is possible to write down by in- speetion the real part %(0o) that is associated with this function. Thus,

N

%(o•)= 1 +1 +(oo/oov) •' ' (24) corresponding to a complex index of refraction

N

n(•o)= 1 + iM(oo/oo•,)+ 1 -i(oo/oo•,) ' (25) which satisfies all of the conditions imposed on n(•o).

Equation 24 shows that the absorption associated with MgSO4 requires that there be a modest velocity dispersion in sea water. Equation 24 predicts that v•h(oo) - v•(0) at atmospheric pressure, 20 øC, and S = 35 parts per thousand is near 0.03 m/sec. This dispersion is larger than the errors of precise velocity measurements of sea water but smaller than the discrepancy between empirical formulas and experimental data stated by DelGrosso and Mader. •0

IV. CONCLUSION

The usefulness of the dispersion relationships in acoustics appears to have been overlooked. When experimental values of both velocity and attenuation are available, their mutual compatibility can be checked. If, as is more frequently the case, there are only attenuation data, the magnitude of the velocity dispersion can be predicted. The reader may have some hesitancy in integrating the index of refraction over an infinite frequency range when, of necessity, acoustic waves can exist only over a finite region. This point is discussed in detail by Ginzberg 4 who limits integrals much as those in Eqs. 1-5 to a finite range of frequency and introduces the corresponding error term. The same difficulty arises in electrical circuit theory where the lumped constant approximations are valid only at finite frequencies, but this does not impair the usefulness of dispersion theory in circuit theory.

ACKNOWLEDGMENTS

The author wishes to thank Dr. Loyd D. Hampton who suggested this investigation and who has kindly granted permission to reproduce Fig. 1 from his work. The author is also indebted to Aubrey Anderson for many helpful conversations. This work was partially sup- ported by ONR, Code 483.

iN. G. van Kampen, "S-Matrix and Causality Condition. I. Maxwell Field," Phys. Rev. 89, 1072-1079 (1953).

2john S. Toll, "Causality and the Dispersion Relation: Logical Foundations," Phys. Rev. 104, 1760-1770 (1956).

3M. L. Goldberger, "Introduction to the Theory and Applica- tions of Disp. ersion Relations," in Relations de Dispersion et Particules Elementaires, edited by Cecile DeWitte (Hermann, Paris, 1960), pp. 15-157.

4V. L. Ginzberg, "Concerning the General Relationship between Absorption and Dispersions of Sound Waves," Sov. Phys.- Acoust. 1, 32-41 (1955).

5Loyd D. Hampton, "Acoustic Properties of Sediments," J. Acoust. Soc. Am. 42, 882-890 (1967).

6Edwin L. Hamilton, "Elastic Properties of Marine Sediments," J. Geophys. Res. 76, 579-604 (1971).

?Arthur Erd•lyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. I, p. 309.

Sy. Igarashi, "In Situ High-Frequency Acoustic Propagation Measurements in Marine Sediments in the Santa Barbara

Shelf, California," Report NUC TP 334, Naval Undersea Cen- ter, San Diego, Calif. (Jan. 1973).

9M. Schulkin and H. W. Marsh, "Sound Absorption in Sea Water," J. Acoust. Soc. Am. 34, 864-865(L) (1962).

løV. A. DelGrosso and C. W. Mader, "Speed of Sound in Sea- Water Samples," J. Acoust. Soc. Am. õ2, 961-974 (1972).

J. Acoust. Soc. Am., Vol. 55, No. 3, March 1974


dispersion relationships in sediments and sea water

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