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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 60 (2012) 1791–1813

0022-50

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n Corr

E-m

journal homepage: www.elsevier.com/locate/jmps

Acoustic radiation force in tissue-like solids due tomodulated sound field

Egor V. Dontsov, Bojan B. Guzina n

Department of Civil Engineering, University of Minnesota, United States

a r t i c l e i n f o

Article history:

Received 14 April 2011

Received in revised form

15 December 2011

Accepted 15 April 2012Available online 23 May 2012

Keywords:

Acoustic radiation force

Nonlinear acoustics

Modulated ultrasound

Dual time scale method

96/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.jmps.2012.04.006

esponding author. Tel.: þ1 612 626 0789; fa

ail address: guzina@wave.ce.umn.edu (B.B. G

a b s t r a c t

The focus of this study is the sustained body force (the so-called acoustic radiation

force) in homogeneous tissue-like solids generated by an elevated-intensity, focused

ultrasound field (Mach number¼Oð10�3)) in situations when the latter is modulated

by a low-frequency signal. This intermediate-asymptotics problem, which bears

relevance to a number of emerging biomedical applications, is characterized by a

number of small (but non-vanishing) parameters including the Mach number, the ratio

between the modulation and ultrasound frequency, the ratio of the shear to bulk

modulus, and the dimensionless attenuation coefficient. On approximating the

response of soft tissues as that of a nonlinear viscoelastic solid with heat conduction,

the featured second-order problem is tackled via a scaling paradigm wherein the

transverse coordinates are scaled by the width of the focal region, while the axial and

temporal coordinate are each split into a ‘‘fast’’ and ‘‘slow’’ component with the twin

aim of: (i) canceling the linear terms from the field equations governing the propagation

of elevated-intensity ultrasound, and (ii) accounting for the effect of ultrasound

modulation. In the context of the focused ultrasound analyses, the key feature of the

proposed study revolves around the dual-time-scale treatment of the temporal variable,

which allows one to parse out the contribution of ultrasound and its modulation in the

nonlinear solution. In this way the acoustic radiation force (ARF), giving rise to the

mean tissue motion, is exacted by computing the ‘‘fast’’ time average of the germane

field equations. A comparison with the existing theory reveals a number of key features

that are brought to light by the new formulation, including the contributions to the ARF

of ultrasound modulation and thermal expansion, as well as the precise role of

constitutive nonlinearities in generating the sustained body force in tissue-like solids

by a focused ultrasound beam.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Garnering keen interest since its first observation by Faraday (1831) who detected the circulation of air caused by avibrating plate, the radiation force of sound is a phenomenon that has been intensely studied for over a century (Rayleigh,1884; Rudenko and Soluyan, 1977; Lighthill, 1978; Hamilton and Blackstock, 1998). In simple terms, the concept of theacoustic radiation force (ARF) describes the generation of ‘‘low’’-frequency mean motion in the medium when a ‘‘high’’-frequency, time-harmonic field of sufficient intensity is propagated through it. This downward leakage of energy across the

ll rights reserved.

x: þ1 612 626 7750.

uzina).

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–18131792

frequency spectrum, brought about by an intricate interplay between the (kinematic and constitutive) nonlinearities of theproblem and dissipative and/or heterogeneous nature of the propagation medium, has been known to cause bothstreaming in fluids (Faraday, 1831) and sustained deformation in soft, tissue-like solids (Sarvazyan et al., 1998; Bercoffet al., 2004; Palmeri et al., 2005).

On the medical forefront, the phenomenon of the ARF has been increasingly exploited by way of elevated-intensity,focused ultrasound beams toward a broad spectrum of applications including tissue elasticity and viscoelasticity imaging,targeted drug delivery, and monitoring of ablation therapy (Sarvazyan et al., 2010; Sarvazyan, 2010). In a number of theseapplications, the focused ultrasound field is modulated with either transient or steady-state signal to: (i) facilitate theobservations of the ARF-induced tissue motion (Bercoff et al., 2004; Liu and Ebbini, 2008), (ii) expose the rate effects intissue’s constitutive response (Palmeri et al., 2005), and (iii) minimize the risk of tissue damage from local heating effects(Sarvazyan et al., 1998). For clarity, it is noted that the term ‘‘ARF’’ is in the literature used to denote two distinctphenomena affiliated with the propagation of elevated-intensity sound, namely that of a sustained, i.e. mean body force ina homogeneous medium (Rayleigh, 1884; Rudenko and Soluyan, 1977; Rudenko et al., 1996), and the mean force on aheterogeneity, i.e. obstacle caused by the nonlinear scattering effects (e.g. Hasegawa and Yosioka, 1969). To provide a focusfor the discussion, the remainder of this paper deals exclusively with the generation of the ARF in homogeneous media.

To motivate the study of the ARF, one may note that the computation of the full nonlinear response of either solid orfluid medium subjected to elevated-intensity sound field is generally an arduous task due to the presence of a small timescale affiliated with the carrier sound signal. In many situations, however, sufficient information about the processexamined, effect sought, or medium interrogated can be obtained by dispensing with the fine time scale and focusing onthe mean ‘‘backbone’’ response instead. In the case of dissipative fluids, this is effectively accomplished via an asymptotictreatment (Rayleigh, 1884; Rudenko and Soluyan, 1977; Lighthill, 1978; Hamilton and Blackstock, 1998) which formulatesthe problem in terms of the mean fluid motion, reckoned per period of sound vibrations. Here it is noted that: (i) the time-averaged field equations (which happen to describe fluid streaming) are inherently simpler than their instantaneouscounterparts, and (ii) the reckoned balance of linear momentum features an effective body-force term, termed the ARF,which drives the mean motion and is computed as a time average of the leading nonlinear terms in the original equation.In the case of tissue-like solids the situation is similar (Sarvazyan et al., 1998, 2010), albeit with an important distinctionthat the equations governing the mean tissue response are linear owing to the absence of streaming. Irrespective ofthe nature of the propagation medium, however, the key challenge in exacting its mean motion revolves around thecomputation of the ARF for the particular problem at hand.

Among the aforesaid diversity of biomedical applications of the ARF, perhaps the most widespread – and the one thatmotivates this study – deals with tissue (visco-) elasticity imaging where an elevated-intensity, focused ultrasound field(Mach number ¼ Oð10�3)) is used to ‘‘push’’ inside the tissue and thus achieve non-invasive internal palpation (Bercoffet al., 2004; Liu and Ebbini, 2008; Guzina et al., 2009; Sarvazyan, 2010). In this class of diagnostic ultrasound techniques,the carrier signal (MHz rate) is often modulated by a low-frequency envelope (kHz rate) (Fatemi and Greenleaf, 1998;Bercoff et al., 2004; Barannik et al., 2004; Guzina et al., 2009) both to facilitate the motion measurements and to confinetheir support to a neighborhood of the focal point which simplifies the interpretation. In the landmark study of the effectsof ultrasound modulation on the ARF (Sarvazyan et al., 1998), the germane radiation force in a tissue-like solid wasapproximated as a product between the non-modulated expression for the ARF (computed for a viscous fluid) and thesquared modulation envelope. Unfortunately, such ‘‘separation’’ approach becomes inadequate when the modulationfrequencies reach kHz range (assuming ultrasound frequencies on the order of MHz) as demonstrated in Dontsov andGuzina (2011a) via the example of plane waves. Further, the ARF generated in a tissue-like solid may not be describable bythat induced in a viscous fluid owing to irreconcilable differences between the respective (nonlinear) constitutivebehaviors. Recognizing the latter discrepancy, recent studies in Ostorvsky et al. (2007), Sarvazyan et al. (2010) offer apreliminary examination of the ARF in a nonlinear soft solid assuming i) no effects of ultrasound modulation, and ii) an ad

hoc, trial variation of the ultrasound-scale displacement field as a tool to illustrate the role of the Landau’s third-ordermoduli (Landau and Lifshitz, 1986) in the generation of the ARF.

To help bridge the gap between the practice and theory of the ARF in soft tissues, this study aims to derive a rigorousexpression for the acoustic radiation force due to modulated, focused ultrasound field in tissue-like solids assuming that theratio between the modulation and carrier frequency is on the order of the Mach number. To account for both the rate-dependent nature and ultrasound-induced heating of a soft tissue, its constitutive behavior is approximated as that of anonlinear elastic solid with heat conduction and viscosity (Coleman and Noll, 1963). For generality, the analysis considersa wide range of ultrasound beam geometries reported in the literature (Fatemi and Greenleaf, 1998; Sarvazyan et al., 1998,2010; Guzina et al., 2009) that transcend the limitations of past treatments (Sarvazyan et al., 1998; Ostorvsky et al., 2007;Sarvazyan et al., 2010), yet conform with the customary premise of nearly-planar wave propagation within the focal region(Sarvazyan et al., 1998). Following the recent treatment of the affiliated plane-wave problem for Newtonian fluids(Dontsov and Guzina, 2011a), the effects of ultrasound modulation are tackled via a dual-time-scale approach which splitsthe temporal variable into its ‘‘fast’’ i.e. ultrasound-scale, and ‘‘slow’’ i.e. modulation-scale component. On computing the‘‘fast’’ time averages of the nonlinear field equations, the effect of an elevated-intensity, modulated ultrasound beam on atissue-like solid is synthesized via (i) the ARF which signifies the mean mechanical action, and (ii) apparent heat sourceterm which represents the mean thermal action. Recognizing the differences between the Eulerian and Lagrangian meanmotion (Lighthill, 1978), the formulation is developed for both types of averages. A comparison with the existing theories

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–1813 1793

(Sarvazyan et al., 1998; Ostorvsky et al., 2007) reveals a number of key features that are brought forward by the newformulation, including the contributions to the ARF of ultrasound modulation and thermal expansion, as well as the exactrole of constitutive nonlinearities in generating the mean body force in a tissue-like solid by the focused ultrasound beam.The developments are illustrated via an analytical example which provides clues into apparent discrepancies between theexisting estimates of the ARF (Sarvazyan et al., 1998) and experimental observations.

2. Preliminaries

To help understand the transfer of momentum and energy from an elevated-intensity, focused ultrasound field –propagated through a soft tissue – to the later, this section recalls the underpinning balance laws and establishes thenecessary constitutive model for soft tissues. Due to the fact that such generated motion may entail significant strains in atissue near the focus of the beam, the problem is for generality tackled from both Lagrangian and Eulerian viewpoints.

Dimensional platform. For the analysis of wave motion in soft tissues that are herein modeled as thermo-viscoelasticsolids, the problem variables and parameters can be conveniently normalized by the mass density Ro and absolutetemperature T o of a soft tissue at rest; the small-amplitude compressional wave speed in a tissue co, and the (carrier)ultrasound frequency O. With such scaling scheme in place, all quantities appearing hereon are understood as beingdimensionless.

2.1. Balance laws

In what follows, let x and xo denote the current and reference position of a material point whereby

uðxo,tÞ ¼ x�xo, ð1Þ

signifies the displacement in a tissue. With such notation, all thermomechanical quantities of interest can be written ineither Eulerian or Lagrangian frame of reference, with the latter description being indicated by the ‘‘hat’’ symbol. In termsof the displacement and velocity fields, one accordingly has

uðx,tÞ ¼ uðxo,tÞ, vðx,tÞ ¼ vðxo,tÞ ¼@u

@t: ð2Þ

Owing to the fact that the expression for the acoustic radiation force inherently depends on the choice of the referenceframe (Sarvazyan et al., 2010), the ensuing developments aim to expose the effect of ultrasound modulation on the ARF inboth descriptions.

2.1.1. Eulerian frame of reference

On lettingrð�Þ denote the gradient with respect to x ‘‘to the right’’ (Malvern, 1969), one may write the balances of mass,linear momentum, and energy as

@r@tþr � ðrvÞ ¼ 0,

r @v

@tþv � rv

� �¼r � r,

r @U

@tþv � rU

� �¼�r � qþr : rv, ð3Þ

in Eulerian coordinates, where r denotes the mass density; r is the Cauchy stress tensor; U denotes the specific internalenergy; q is the heat flux, and A : B� trðAT

� BÞ for any two second-order tensors A and B where ‘‘T’’ indicates tensortranspose.

2.1.2. Lagrangian frame of reference

The counterparts of (3) in the Lagrangian frame of reference (Malvern, 1969) can be written as

r ¼ 1

detðF Þ, F ¼ IþrTu,

@2u

@t2¼r � P , P ¼ detðF ÞF

�1� r,

@U

@t¼�r � ðdetðF ÞF

�1� qÞþ P :

@FT

@t, ð4Þ

where rð�Þ indicates differentiation with respect to xo, F is the deformation gradient, I is the second-order identity tensor,and P denotes the first Piola-Kirchhoff stress tensor.

To close either system (3) or (4), one must provide the constitutive equations for r, U and q which are established next.

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–18131794

2.2. Thermo-viscoelastic constitutive framework for soft tissues

To study the transfer of momentum and energy from a modulated (elevated-intensity) ultrasound field to a soft tissue,it is convenient to approximate the latter as an elastic material with heat conduction and viscosity (Coleman and Noll, 1963)whose constitutive behavior is determined in terms of five response functions: U, T, R, J and q, namely

U ¼UðF ,SÞ,

T ¼ TðF ,SÞ,

r¼RðF ,SÞþ JðF ,SÞ : rv,

q¼ qðF ,S,rTÞ, ð5Þ

where S denotes the specific entropy, and T is the absolute temperature. Here it is noted that (5c) can be loosely affiliatedwith the Kelvin-Voigt mechanical analog due to the fact that RðF ,SÞ is the (nonlinear) thermoelastic part of the Cauchystress tensor, while J : rv represents its viscous component, assumed to be linear in rv for an assumption to the contrarywould necessitate the inclusion of acceleration gradients in the argument list for r (Coleman and Noll, 1963). To satisfy theClausius–Duhem inequality, (5) must be such that

T ¼@U

@S

� �F

, R¼ r @U

@F

� �S

� FT, rv : J : rvZ0, rT � qr0, ð6Þ

assuming that the temperature relation is invertible (Coleman and Noll, 1963).Comment on the constitutive format. Considering the particular structure of the assumed viscoelastic relationship (5c), it

is worth noting that the nature of the rate-dependent constitutive behavior of soft tissues over a broad frequency spectrum(kHz to MHz range) remains an open question. In the MHz frequency range, (linear) ultrasound studies indicate that thetissues’ constitutive behavior in shear is consistent with that of the Kelvin–Voigt model, while their bulk response is moreclosely described by the Zener (i.e. standard linear solid) model (Ahuja, 1979). In the lower (quasi-static to kHz) frequencyrange, on the other hand, rate-dependent models for the constitutive response of soft tissues in shear vary widely (Guzinaet al., 2009; Greenleaf et al., 2003; Schmitt et al., 2008; Sinkus et al., 2005a), while the viscoelastic description of their bulkbehavior is virtually nonexistent. In this setting, a generalization of (5c) that may be better suited for finely matching anyparticular set of constitutive observations at both (ultrasound and modulation) time scales can be formulated within theframework of materials with memory (Coleman, 1964), wherein U, r and q are assumed to depend on the time histories ofF , S andrT . An account for such behavior, however, would result in a integro-differential formulation that is not amenableto a compact analytical treatment. As a result the assumed viscoelastic framework, that is consistent with earlier ARFstudies (Rudenko et al., 1996; Sarvazyan et al., 1998), is deemed adequate for its tractability and the lack of comprehensive(broadband) experimental data on the rate-dependent behavior of soft tissues.

Isotropic solid. Owing to the fact that the isotropic solids contain �I in their symmetry group (Truesdell and Noll, 2004),it can be shown (Coleman and Noll, 1963) that (5d) is subject to an additional constraint of antisymmetry with respect torT , namely

qðF ,S,rTÞ ¼ qðF ,S,�rTÞ: ð7Þ

Under the premise of material isotropy, one may further reduce (5a) as

U ¼UðI1ðEÞ,I2ðEÞ,I3ðEÞ,SÞ, I1 ¼ trðEÞ, I2 ¼12½ðtrðEÞÞ2�trðE2

Þ�, I3 ¼ detðEÞ, ð8Þ

where In (n¼ 1;2,3) denotes the nth invariant of the Green-Lagrangian strain tensor E¼ 12ðF

T� F�IÞ. On substituting (8) in

(6), one finds

T ¼@U

@S, R¼ rF �

@U

@I1Iþ

@U

@I2ðI1ðEÞI�EÞþ

@U

@I3ðI2ðEÞI�I1ðEÞEþE2

Þ

" #� FT, ð9Þ

where the dependence on entropy is embedded in the derivatives of U . To facilitate the specification of U , it can be shownon the basis of (9) and basic thermodynamic considerations that the mixed derivative of specific internal energy at theinitial (equilibrium) state ‘‘o’’ permits representation

@2U

@S@E

!o

¼@T

@E

� �S,o

¼@½r�1F�1

�R � F�T�

@S

!E,o

¼�bT

cpI, bT ¼ aTK , ð10Þ

where aT is the coefficient of volumetric thermal expansion, K denotes the linear elastic bulk modulus, and cp is thespecific heat capacity at constant stress. From (10), one finds that the linearization of the second of (9) results in a linearthermoelastic relationship

R�ro ¼ K�2

3G

� �ðr � uÞIþGðruþrTuÞ�

bT

cpðS�SoÞI, ð11Þ

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–1813 1795

where K ¼ 1�43G and G denote respectively the tissue’s bulk and shear modulus, while So and ro are respectively the

specific entropy and Cauchy stress tensor evaluated at the initial state. To preserve the isotropy of the material in aprestressed reference configuration, it is hereon assumed that ro ¼�poI, i.e. that the initial state of stress is hydrostatic.For completeness, it is noted from (11) that K signifies the bulk modulus at constant entropy, as opposed to the bulkmodulus at constant temperature (KT) as is usually assumed in linear thermoelasticity (Malvern, 1969). The differencebetween the two definitions is, however, typically small since K=KT ¼ cp=cv � 1 where cv is the specific heat capacity atconstant strain.

3. Governing equations for small Mach number

With reference to the adopted scaling scheme, one may introduce the Mach number affiliated with the ultrasound wavein a soft tissue as

e¼ vmax �ffiffiffi2p

I1=2max, ð12Þ

where vmax is the maximum (dimensionless) particle velocity, and Imax is the associated wavefield intensity.

3.1. Smallness of problem parameters

In diagnostic ultrasound applications involving the acoustic radiation force, the Mach number is typically on the orderof 10�3 (Sarvazyan et al., 1998) which supports the hypothesis e51 commonly made in the ARF analyses (Sarvazyan et al.,1998; Rudenko et al., 1996). In the context of nonlinear solid mechanics, one finds from (12) and the definition of Green-Lagrangian strain that E¼OðeÞ (see Section 3.4), i.e. that the maximum strain generated by the ultrasound wave is on theorder of 0.1%. While such level of strain normally does not warrant the deployment of finite-deformation kinematics in thecontext of static and quasi-static processes, numerous experimental studies have demonstrated that e¼ Oð10�3

Þ issufficient to produce observable nonlinear effects in ultrasound wave propagation (Sarvazyan et al., 1998).

For an effective asymptotic analysis of (4)–(9) when e51, it is noted that the dimensionless attenuation coefficient a insoft tissues takes values in the range ð0:525Þ � 10�3 at MHz-rate ultrasound frequencies (Papadakis, 1999; Ahuja, 1979),which implies a¼OðeÞ. To cater for medical imaging applications where the ultrasound modulation frequency is regularlyon the order of kHz (Fatemi and Greenleaf, 1998), the proposed analysis further assumes om ¼OðeÞ for this key parameter,considered to be oðeÞ in previous treatments (Rudenko and Soluyan, 1977). With such hypotheses, one may convenientlyrescale the attenuation coefficient and modulation frequency as

~a :¼ ae¼ Oð1Þ, ~om :¼

om

e¼ Oð1Þ, e51: ð13Þ

Recalling (5), it is noted that the assumption about the smallness of a further implies that the combined dissipation effectdue to viscosity and heat conduction must be OðeÞ. Accordingly, one may tentatively set

~J ðF ,SÞ :¼JðF ,SÞ

e¼Oð1Þ, ð14Þ

a hypothesis that will be substantiated later once the constitutive model for soft tissues, synthesized via R and J, isspecified.

In the context of (11), estimates of the linear-elastic shear modulus in soft tissues vary widely and have been reportedin the range 103

2106 Pa (Sarvazyan et al., 1998; Schmitt et al., 2008; Sinkus et al., 2005a). Given the fact that thecompressional-wave modulus of soft tissues, Roc2

o, is close to the bulk modulus of water (2:2� 109 Pa), it follows on thebasis of the assumed scaling scheme, that G does not exceed OðeÞ. As a result, one may safely accompany (13) and (14) byintroducing the rescaled shear modulus

~G :¼G

e rOð1Þ: ð15Þ

In this setting, situations where ~G51 can be dealt with by dropping the terms containing ~G as needed.For an in-depth analysis of the thermomechanical problem at hand, it is further noted that the coefficient of thermal

conductivity, featured in the Fourier law (i.e. the leading-order approximation of (5d)), is normally k� 3� 10�7¼Oðe2Þ in

soft tissues (NCRP, 1992). In the context of (3c), (5), (6) and (14), this implies that the dissipation due to heat conduction issmall relative to that induced by material viscosity. This is in contrast to the customary hypothesis underpinning theanalysis of nonlinear wave propagation in fluids (Hamilton and Blackstock, 1998), that is the basis for the existing theoriesof the ARF (Rudenko and Soluyan, 1977; Rudenko et al., 1996), stating that the two dissipation mechanisms are of the ofthe same order. To effectively build on the ARF solution for non-modulated ultrasound beams (Rudenko and Soluyan,1977; Rudenko et al., 1996), it is hereon assumed that

~k :¼ke rOð1Þ, ð16Þ

which allows one to formally retain the terms affiliated with heat conduction (and thus induced energy dissipation) in theasymptotic analysis of the thermomechanical system. Of course, once the final expression for the ARF is obtained, (16)

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–18131796

does not preclude substituting a realistic value for k. With (13)–(16) in place, the assumptions on the order of problemparameters can be completed by postulating the initial hydrostatic stress, ro ¼�poI in (11) to be commensurate with theatmospheric pressure whereby ~po :¼ po=erOð1Þ, and taking the specific heat capacity cp and the volumetric coefficient ofthermal expansion aT to be Oð1Þ (e.g. cp � 0:7 and aT � 0:1 at 37 1C for water).

3.2. Eulerian description

To aid the asymptotic treatment of the ARF generated by a modulated ultrasound wave when e51, consider the small-amplitude perturbations of the reference (thermodynamic equilibrium) state in terms of the displacement vector u,particle velocity v, heat flux q, mass density r, Cauchy stress tensor r, absolute temperature T, and specific entropy S

written as

u¼ e ~u, v¼ e ~v , q¼ e2 ~q, r¼ 1þe ~r, r¼ roþe ~r, T ¼ 1þe ~T , S¼ Soþe2 ~S, ð17Þ

where constants ro and So signify respectively the equilibrium (initial) values of the stress tensor and specific entropy. Byvirtue of (1), (2) and (7), the scaled system kinematics takes the form

~u ¼x�xo

e , ~v ¼ ðI�erT ~uÞ�1�@ ~u

@t: ð18Þ

On the basis of (5), (6), (17) and (18), balance laws (3) can be rewritten as

@ ~r@tþr � ðð1þe ~rÞ ~vÞ ¼ 0,

ð1þe ~rÞ @~v

@tþe ~v � r ~v

� �¼r � ~r,

ð1þe ~rÞð1þe ~T Þ @ ~S

@tþe ~v � r ~S

!¼�r � ~qþer ~v : ~J : r ~v : ð19Þ

Order of the perturbation quantities. From the definition of the Mach number in (12), it is clear that ~v ¼Oð1Þ. Comparingthe leading terms in (18) and (19), one consequently finds that ~u ¼Oð1Þ, ~r ¼Oð1Þ and ~r ¼Oð1Þ. In Section 3.4, it will befurther shown that likewise ~T ¼Oð1Þ. Under the latter restriction and hypothesis (16), it will also be demonstrated that~qrOð1Þ and consequently ~SrOð1Þ by way of (19c). In this setting, one may safely proceed with the asymptotic analysis byconsidering all (scaled) perturbation quantities in (17) to be on the order of unity.

3.3. Lagrangian description

On the basis of (4)–(6) and (17), the counterparts of (18) and (19) in the Lagrangian frame of reference can be written as

~u ¼x�xo

e, ~v ¼

@ ~u

@t, ð20Þ

and

1þe ~r ¼ 1

detðF Þ, F ¼ IþerT ~u ,

@2 ~u

@t2¼r � ~P , ~P ¼ detðF ÞF

�1� ~r ,

ð1þe ~T Þ@~S

@t¼�r � ðdetðF ÞF

�1� ~q Þþe detðF ÞðF

�T� r ~v Þ : ~J : ðF

�T� r ~v Þ, ð21Þ

By employing the reasoning as in Section 3.2, it can be similarly shown that all perturbation quantities and thus theleading terms in (21) are, at most, on the order of unity — a feature that is critical in exposing the second-order behavior ofthe system (underpinning the generation of the ARF) when e51.

3.4. Approximation of the constitutive behavior

Eulerian frame of reference. For a meaningful treatment of (19), it is critical to specify the response functions in (5),characterizing the isotropic elastic material with heat conduction and viscosity. Owing to the fact that the internal energydoes not appear explicitly in the rescaled field equations, the focus is hereon made on (5b)–(5d). In the context of thepresent problem, such specification can be effected without the loss of generality by recalling (8)–(10) and expanding

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–1813 1797

(5b)–(5d), up to a suitable order, with respect to e. To this end, let

~E :¼E

e ¼1

2ðFT� r ~uþrT ~u � FþeFT

� r ~u � rT ~u � FÞ ¼ Oð1Þ, Inð~EÞ ¼

InðEÞ

en¼Oð1Þ, ð22Þ

denote the scaled strain tensor and its invariants ðn¼ 1;2,3Þ, where F ¼ ðI�erT ~uÞ�1¼ Oð1Þ. In what follows, an implicit

assumption is made that the response functions in (5) are well behaved in the sense that the partial derivatives of eachconstitutive relationship, computed at the initial (equilibrium) state, are continuous and such that

@mþnF

@Xm@Yn

� �o

��������rO max

X

@F

@X

� �o

� �, F¼U,T ,r,q, X ,Y ¼ F ,S,rv,rT, mþnZ2, ð23Þ

as applicable. With reference to (6) this in particular implies that ð@U=@SÞF and ð@U=@FÞS, and thus all derivatives of U withrespect to its arguments, are at most Oð1Þ at the equilibrium state. Physically, hypothesis (23) ensures that the higher-order derivatives in (23), multiplying the nonlinear terms in the asymptotic expansion, are bounded in order by the leadingterm which guarantees that the response of the system is predominantly linear under small perturbations about theequilibrium state.

Since the ARF is known to stem from the quadratic (and bi-linear) terms featuring the perturbation quantities (Rudenkoand Soluyan, 1977), one may first invoke (8), (9a), (10), (17) and (22) and expand the scaled perturbation component of(5b) in e as

~T ¼e2 @2U

@S@E

!o

: ~Eþe @2U

@S2

!o

~Sþe ~E :@3U

@S@E2

!o

: ~E

¼e2

�bT

cpr � ~u�e bT

2cp

� �r ~u : rT ~uþe @3U

@S@I21

!o

þ@2U

@S@I2

!o

" #ðr � ~uÞ2�e bT

cpþ

@2U

@S@I2

!o

" #r

S ~u : rS ~uþe @2U

@S2

!o

~S, ð24Þ

where rS ~u :¼ 12ðr ~uþr

T ~uÞ, and ‘‘¼en

’’ denotes equality with an OðenÞ residual. Clearly, (24) verifies the claim that ~T ¼Oð1Þmade in Section 3.2.

Similarly, (9b) can be scaled by the Mach number and expanded in e using (22) as

~R :¼R�ro

e ¼e2

ð1�2e ~GÞr � ~uIþ2e ~GrS ~uþe 1

2ðBþ3CþD�2Þðr � ~uÞ2�

1

2ðCþDÞrS ~u : rS ~u

�

þr ~u : rT ~uþ1

2r ~u : r ~u�

bT

cp

~S

�I�e½ðCþD�2Þr � ~uI�Drs ~u� � rs ~u, ð25Þ

B :¼@3U

@I31

!o

, C :¼@2U

@I1@I2

!o

, D :¼@U

@I3

!o

, ð26Þ

see Appendix A.1. Note that the expansions of hyperelastic constitutive relationships similar to (25) (albeit without theentropy contribution) can be found elsewhere in the literature (e.g. Hamilton and Blackstock, 1998; Ostorvsky et al., 2007),featuring the so-called Landau’s third-order moduli (Landau and Lifshitz, 1986) that can be expressed as linearcombinations of the featured nonlinearity parameters B,C and D. By way of (14), (17) and (25), constitutive relationship(5c) reads

~r ¼ ~RðF ,SÞþe~J ðF ,SÞ : r ~v , ð27Þ

which demonstrates that ~r can be approximated with accuracy Oðe2Þ by retaining only the leading term in the expansionof ~J . To this end, one may invoke the premise of material isotropy whereby the leading-order expansion of the viscosityterm in (27) becomes

~J ðF ,SÞ : r ~v ¼e ~mB�

2

3~m

� �ðr � ~vÞIþ2 ~mrS ~v , ~mB :¼

mB

e, ~m :¼

me

, ð28Þ

where constants mBZ0 and mZ0 denote respectively the viscosities of the solid in bulk and shear (Coleman and Noll,1963). In support of (14) and (28), it is noted that the studies of ultrasound wave dispersion and attenuation in soft tissues(Ahuja, 1979) indicate that ~mB ¼Oð1Þ and ~mrOð1Þ as implied by the referenced formulas.

On recalling (7), the heat flux relationship (5d) can next be expanded with respect to its arguments by retaining only theterms that contain the odd powers of rT. Within the framework of material isotropy and (23), such expansion yields

~q ¼e2

�r ~T � ½ ~kI�erT ~u : ~A�, ð29Þ

where, consistent with the definition of the scaled coefficient of thermal conductivity as in (16), one has

~kI :¼ �1

e@q

@rT

� �o

, ~A :¼1

e@2q

@F@rT

� �o

, ð30Þ

under the constraint that 9 ~A9rOð ~kÞ. In support of the latter restriction, it is noted that assuming 9 ~A94Oð ~kÞ would violatenot only (23), but also the Clausius–Duhem inequality for the second-order tensor within the square brackets in (29) could

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–18131798

not be guaranteed to be positive-definite under an arbitrary choice of rT ~u. On the basis of (16), (24) and (29) it nowfollows that ~qrOð1Þ and, by way of (19c), that ~SrOð1Þ as asserted in Section 3.2.

In concluding the asymptotic treatment of (5), one may note that (19), (24)–(29) yield a closed system of equations,each expanded up to the second order, with an explicit set of constitutive equations in terms of ~T , ~r and ~q that isapplicable to a generic (isotropic) elastic solid with heat conduction and viscosity (Coleman and Noll, 1963).

Lagrangian frame of reference. To close the system of field equations (20) and 21, it is necessary to provide theLagrangian analogues of (24)–(29). To this end, one finds from (22) that

~E :¼E

e¼

1

2ðr ~uþrT ~uþe r ~u � rT ~u Þ ¼ Oð1Þ, ð31Þ

whose invariants obey scaling equivalent to that in (22). In this setting, temperature expansion (24) can be rewritten using(22) and (31) as

~T ¼e2

�bT

cpr � ~uþe bT

2cp

� �r ~u : rT ~uþe @3U

@S@I21

!o

þ@2U

@S@I2

!o

" #ðr � ~u Þ2

�e bT

cpþ

@2U

@S@I2

!o

" #r

S ~u : rS ~uþe @2U

@S2

!o

~S : ð32Þ

Similarly, the first Piola–Kirchhoff stress tensor featured in (21b) can be expanded using (27) as

~P ¼ ~Pþe detðF ÞF�1� ~J ðF ,SÞ : ðF

�T� r ~v Þ, ð33Þ

where ~P :¼ detðF ÞF�1� ~R is given by

~P ¼e2

ð1�2e ~GÞr � ~uIþ2e ~GrS ~uþe 1

2ðBþ3CþDÞðr � ~u Þ2�

1

2ðCþDÞrS ~u : rS ~u :

�

þ1

2r ~u : r ~u�

bT

cp

~S

�I�e½ðCþDÞr � ~uI�Drs ~u � � rs ~uþer � ~ur ~u , ð34Þ

as shown in Appendix A.1, and

detðF ÞF�1� ~J ðF ,SÞ : ðF

�T� r ~v Þ ¼

e ~mB�23~m

� ðr � ~v ÞIþ2 ~mrS ~v , ð35Þ

see also (28). Finally, expansion of the heat flux (29) can be rewritten in the Lagrangian frame of reference as

~q ¼e2

�r ~T � ½ ~kI�erT ~u : ðI4þ~AÞ�, ð36Þ

where I4 is the fourth-order identity tensor and ~A is given by (30).Limiting case of (27) for a Newtonian fluid. To facilitate the comparison of the results of this study with those of its

predecessor, obtained for fluids under the premise of plane-wave propagation (Dontsov and Guzina, 2011a), it isinstructive to consider the degenerate case of (27) for a Newtonian fluid. Assuming that bT,cp, ~mB and ~m characterizing theelastic solid can be assigned to the Newtonian fluid, the question remains as to the choice of constants ~G,B,C and D thatwould permit such degeneration. A direct comparison between (25) and its fluid companion ( ~Rfl ¼� ~pI, where ~p signifiesthermodynamic pressure, see e.g. Hamilton and Blackstock, 1998), demonstrates that

~G ¼ 0, B¼�ð2bþ3Þ, C ¼ 2, D¼ 0, ð37Þ

where b is the coefficient of acoustic nonlinearity (Hamilton and Blackstock, 1998) for the fluid.

4. Formulation and scaling of the focused beam problem

To understand the effect of low-frequency modulation on the generation of the ARF, the next step is to formulate theboundary value problem that is consistent with the pertinent applications of elevated-intensity ultrasound (e.g. Fatemiand Greenleaf, 1998) toward remote palpation of soft tissues. For simplicity, this is accomplished by making reference tothe Cartesian coordinate system ðx1,x2,zÞ whose z-axis coincides with that of the ultrasound transducer as in Fig. 1. Ondenoting the featured transverse coordinates jointly by xi, i¼ 1;2, the geometry of a focused ultrasound transducer withshape G can accordingly be synthesized via its focal distance d (i.e. the radius of curvature) and transverse dimensions ai asindicated in Fig. 1. In this setting, the transducer’s action can be formulated, in the Lagrangian frame of reference, viaDirichlet boundary condition

~u 9G ¼KðxoÞMðetÞsinðtÞ, xo 2 G, ð38Þ

where ~u is the scaled displacement vector, see (17); G denotes the surface of a transducer; prescribed function K signifiesthe so-called profile of the beam; and M¼Oð1Þ is the ultrasound modulation envelope such that M� 0 for tr0 anddMðetÞ=dðetÞ ¼Oð1Þ for t40, i.e. M is a smooth modulation signal with quiescent past and dominant (dimensionless)frequency om ¼ OðeÞ. Here it is assumed that the oscillating transducer is the sole source of excitation for the problem,whereby the ‘‘left’’ boundary condition (38) is complemented by the radiation condition at z¼1. For a comprehensive

wi

z

xi

d

ai

Γ

Fig. 1. Schematics of a transducer used to generate the focused ultrasound field (left), which in turn gives rise to the ARF within the focal region (right).

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–1813 1799

study of the featured thermomechanical problem, it is further assumed that the transducer’s boundary is thermallyinsulated, i.e. that ~q 9G ¼ 0, where ~q denotes the scaled heat flux.

4.1. Scaling affiliated with geometry of the beam

Since the dimensionless ultrasound frequency in this study equals unity, the geometry of a focused ultrasound beam inFig. 1 is governed by only three dimensionless parameters, namely ai (i¼ 1;2) and d, or alternatively, by fi :¼ d=ð2aiÞ and d,where fi is the so-called focal ratio computed with respect to ai. For an effective asymptotic treatment of the featuredboundary value problem, it is noted that the ultrasound transducers used to generate the ARF typically have the aperturei.e. diameter between 1 and 4.5 cm, focal distance 2–9 cm and carrier frequency 1–5 MHz (Fatemi and Greenleaf, 1998;Giannoula and Cobbold, 2009). In this setting, one finds that the focal ratios affiliated with ARF-generating transducers arenormally within the interval f¼ 1:525, while their (dimensionless) focal distances are in the range d¼ 30021500 (Fatemiand Greenleaf, 1998; Giannoula and Cobbold, 2009).

Focal region. In theory and applications of the acoustic radiation force (Sarvazyan et al., 1998), of critical interest is the‘‘cigar’’-shaped zone D of coalescence of ultrasound waves (indicated by the shaded area in Fig. 1), where the Mach numberreaches sufficiently high values so that the sought nonlinear effect is significant. Given possible disproportionality of thesize of this region relative to other characteristic length scales of the problem, however, it is essential to moderate itsdimensions by a suitable power of the Mach number in a way similar to transformation (17). Assuming axisymmetrictransducer geometry and denoting the affiliated intensity of the ultrasound beam as Iðr,zÞ where r¼ ðx2

1þx22Þ

1=2, one mayfirst define the half-width of the focal zone, w, as the radial distance such that Iðw=2,dÞ ¼ 1

2Ið0,dÞ where z¼d is the focaldepth. Similarly the half-length, ‘, of the focal zone is defined as ‘¼ z2�z140 (z1odoz2) such that Ið0,z1Þ ¼

Ið0,z2Þ ¼12Ið0,dÞ. With such definitions, the half-width and half-length of the focal zone can be estimated via linear

acoustic (Rayleigh integral) calculations as

w� 2pf, ‘� 14pf2: ð39Þ

On recalling the premise that e¼ Oð10�3Þ in the focal region, its half-width can be accordingly scaled as

~w i ¼ elwi ¼Oð1Þ, l¼logð2pfÞ�log e 4

1

4, i¼ 1;2, ð40Þ

for focal numbers fZ1. Here it is noted that the lower bound on l in (40) is critical for the development of a compactexpression for the ARF. In particular, it will be shown that the relative approximation error of the ARF behaves asOðeminf1;4l�1gÞ, whereby l¼ 1

4 is a threshold beyond which a simple expression for the ARF ceases to apply.Scaling of transverse coordinates. In the analysis of the focusing effects such as those of interest in this study, it is next

reasonable to assume that inside D, the partial derivatives of any thermomechanical quantity with respect to thetransverse coordinates xi scale with the width of the focal region. On denoting the compound state vector solving (20)and (21) as

Q ðxi,z,tÞ :¼ ð ~r, ~u, ~v , ~r, ~q, ~T , ~SÞðxi,z,tÞ, ð41Þ

this in particular implies that

@Q

@ðxi=wiÞ

¼OðQ Þ, x 2 D

rOðQ Þ, x =2 D

(, ð42Þ

which assumes that the germane partial derivatives diminish outside D due to widening of the beam. On the basis of(40)–(42), the scaled transverse coordinates (in Eulerian formulation) can be introduced as

wi :¼ elxi,@Q

@wi

rOðQ Þ: ð43Þ

Note that scaling (43) with l¼ 12 coincides with that underpinning the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation

(Hamilton and Blackstock, 1998), which is commonly used to describe focused ultrasound beams. As a rule, the latter

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–18131800

analyses postulate the so-called ‘‘narrow beam’’ geometry that can be affiliated with focal numbers f� 5. In contrast, thepresent framework caters for configurations with fZ1 that are more representative of the ARF transducers reported in theliterature. For completeness, one should also observe that conditions (43) imply that the featured thermomechanicalquantities vary slowly perpendicular to the beam axis (over distances of one ultrasound wavelength); in this context, suchwaves are sometimes referred to as the quasi-plane waves (Hamilton and Blackstock, 1998).

In light of (43), one finds that the right-hand side of the balance of linear momentum (19b) behaves, to the leadingorder, as r � ~r ¼rr � ~u ¼ ðOðelÞ,OðelÞ,Oð1ÞÞ. This in turn implies that ~v , and thus ~u, are vector fields whose components inthe ðx1,x2,zÞ coordinate frame behave respectively as ðOðelÞ,OðelÞ,Oð1ÞÞ. Accordingly, the ensuing asymptotic analysis can befacilitated by writing

~u ¼ ð ~u i, ~uzÞðxi,z,tÞ :¼ ðel ~u i, ~uzÞðwi,z,tÞ, ~u i, ~uz ¼Oð1Þ,

~v ¼ ð ~v i, ~vzÞðxi,z,tÞ :¼ ðel ~v i, ~vzÞðwi,z,tÞ, ~v i, ~vz ¼Oð1Þ, ð44Þ

where ~u i stands for the ordered pair ð ~u1, ~u2Þ, ~u i for ð ~u1, ~u2Þ, and so on. An important restriction on (43) and (44) is that theystrictly apply to the quantities associated with the ultrasound time scale and, in general, may not be applicable to themean motion induced by the ARF.

In preparation of the ensuing discussion, it is noted the counterparts of (41)–(44) in the Lagrangian frame of referencecan be obtained by superseding fxi,z,wi,Q , ~u i, ~uz, ~vi, ~vzg respectively by fxio,zo,wio,Q , ~u i, ~u z, ~v i, ~v zg, where Q :¼ ð ~r , ~u , ~v , ~r , ~q , ~T , ~S Þ.

4.2. ‘‘Slow’’ time variable

The key difficulty in exposing the time-averaged behavior of the state vector Q or Q for the modulated ultrasoundproblem resides in the presence of disparate time scales in the boundary condition (38). To tackle the problem, considerthe reparameterized Lagrangian state vector Q ðwio,zo,tÞ ¼ Q ðxio,zo,tÞ, which solves (21) and (32)–(36) subject to boundarycondition (38) and the radiation condition at infinity. Recalling that (38) is the sole source of excitation for the problem,consider next the phase-shifted solution Q fðwio,zo,tÞ which is affiliated with the perturbed boundary condition

Qu

fðwio,zo,tÞ9G ¼KðxoÞMðetÞsinðtþfÞ, xo 2 G, ð45Þ

where Qu¼ ~u is the ~u-component of the state vector Q , and f is the phase-shift parameter applied to the carrier

ultrasound signal. To help parse out the ‘‘ultrasound’’ time in the solution, one may introduce the ‘‘slow’’ time variablet :¼ et and auxiliary vector function Gðwio,zo,t,tÞ so that

Gðwio,zo,t,tþfÞ :¼ Q fðwio,zo,tÞ, ð46Þ

holds everywhere in space and time, see also Dontsov and Guzina (2011a) in the context of modulated plane waves. On thebasis of (45) and (46), one finds that

Guðwio,zo,t,tþfÞ9G ¼KðxoÞMðetÞsinðtþfÞ, xo 2 G, ð47Þ

where Gu¼ ~u . On the basis of (47) and earlier premise dMðtÞ=dt¼Oð1Þ, one finds that @G

u=@t¼Oð1Þ on the boundary.

Even though the governing field equations (21) are nonlinear, one can safely assume that the overall solution retains suchsmoothness throughout the domain, i.e. that @G=@t¼Oð1Þ everywhere due to the facts that the problem is mildly nonlinear(e51) and that the material is dissipative which precludes the existence of characteristic surfaces, i.e. shocks. As a result, itfollows from (47) that G is periodic in ‘‘fast’’ time t, i.e. that

Gðwio,zo,t,tþ2pÞ ¼ Gðwio,zo,t,tÞ: ð48Þ

Here one may note that reparameterization (46) amounts to the method of dual time scales that is used toward theasymptotic treatment of differential equations involving ‘‘fast’’ and ‘‘slow’’ processes, see e.g. Pearson (1981) in the contextof nonlinear wave problems.

4.3. ‘‘Slow’’ space variable

Under the quasi-plane wave assumption (43), the displacement response of the featured (tissue-like) solid due tofocused ultrasound beam is dominated by its axial component, ~u z, as indicated via (44). In this setting it can be shownfrom (20), (21a), and (34) and (35) that the balance of linear momentum (21b) (that is central to the definition of the ARF)can be written, to the leading order, as

@2 ~u z

@t2�@2 ~u z

@z2o

¼OðeWÞ, W¼minf1;2lg: ð49Þ

In light of (46), wave equation (49) now reads

@2Guz

@t2þ2e@

2Guz

@t@tþe2@

2Guz

@t2�@2G

uz

@z2o

¼OðeWÞ, ð50Þ

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–1813 1801

where Guz¼ ~u z is the ~u z-component of the state vector G. An effective way to cancel the Oð1Þ-terms in (50) and thus aid its

OðeÞ-asymptotic analysis is to introduce the ‘‘slow’’ axial coordinate zo :¼ eWzo and an assistant function

Rðwio,zo,zo,t,tÞ :¼ Gðwio,zo,t,tÞ ¼ Q ðwio,zo,tÞ, ð51Þ

that is periodic in ‘‘fast’’ time (see (48) and (51)) and subject to the one-way-wave-equation constraint, i.e.

Rðwio,zo,zo,t,tþ2pÞ ¼ Rðwio,zo,zo,t,tÞ,@R

@zo¼�

@R

@t: ð52Þ

With reference to (50), the new function is such that Ruz¼ ~u z satisfies the governing equation

2e@2R

uz

@t@t �2eW @2R

uz

@zo@zoþe2@

2Ruz

@t2�e2W@

2Ruz

@z2o

¼OðeWÞ: ð53Þ

From (53) one may note that the introduction of the ‘‘slow’’ space variable via (51) and (52), commonly used in derivingthe Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation of nonlinear acoustics (Hamilton and Blackstock, 1998) (under thepremises of no modulation and ‘‘narrow’’ beam geometry for which l¼ 1

2) W¼ 1), helps to balance the leading terms as iteliminates the Oð1Þ-entries from the balance of linear momentum (50).

For future reference, it is recalled that the ‘‘slow’’ spatiotemporal coordinates in (51) are given by

wio ¼ elxio, zo ¼ eWzo, t¼ et, l¼logð2pfÞ� log e , W¼minf1;2lg, ð54Þ

where fZ1 is the focal number of the ultrasound beam. A comparison between the argument lists of R and Q in (51) alsoreveals that R splits both axial and temporal coordinate into their ‘‘slow’’ i.e. macro (zo,t) and ‘‘fast’’ i.e. micro (zo,t)components. This is in contrast to the treatment of the KZK equation (Hamilton and Blackstock, 1998) that entails neithersplitting nor scaling of the temporal coordinate. Here it is also useful to observe that the scaling of ‘‘slow’’ spatialcoordinates (wio,zo) given by (54) is transversely isotropic. This particular feature of the scaled solution is driven by:(i) axially-symmetric estimate (40) of the width of the focal zone, and (ii) ‘‘cigar’’ shape of the focal region, see Fig. 1.

Eulerian formulation. As indicated earlier, the expression for the ARF is strongly dependent on the choice of the referenceframe. To expose the differences between the material and spatial description of the problem, the Lagrangian state vectorR can alternatively be recast in terms of its Eulerian companion R as

Rðwio,zo,zo,t,tÞ ¼Rðwi,z,z,t,tÞ, wi ¼ elxi, z¼ eWz, t¼ et, ð55Þ

which relates to the Eulerian state vector as

Rðwi,z,z,t,tÞ ¼Q ðwi,z,tÞ: ð56Þ

On the basis of (20) and (55), one can show that R preserves the periodicity and reciprocity properties of R , namely

Rðwi,z,z,t,tþ2pÞ ¼ Rðwi,z,z,t,tÞ,@R

@z¼�

@R

@t: ð57Þ

Using (55) and Taylor expansion of R around ðwio,zo,zo,t,tÞ, it is also noted for future reference that

Rðwio,zo,zo,t,tÞ ¼e

Rðwio,zo,zo,t,tÞ: ð58Þ

4.4. ‘‘Fast’’ time averages

With the above definitions, the Eulerian ‘‘fast’’ time average of the state vector (55) can be introduced as

/RSEðwi,z,tÞ :¼ 1

2p

Z tþp

t�pRðwi,z,z,t,t0Þ dt0: ð59Þ

To justify the reduced argument list on the left-hand side of (59), it is noted using (57) that

@/RSE

@z¼

1

2p

Z tþp

t�p

@R

@zðwi,z,z,t,t0Þ dt0 ¼ �

1

2p

Z tþp

t�p

@R

@t0ðwi,z,z,t,t0Þ dt0 ¼ 0,

@/RSE

@t¼

1

2p ðRðwi,z,z,t,tþpÞ�Rðwi,z,z,t,t�pÞÞ ¼ 0:

With reference to the instantaneous relationship (56), the Eulerian ‘‘fast’’ time average of the ‘‘mother’’ state vectorQ ðwi,z,tÞ can be computed respectively as /QSE ¼/RSE. Accordingly, the Eulerian time averages of @Q=@xi, @Q=@z and@Q=@t can be shown via (56) and (57) to read

@Q

@xi

�E

¼ el @Q

@wi

�E

¼ el @R

@wi

�E

¼ el@/QSE

@wi

,

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–18131802

@Q

@z

�E

¼ �@R

@tþeW@R

@z

�E

¼ eW@/RSE

@z¼ eW@/QSE

@z,

@Q

@t

�E

¼@R

@tþe@R

@t

�E

¼ e@/RSE

@t¼ e@/QSE

@t: ð60Þ

The Lagrangian ‘‘fast’’ time average, hereon denoted by / �SL, is introduced by the same logic as / �SE in (59). Forcompleteness, it is noted that /@Q =@ð�ÞSL where ‘‘�’’¼ xio,zo,t satisfy the relationships that are equivalent to (60). In whatfollows, it is shown that the use of the ‘‘fast’’ temporal averages is critical for obtaining a meaningful expression for the ARFdue to modulated ultrasound field.

Remark. From (59) and its equivalent in terms of / �SL, one may note that the Eulerian and Lagrangian ‘‘fast’’ timeaverages of any thermomechanical quantity can be affiliated with respective transformations

/gð�,z,tÞSE ) /gSEð�,z,tÞ, z¼ eWz, t¼ et,

/gð�,zo,tÞSL ) /gSLð�,zo,tÞ, zo ¼ eWzo, t¼ et,

where z, zo and t denote the virgin axial and temporal coordinates. Consistent with this observation, all ensuingoccurrences of z, zo and t are understood in the sense of virgin time and space unless stated otherwise.

5. Acoustic radiation force

With reference to Fig. 1, it is noted that the geometry of the domain (except that affiliated with transducer’s boundary G)does not enter the problem of ARF generation, and will hereon be considered arbitrary. The only restriction on the domain isthat it should be sufficiently large so that the wave reflections from its boundaries are sufficiently weak, i.e. that their Machnumber is significantly smaller than e within the focal region.

5.1. Eulerian formulation

To facilitate the computation of the ARF in the Eulerian frame of reference, one may recall (25) and (28) and decomposethe Cauchy stress tensor (27) as ~r ¼ ~rlin

þ ~rnon, where

~rlin¼ ð1�2e ~GÞðr � ~uÞIþ2e ~GrS ~u�e bT

cp

~SIþe ~mB�2

3~m

� �r �

@ ~u

@t

� �Iþ2e ~mrS @ ~u

@t, ð61Þ

and ~rnon denote respectively its linear and nonlinear components. With reference to (18) and (61), one may now apply the‘‘fast’’ time average (59) to the rearranged balance of linear momentum (19b) as

@2 ~u

@t2�r � ~rlin

�E

¼ r � ~rnon�e ~v � r ~v�e ~r @ ~v

@tþe ~v � r ~v

� ��e @

@trT ~u �

@ ~u

@tþe ~g

� � �E

, ð62Þ

whose right-hand side signifies the sought ARF. Here ~g ¼ Oð1Þ is the residual in the expansion ~v ¼ ðIþerT ~uÞ � ð@ ~u=@tÞþe2 ~g ,see also (18); like other variables, this residual is periodic in ‘‘fast’’ time so that /@ ~g=@tSE ¼ OðeÞ. As can be seen from (62),the acoustic radiation force stems from both constitutive and geometric nonlinearities of the problem — a feature that willbe examined in greater detail shortly.

With reference to (59), let / ~uSEðwi,z,tÞ denote the ‘‘fast’’ time average of the displacement vector, and let

L/ ~uSE ¼ ðLi,LzÞ/ ~uSE :¼ e2 @2/ ~uSE

@t2�ð1�e ~GÞrr �/ ~uSE�e ~Gr

2/ ~uSE�e2 ~mBþ1

3~m

� �rr �

@/ ~uSE

@t�e2 ~mr2 @/ ~uSE

@t, ð63Þ

be an extension of the (linear) Navier operator for viscoelastic solids, where

r/ �SE ¼@

@xi

,@

@z

� �/ �SE ¼ el @

@wi

,eW @@z

� �/ �SE: ð64Þ

With the aid of (60) and (63), (62) can be rewritten in component form as

Li/ ~uSEþe1þl bT

cp

@/ ~SSE

@wi

¼ ~fE

i , i¼ 1;2,

Lz/ ~uSEþe1þW bT

cp

@/ ~SSE

@z¼ ~f

E

z , ð65Þ

where ~fE

i and ~fE

z and are the respective Cartesian components of the ARF. On recalling the definition of the scaled (Oð1Þ)instantaneous quantities ~u i, ~uz, ~v i and ~vz in (44), one finds from (19b) that

~fE

i ¼e1þ lþ W

el@/ ~snon

ji SE

@wj

þeW@/~snon

zi SE

@z�e1þl ~r@

~vi

@t

�E

�e1þl ~vz@ ~v i

@z

�E

,

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–1813 1803

~fE

z ¼e2þ W

el@/ ~snon

jz SE

@wj

þeW@/~snon

zz SE

@z�e ~r @ ~vz

@tþe ~vz

@ ~vz

@z

� � �E

�e2 @

@t@ ~uz

@t

@ ~uz

@z

�E

�e ~vz@ ~vz

@z

�E

�e1þ2l ~v j

@ ~vz

@wj

* +E

, ð66Þ

where i,j¼ 1;2; ~snonji , ~snon

jz and ~snonzz are the Cartesian components of ~rnon, and Einstein summation convention is assumed

over repeated index j. By virtue of (25) and (27), one finds that

/ ~snonji SE ¼

e1þ W

eBþ2Cþ1

2

@ ~uz

@z

� �2* +

E

dji,

/ ~snoniz SE ¼/ ~snon

zi SE ¼e1þ lþ W

�e1þl C�2

2

@ ~uz

@z

@ ~uz

@wi

þ@ ~u i

@z

� � �E

,

/ ~snonzz SE ¼

e1þ W

eBþ5

2

@ ~uz

@z

� �2* +

E

,

where dji denotes the Kronecker delta. On substituting this result into (66) and noting via (18) and (19a) that

~r @ ~vz

@tþe ~vz

@ ~vz

@z

� � �E

¼e1þ W

e@/~r ~vzSE

@t þ1

2eW@/

~v2zSE

@zþe2l ~vz

@ ~v j

@wj

* +E

,

~vz@ ~v i

@z

�E

¼eW�

@ ~vz

@z~v i

�E

¼eW @ ~r

@t~v i

�E

¼eW� ~r@

~v i

@t

�E

,

~r ¼eW

�@ ~uz

@z,

the components of the ARF can be rewritten as

~fE

i ¼e1þ lþ W

e1þl Bþ2Cþ1

2

@/ ~r2SE

@wi

, i¼ 1;2,

~fE

z ¼e1þ 2W

�e1þ2l C�2

2

@

@wj

@ ~uz

@z

@ ~uz

@wj

þ@ ~u j

@z

!* +E

þe1þW Bþ5

2

@/ ~r2SE

@z�e1þW@/ ~v

2zSE

@z�e1þ2l@/ ~vz ~v jSE

@wj

: ð67Þ

For a further reduction of ~fE

z , one may find from (18), (19a), (56) and (57b) that

~vz ¼e @ ~uz

@t¼eW�@ ~uz

@z¼eW ~r, ð68Þ

which, in conjunction with (19b) and (61), yields

@

@wj

@ ~uz

@z

@ ~uz

@wj

* +E

¼eW�@

@wj

~uz@2 ~uz

@z@wj

* +E

¼eW�@

@wj

~uz@ ~v j

@t

�E

¼eW @/ ~vz ~v jSE

@wj

,

@

@wj

@ ~uz

@z

@ ~u j

@z

�E

¼eW @

@wj

~r@~u j

@t

�E

¼eW @/ ~vz ~v jSE

@wj

: ð69Þ

By virtue of (68), and (69), (67) simplifies to

~fE

i ¼e1þ lþ W

e1þl Bþ2Cþ1

2

@~IE

@wi

, i¼ 1;2,

~fE

z ¼e1þ 2W

�e1þ2lðC�1Þ@/ ~vz ~v jSE

@wj

þe1þW Bþ3

2

@~IE

@z, ð70Þ

where

~IE :¼ / ~r2SE ¼e / ~p2SE, ~p ¼�1

3 tr ~r: ð71Þ

Note that the introduction of intensity-like quantity ~IE is motivated by the fact that the existing ARF studies focusprimarily on the z-component of the radiation force, that is commonly expressed in terms of the intensity of the ultrasound

field in the z-direction (Hamilton and Blackstock, 1998; Sarvazyan et al., 1998). Given transducer geometry and output, thelatter quantity is computable a priori, with an OðeWÞ error, as the mean squared acoustic pressure.

Entropy contribution. To close the averaged system of equations (65), one must bring into consideration the balance ofenergy (19c) due to the presence of mean specific entropy / ~SSE in the system. Accordingly the ‘‘fast’’ time average of (19c)

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–18131804

can be computed via (28) and (29) as

e@/~SSE

@t¼e1þ W

�e ~vz@ ~S

@z

* +E

�e ~k ð ~Tþ ~rÞ@2 ~T

@z2

* +E

þe ~mBþ4

3~m

� �@ ~vz

@z

� �2* +

E

: ð72Þ

To simplify the right-hand side of the (72), it is noted via (19c) and (57b) that

@ ~S

@z¼eW�@ ~S

@t¼eW� ~k@

2 ~T

@z2: ð73Þ

Using the latter result together with (19a), (24), (57b) and (68), the averaged energy equation (72) reduces to

@/ ~SSE

@t ¼eW ~d~I

n

E, ~In

E :¼@ ~r@t

� �2* +

E

, ~d :¼ ~mBþ4

3~mþ b2

T

c2p

~k, ð74Þ

where d is the so-called diffusivity of sound (see Hamilton and Blackstock, 1998 in the context of Newtonian fluids).Boundary conditions. With the averaged balance laws (65) and (74) written in terms of / ~uSE and / ~SSE now in place,

the next step is to compute the ‘‘fast’’ time average of the boundary condition (38) which simulates the action of anoscillating transducer. In terms of the kinematic quantities, one finds from (38), (58) and (59) that

/ ~uSL

���G¼ 0

/ ~uSL ¼e / ~uSE

9=; ) / ~uSE9G ¼

e0, ð75Þ

In general, (75) is complemented by an appropriate set of (mean) boundary conditions germane to the ‘‘macro’’ domain ofinterest, e.g. the radiation condition at infinity. For clarity, it is emphasized that these ‘‘macro’’ boundary conditions do notenter the computation of the ARF (i.e. the apparent mean body force) as long as the Mach number affiliated with suchreflections, arriving at the focal region, is significantly smaller than e.

With reference to the thermal boundary condition ( ~q 9G ¼ 0) postulated earlier, it is noted that the mean specificentropy solves the initial value problem given by (74) and the quiescent past condition. From this observation, it is clearthat the ‘‘fast’’ time average of the heat flux does not enter the formulation and is hereon disregarded. This peculiarfeature of the mean solution stems from the fact that the characteristic diffusion time (reckoned with reference to the‘‘macro’’ length scale z) is much larger than its modulation counterpart. In particular, one finds from (72) that~k/r2 ~TSE ¼ oð@/ ~SSE=@tÞ, even though ~kr2 ~T ¼Oð@ ~S=@tÞ in terms of the instantaneous behavior, see also Dontsov andGuzina (2011a) in the context of plane waves. In this setting, it is also noted that the system given by (65) and (74) is semi-

coupled in that / ~SSE can be solved independently from (74), and then substituted into (65) as an additional source term.

5.2. Lagrangian formulation

To study the generation of the ARF in the Lagrangian frame of reference, one needs to apply the Lagrangian counterpartof (59) to the balance of linear momentum (21b). Following the same procedure as in Section 5.1, the ‘‘fast’’ time average of(21b) can be computed as

Li/ ~uSLþe1þl bT

cp

@/ ~SSL

@wio

¼ ~fL

i , i¼ 1;2,

Lz/ ~uSLþe1þW bT

cp

@/ ~SSL

@zo¼ ~f

L

z , ð76Þ

where Li/ �SL and Lz/ �SL are given by the Lagrangian version of (63), while ~fL

i and ~fL

z are the respective Lagrangiancomponents of the ARF given by

~fL

i ¼e1þ lþ W

e1þl Bþ2Cþ1

2

@~IL

@wio

, i¼ 1;2,

~fL

z ¼e1þ 2W

�e1þ2lðC�1Þ@/ ~v z ~v jSL

@wjo

þe1þW Bþ3

2

@~IL

@zo, ð77Þ

where ~IL :¼ / ~r2SL approximates the Lagrangian intensity.

Entropy contribution and boundary conditions. To close (76), the Lagrangian ‘‘fast’’ time average of the balance of energy(21c) can be computed with the aid of (21a), (21b), (32)–(36) and (52) as

@/ ~SSL

@t¼eW ~d~I

n

L , ~In

L :¼@ ~r@t

!2* +L

: ð78Þ

Note that the averaged balance equations (76) and 78 govern the mean response of a solid in the Lagrangian frame ofreference, expressed in terms of the displacement vector and specific entropy. This system is complemented by the (mean)

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–1813 1805

kinematic boundary condition on the transducer’s face, which can be found on the basis of (38) and the Lagrangiancounterpart of (59) to read

/ ~uSL9G ¼ 0: ð79Þ

For completeness it is noted, making reference to the discussion concluding Section 5.1, that the thermal boundarycondition does not enter the problem, and that (79) is in general accompanied by a suitable set of ‘‘macro’’ boundaryconditions dependent on the geometry of the domain.

5.3. Reduced expression of the ARF in terms of intensity derivatives

As examined in Section 5.1, the spatial description of the ARF depends not only on the intensity of the sound beam ~IE

but also on the quantity @=@wj/ ~vz ~v jSE which requires additional knowledge of the velocity field, produced by themodulated ultrasound beam. A similar comment applies to the results in Section 5.2 where the radiation body force isexpressed in terms of ~IL and @=@wjo/ ~v z ~v jSL. Fortunately, in most situations @=@wj/ ~vz ~vjSE (resp. @=@wjo/ ~v z ~v jSLÞ can berelated to the derivatives of ~IE (resp. ~IL) as shown next, for simplicity, in the Lagrangian frame of reference. In particular, onmultiplying the z-component of (21b) by @ ~u z=@zo, one finds from (34) and (35) that

@ ~u z

@zo

@2 ~u z

@t2¼e1þ W @ ~u z

@zo

@2 ~u z

@z2o

þe2l @2 ~u i

@zo@wio

þeBþ3

2

@

@zo

@ ~u z

@zo

!2

�e bT

cp

@ ~S

@zoþe ~mBþ

4

3~m

� �@2 ~v z

@z2o

24

35: ð80Þ

As shown in Appendix A2, the ‘‘fast’’ time average of (80) can be reduced as

e1�2l@~IL

@t þeW�2l@

~IL

@zoþe1�2l ~d~I

n

Lþ@/ ~v z ~v iSL

@wio

¼e2W�2l

0, ð81Þ

which allows the @/ ~v z ~v iSL=@wio term, which enters the ARF expression, to be expressed in terms of intensity derivatives. From(77) and (81), one in particular finds that the ARF formula in the Lagrangian frame of reference admits vector representation

~fL¼

e1þ Wþ ðl,WÞ

eðC�1�bÞro~ILþe2ðC�1Þ ~d~I

n

Lþ@~IL

@t

!ez, ro ¼ el @

@wio

,eW @

@zo

� �, ð82Þ

where ~fL¼ ð~f

L

i , ~fL

zÞ; ez is the unit vector in the z-direction; b¼�ðBþ3Þ=2 is the coefficient of acoustic nonlinearity introduced in(37), and symbol ‘‘e1þWþðl,WÞ’’ indicates that ~f

L

i and ~fL

z are approximated with respective accuracies Oðe1þlþWÞ and Oðe1þ2WÞ. In asimilar fashion, the Eulerian expression for the radiation body force can be obtained as

~fE¼

e1þ Wþ ðl,WÞ

eðC�1�bÞr~IEþe2ðC�1Þ ~d~In

Eþ@~IE

@t

!ez, r¼ el @

@wi

,eW @@z

� �: ð83Þ

With reference to (82) and (83), it is noted that the restriction l414 introduced in (40) becomes important at this point.

In particular when lo14, the Oðe2Þ axial component of the ARF (which is proportional to C�1 and is the main focus

of biomedical applications) becomes smaller than its approximation accuracy Oðe1þ2WÞ ¼Oðe1þ2 minf1;2lgÞ. In this case (82)and (83) cease to be useful and their ‘‘mother’’ counterparts, given respectively by (87) and (80), must be used instead.

6. Comments on the result

Eulerian versus Lagrangian formulation. From the results in Section 5, it is apparent that there is no formal differencebetween the Lagrangian and Eulerian formulations of the boundary value problem governing the mean displacement andspecific entropy. Given the fact that the analysis of the ARF is essentially a second-order problem, this equivalence ishowever far from obvious, and could not have been asserted without a detailed analysis. Indeed, it is noted that the paritybetween the mean Lagrangian and Eulerian formulations is strongly dependent on the choice of independent variables. Inparticular, one finds from (18) and (71) that

/ ~v zSL ¼ e@/ ~u zSL

@t, / ~vzSE ¼

e1þ 2l

e@/~uzSE

@tþe ~vz

@ ~uz

@z

�E

¼e1þ W

e@/~uzSE

@tþe~IE,

which demonstrates that the Lagrangian and Eulerian variants of the boundary value problems written in terms of / ~vSand / ~SS (in lieu of / ~uS and / ~SS) are in fact different. For completeness, it is noted that similar differences (between theEulerian and Lagrangian averages of the same quantity) appear when dealing with the mean mass density or pressure innonlinear acoustics, which leads to a well studied distinction between the so-called Rayleigh and Langevin radiationpressures on an obstacle exposed to elevated-intensity ultrasound (Hamilton and Blackstock, 1998; Hasegawa et al., 2000).

Limiting case of plane-wave propagation. For verification purposes, it is important to consider the limiting case of planewaves, and compare it with the existing solution in Dontsov and Guzina (2011a). On the basis of the foregoingdevelopments, the sought plane-wave limit can be immediately computed by setting @ð�Þ=@wi ¼ 0 and W¼ 1. Under this

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–18131806

restriction, one finds by way of the Eulerian statement of (81) and (83) that (65) reduces to

Lz/ ~uSEþe2 bT

cp

@/ ~SSE

@z¼e3

e2b ~d~In

Eþ@~IE

@t

!,

where the acoustic radiation force, given by the term on the right-hand side, coincides with the result in Dontsov andGuzina (2011a), obtained assuming propagation of modulated plane waves through a Newtonian fluid.

Computation of intensity-like quantities. Taking (83) as an example, it is clear that a prerequisite for computing the ARF is theknowledge of intensity-like quantities ~IEðx,tÞ ¼/ ~r2SE and ~I

n

Eðx,tÞ ¼/ð@ ~r=@tÞ2SE within the focal region. This in turn requiressolving the original nonlinear system (19), at the ultrasound time scale, for a given set of transducer characteristics and tissueproperties. Due to the smallness of the germane shear modulus, however, ~IE and ~I

n

E are commonly computed from anapproximate ultrasound solution which solves the linearized version of (19) for a viscous fluid (Bercoff et al., 2004; Palmeri et al.,2005). In dealing with the modulated beam problem, it can be in particular shown (Dontsov and Guzina, 2011b) that

~IEðx,tÞ ¼ ~IE,nmðxÞM2ðet�ezÞ, ~I

n

Eðx,tÞ ¼ ~In

E,nmðxÞM2ðet�ezÞ, ð84Þ

under the premise of linear approximation, where ~IE,nm and ~In

E,nm are time-invariant intensity fields corresponding to the non-modulated beam problem. To incorporate the non-linear effects into the computation of ~IE and ~I

n

E, the underpinning ultrasoundsolution (in this case ~r) can alternatively be solved from the so-called KZK equation for viscous fluids (Rudenko et al., 1996;Sarvazyan et al., 1998; Hamilton and Blackstock, 1998), which approximates (19) within the focal region under the premise ofquasi-plane waves, given by (44) with l¼ 1

2. An extension of the KZK equation to modulated ultrasound fields, that is beyond thescope of this study, is examined in Dontsov and Guzina (2011b). In either (linear or nonlinear) approach, the approximations of~IE and ~I

n

E are accordingly computed assuming acoustic properties (sound speed, attenuation, and the coefficient of acousticnonlinearity as applicable) that are representative of a given tissue type.

Relationship between ~IE and ~In

E. In situations where ~IE and ~In

E are computed on the basis of the linear acoustic solution, it

follows from (38) and linearity of the problem that @ ~r=@tðx,tÞ ¼e ~rðx,tþp=2Þ whereby ~IE ¼

e ~In

E. If the two quantities are

computed from a non-linear solution (e.g. that solving the KZK equation), on the other hand, one may expand the resulting

mass density fluctuations in ‘‘fast’’ time Fourier series as ~r ¼P1

k ¼ 1~rðkÞ, where ~rðkÞ denotes the kth solution component

with dominant dimensionless frequency equaling k, so that @ ~rðkÞ=@tðx,tÞ ¼e

k ~rðkÞðx,tþp=2Þ. On recalling the definition of the‘‘fast’’ time average (59), one finds that

IE ¼X1k ¼ 1

~rðkÞ !2* +

E

¼X1k ¼ 1

~IðkÞ

E , ~IðkÞ

E ¼/ð ~rðkÞÞ2SE,

~d~In

E ¼e ~d

X1k ¼ 1

k ~rðkÞ !2* +

E

¼X1k ¼ 1

~dk2~IðkÞ

E : ð85Þ

At this point one may note that a linear plane wave, propagating at frequency k through an isotropic Kelvin-Voigt material

whose constitutive behavior is given by (61), will experience exponential decay e�e ~aðkÞz in the direction of propagation,

where ~aðkÞ ¼ 12~dk2, see Ricker (1977). Accordingly, one finds from (85) that

~d~In

E ¼e X1

k ¼ 1

2 ~aðkÞ~I ðkÞE :¼ 2 ~a~IE, ~a ¼ 1~IE

X1k ¼ 1

~aðkÞ~I ðkÞE , ð86Þ

where ~a denotes an effective attenuation coefficient for the nonlinear ultrasound problem at hand, that accounts fordisparate attenuation rates affiliated with higher-order harmonics and scales as ~a ¼ a=e¼Oð1Þ, see (13).

Compact formulation of the mean equation of motion. As examined in Section 6, there is formally no difference betweenthe averaged governing equations written in the Eulerian and Lagrangian frames of reference, as long as the formulation iswritten in terms of / ~uS and / ~SS. In this case it is no longer necessary to track both formulations, and one may proceedwith the Eulerian description only. On dispensing with all scaling and residuals (but keeping everything dimensionless),the balance of linear momentum (65) can be rewritten by virtue of (17), (71), (74), (83) and (86) as

L/uSE ¼ f Epotþf E

dir, ð87Þ

where the potential and directional body force fields are given respectively by

f Epot ¼r ðC�1�bÞIE�

bT

cp

Z t

02aIE dt0

� �, f E

dir ¼ ðC�1Þ 2aIEþ@IE

@t

� �ez: ð88Þ

Here r¼ ð@=@xi,@=@zÞ, and the scaling-free Navier operator is given by

L/uSE ¼@2/uSE

@t2�ð1�GÞrr �/uSE�Gr2/uSE� mBþ

1

3m

� �rr �

@/uSE

@t�mr2 @/uSE

@t: ð89Þ

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–1813 1807

Note that the entropy term, featured in (65), is treated as a source term and accordingly moved to the right-hand side of(87) which is composed of two distinct components. The first, i.e. potential component f E

pot, causes only irrotational motion(in the absence of domain boundaries), while the second term f E

dir, representing the body force in the direction of theultrasound beam, also causes shear motion. Since the local interrogation of the tissue’s shear modulus is of primaryinterest in many ARF applications, the potential part of the mean body force is commonly disregarded, and the term ‘‘ARF’’is reserved for its directional component (Sarvazyan et al., 1998; Palmeri et al., 2005; Sarvazyan et al., 2010).

Tissue homogeneity. The basic premise underpinning the proposed formula for the ARF is that the tissue-like solid ishomogeneous as examined in Section 1. In the context of (88), however, it is important to recognize that the featuredrestriction of tissue homogeneity applies only to the focal region, see Fig. 1. In particular if the tissue is heterogeneousoutside of the focal region, formulas (88) still apply, except for the fact that the ‘‘global’’ tissue heterogeneity must beaccounted for when computing the intensity field IE. In ultrasound applications, the latter difficulty is often circumventedby making use of the linear acoustic approximation of IE and the fact that the small-amplitude sound speed varies littleacross a wide range of soft tissues (Hamilton and Blackstock, 1998).

Generation of shear waves. On the basis of (87), one may decompose the mean displacement field as

/uSE ¼/uSpotE þ/uSdir

E , ð90Þ

where /uSpotE and /uSdir

E are generated respectively by f Epot and f E

dir. Clearly, /uSpotE is an irrotational field and thus

associated with long-wavelength compressional waves. To understand the nature of /uSdirE , on the other hand, one finds

by way of (66) and (88) that 9r � f Edir9=Jr � f E

dirJ¼OðeW�lÞ51 when 14olo1. On employing the Helmholtz potential

representation of the linear visco-elastodynamic field /uSdirE , it can be shown that the latter motion is predominantly

solenoidal, i.e. associated with short-wavelength shear wave motion, whereby

@2/uSdirE

@t2�Gr2/uSdir

E �mr2 @/uSdir

E

@t¼ f E

dir, ð91Þ

again omitting the accuracy of approximation for simplicity. The z-component of (91), that is commonly used to describethe propagation of shear waves generated emanating from the focal region (Sarvazyan et al., 1998), can accordingly bewritten as

@2/uzSdirE

@t2�Gr2

?/uzSdirE �mr

2?

@/uzSdirE

@t¼ f E

z,dir, f Ez,dir ¼ ðC�1Þ 2aIEþ

@IE

@t

� �, ð92Þ

in the case of a modulated ultrasound beam, where the Laplacian operator is approximated by its transverse componentr2? ¼ @

2=@xi@xi ¼r2þOðe2WÞ. In an experimental setting, /uzSE (i.e. the total axial motion) is typically monitored by a low-

intensity, imaging ultrasound transducer (or an array thereof) (Guzina et al., 2009; Bercoff et al., 2004; Sinkus et al., 2005b). Inmany situations, /uzSdir

E can be extracted from /uzSE via either suitable high-pass filtering in the radial wavenumber domain,or natural separation in the time domain for short pulses (Bercoff et al., 2004) which permits the use of (92). If, on the otherhand, the extraction of /uzSdir

E is not possible, the observations of /uzSE must be interpreted via the coupled system (87). Inthis case, however, the specification of the ARF entails three additional material parameters, namely the coefficient of acousticnonlinearity b given by (37), the coefficient of thermal expansion bT, and the specific heat capacity at constant pressure cp.

Comparison with the existing model. The existing theory of the ARF (Rudenko et al., 1996) dealing with focusedultrasound beams is derived assuming Newtonian fluid as the propagation medium, and non-modulated ultrasound field. Tostudy the problem of ARF generation and consequent shear wave motion in soft tissues due to modulated ultrasound, onthe other hand, the foregoing theory is modified in a somewhat heuristic fashion by: (i) transplanting the ARF, computedfor Newtonian fluids, to soft solids, and (ii) multiplying the ARF computed for non-modulated ultrasound field by thesquared modulation envelope, see Sarvazyan et al. (1998). Formally, these two modifications result in the mean balance oflinear momentum whose z-component can be written as

@2/wzSdirE

@t2�Gr2

?/wzSdirE �mr

2?

@/wzSdirE

@t¼ gE

z,dir, gEz,dir ¼ 2aIE,nm M2

ðetÞ, ð93Þ

where /wzSdirE is an approximation of the mean response due to modulated ultrasound beam, and IE,nm the intensity of the non-

modulated ultrasound field (Sarvazyan et al., 1998). To facilitate the comparison between (92) and (93), the emphasis is hereonmade on the linear approximation of the intensity field which permits, by way of (84), the respective body force terms to bewritten as

f Ez,dir ¼ ðC�1Þ2IE,nmðaM2

ðet�ezÞþeM0ðet�ezÞMðet�ezÞÞ,

gEz,dir ¼ 2aIE,nmM2

ðetÞ, ð94Þ

whereMð�Þ is the prescribed modulation envelope, and IE,nm ¼ IE,nmðxÞ. In this setting, the key differences between the proposedmodel and its predecessor can be summarized as follows.

�

With reference to (87) and (90), f Ez,dir and /uzSdir

E in (92) represent the respective directional components of theradiation body force and induced mean displacement in the z-direction. From the discussion in Sarvazyan et al. (1998),

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–18131808

it is unclear whether the quantities herein denoted by gEz,dir and /wzSdir

E in (93) in fact signify the total quantities ortheir directional components. For the sake of discussion, it is hereon assumed that the latter is the case.

� In the case of no modulation (M� 1), the expressions for f E

z,dir and gEz,dir are identical except for the factor ðC�1Þ, where

C is the hyperelastic nonlinearity coefficient given by (26). Given the facts that the formula for gEz,dir is derived assuming

Newtonian fluid as the propagation medium and that C-2 in the limit of a Newtonian fluid, see (37), one finds thatðC�1Þ (rather than unity) is the appropriate coefficient for the ARF in tissue-like solids described as an elastic materialwith heat conduction and viscosity (Coleman and Noll, 1963).

� Owing to the argument ðet�ezÞ in (94), the radiation body force f E

z,dir propagates within the focal zone with the speed ofsmall-amplitude compressional waves. In contrast, the previous approximation gE

z,dir assumes the radiation force totake the form of a standing wave by enforcing the separation of variables in terms of x and t. For completeness, it isnoted that the analogous finding has been reported in the context of elevated-intensity plane waves propagatingthrough a Newtonian fluid (Dontsov and Guzina, 2011a). In the applications of the ARF, the featured propagation effect(i.e. the difference betweenMðet�ezÞ andMðetÞ) becomes important at modulation frequencies on the order of 10 kHzwhere the change in the phase of f E

z,dir within the focal zone becomes meaningful, see Fatemi and Greenleaf (1998) forexamples of the use of such modulation frequencies.

� In comparison with gE

z,dir, the expression for the ARF given by f Ez,dir contains an additional, modulation-driven term which

is proportional to the time derivative of ultrasound intensity, see also (92). In general, this term becomes importantwhen the dimensionless modulation frequency is on the order of, or larger than, the dimensionless attenuationcoefficient a¼OðeÞ. In the context of plane waves, it was shown (Dontsov and Guzina, 2011a) that the modulation-driven contribution to the ARF can be used to explain the phenomenon of difference-frequency generation (Thuraset al., 1935) of the mean motion in lossless fluids where a¼ 0. In the literature, the phenomena of difference-frequencygeneration and ARF were long thought to be unrelated owing to the common notion that the ARF vanishes inhomogeneous lossless media (Ostorvsky et al., 2007), as indicated by the formula for gE

z,dir in (94).

For completeness, this section concludes by noting that the recent studies in Ostorvsky et al. (2007), Sarvazyan et al.(2010) provide a trial expression for the ARF in soft tissues assuming that: (i) the ultrasound field is not modulated,(ii) the displacement field takes the a priori form wðx,tÞ ¼WðxÞcosðt�zÞez where WðxÞ is a ‘‘slowly’’ varying scalarfunction, and (iii) there is no heat generation. Under such simplifying hypotheses, the authors obtain a counterpart of(93) in terms of the total displacement field /wzSE, featuring the radiation body force as gE

z ¼ gEz,potþgE

z,dir, wheregE

z,potpðC�1�bÞrðW2Þ � ez and gE

z,dirpðC�1Þ@ðW2Þ=@z represent respectively the potential and directional component of

the ARF. Despite the semblance between this result and (87)– (88), the explicit comparison between gEz (Sarvazyan et al.,

2010) and f Ez stemming from (88) is not possible owing to the fact that the unidirectional displacement field wðx,tÞ does

not satisfy the balance of linear momentum, nor it incorporates the specifics of the focused ultrasound beam geometryused to generate the ARF.

7. Analytical example

To illustrate the effect of signal modulation on the ARF, it is first necessary to specify the distribution of ultrasoundintensity affiliated with the modulated beam problem. For simplicity, consider the axisymmetric Gaussian beam source asin Rudenko et al. (1996), Sarvazyan et al. (1998) that is modulated according to (38). In this case, the linear solution forultrasound intensity can be computed from the non-modulated analytical result in Rudenko et al. (1996) and (84) as

IEðx,tÞ ¼ IE,nmðxÞM2ðet�ezÞ ¼ IoM2

ðet�ezÞh2ðzÞ

exp �2az�2r2

a2 h2ðzÞ

" #, hðzÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

z

d

� 2

þ2z

a2

� �2s

, ð95Þ

where r¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

1þx22

qdenotes the radial coordinate, d is the focal distance, and a and Io signify respectively the beam width

(signifying the effective transducer aperture, see Fig. 1) and peak ultrasound intensity at the transducer’s face. To study thevariation of the ARF with modulation frequency,M is taken asMðetÞ ¼HðtÞsinðomtÞ, where H denotes the Heaviside stepfunction and om ¼ OðeÞ. On the basis of (94) and (95), one finds that f E

z,dir and gEz,dir will in general contain multitude of

frequency components, including the static i.e. ‘‘DC’’ component (recall that sin2 y¼ 0:5ð1�cosð2yÞ). To cater forbiomedical applications and to facilitate the comparison with earlier works (Rudenko et al., 1996; Sarvazyan et al.,1998), the remainder of this section focuses on the dominant oscillatory component of the ARF and thus induceddisplacement, namely that at frequency 2om. For simplicity of presentation the Fourier transform of any time-dependentquantity g, evaluated at 2om, is denoted by Fg.

Material and ultrasound parameters. In what follows, the computations are effected assuming carrier ultrasoundfrequency O¼ 2p� 106 rad=sec (1 MHz), mass density Ro ¼ 1000 kg=m3, compressional wave speed co ¼ 1500 m=s,transducer aperture a¼42 (1 cm), focal depth d¼209 (5 cm), frequency of modulation om ¼ 10�3 (1 kHz), attenuationa¼ 1:4� 10�3 (0.5 dB/cm/MHz), shear viscosity m¼ 0:28� 10�3

ð0:1 Pa � sÞ, and shear modulus G¼ 2:2� 10�5 (50 kPa)that are similar to the respective values in Sarvazyan et al. (1998). To make a reasonable assumption on the nonlinearitycoefficient C, on the other hand, one may recall the experimental investigation in Catheline et al. (2003) which estimates

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–1813 1809

the Landau’s third-order elastic moduli in agar-based, tissue mimicking phantoms as

ALandau2 ½�108,�51� kPa, BLandau

2 ½�28,�9� GPa, CLandau2 ½18;71� GPa:

From the definition of Landau moduli (Hamilton and Blackstock, 1998) and (26), on the other hand, one finds that

C ¼�ðRoc2oÞ�1ðALandau

þ 2BLandauÞ ) C 2 ½8;25�:

In light of this result, the ensuing computations assume C¼8 as a representative value.Results. To illustrate the effect of ultrasound modulation on the ARF, Fig. 2 plots the variation of F f E

z,dir and FgEz,dir, at the

geometric focus ðr¼ 0, z¼ d), versus the modulation frequency om. As can be seen from the display, there are two keydifferences between the two solutions. First, f E

z,dir is notably higher than its predecessor (even at om ¼ 0) due to thepresence of coefficient C�1 multiplying the radiation force in soft tissues. Second, the proposed solution exhibits markedvariation with modulation frequency, which is in contrast to the frequency-invariant formula given by gE

z,dir. Withreference to (94), the latter difference arises from (i) argument ðet�ezÞ, rather than ðetÞ, of the modulation functionfeatured in f E

z,dir, and (ii) the presence of the ‘‘derivative term’’ M0M as an additional effect of modulation in the newsolution. Concerning the discrepancy in the magnitude of the two ARF estimates, it is noted that in both Sarvazyan et al.(1998) and Guzina et al. (2009), the amplitude of the observed mean displacements in tissue-mimicking phantomsgenerated by the ARF is roughly five times that predicted by the (visco-) elastodynamic solution assuming gE

z,dir for the bodyforce. This result indeed suggests that an account for the multiplier C�1 may be critical in capturing the phenomenon ofthe ARF in soft tissues.

For completeness, the spatial variation of the radiation body force F f Ez,dir and induced axial displacement F/uzSdir

E isshown in Fig. 3. Here /uzSdir

E is computed with the aid of the Hankel integral transform, which reduces (92) to theequation of a damped harmonic oscillator, see Sarvazyan et al. (1998) for details. From the display, one may observe both(i) spatial variation in the phase of the ARF (not predicted by the previous theory), and (ii) the localization of shear wavesnear the focal depth, z¼d, that is in agreement with earlier result, cf. Sarvazyan et al. (1998). To further highlight thedifferences between the current approach and existing theory, Fig. 4 compares /uzSdir

E solving (92) with its predecessor/wzSdir

E solving (93), each normalized by the magnitude of the respective body force at the focal point (see Fig. 2 for thedefinition of Fz and Gz). The result demonstrates that, despite the normalization by the peak body force, /uzSdir

E =Fz and

Am

plitu

de

Phas

e

6

4

2

00 0.5 1 1.5 2 2.5

x 10-3 x 10-3

3�m �m

x 10-9

Fz Gz

0.2

0.1

0

-0.1

-0.2

0 0.5 1 1.5 2 2.5 3

arg

arg

Fig. 2. ARF versus modulation frequency: comparison between the current model (F f Ez,dir) and its predecessor (FgE

z,dir) in terms of the body force

amplitude (left panel) and phase (right panel) at the focal point.

r/az/d

r/az/d

Re

(u z

dir

E)

/ z

Re

( f

z,di

r)/

zE

Fig. 3. Acoustic radiation force F f Ez,dir (left panel) and induced axial displacement F/uzSdir

E (right panel) in the (r,z) plane at om ¼ 10�3.

Dis

plac

emen

t

Dis

plac

emen

t

Re ( uzdirE

)

Re ( wzdirE

)/Gz

z / d

/ FzRe ( uz

dirE

)

Re ( wzdir

)/Gz

/ Fz1.5

1

0.5

0

-0.5

-1-2 -1 0 1 2

r / a

1

0

-10 0.5 1 1.5 2

Fig. 4. Normalized distribution of axial displacements, F/uzSdirE =Fz and F/wzSdir

E =Gz , in the radial direction (left panel) and axial direction (right panel)

through the focal point.

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–18131810

/wzSdirE =Gz are still distinct due to the mismatch in the peak-normalized spatiotemporal variations of f E

z,dir and gEz,dir

evident from (94).

8. Summary

In this study, a complete expression is derived for the acoustic radiation force (ARF) in homogeneous tissue-like solidsdue to elevated-intensity, focused ultrasound field (Mach number ¼Oð10�3

Þ) that is modulated with ‘‘low’’ frequenciesthat are, relative to the frequency of ultrasound, on the order of the Mach number. In simple terms, the ARF represents aneffective body force for the balance of linear momentum (namely viscosity-appended Navier equations) governing thepropagation of mean ultrasound-induced motion in a tissue. The problem is tackled via a scaling paradigm wherein thetransverse coordinates are scaled by the width of the focal region, while the axial and temporal coordinate are each splitinto a ‘‘fast’’ and ‘‘slow’’ component with the dual aim of: (i) canceling the leading linear terms from the field equationsdescribing the propagation of elevated-intensity ultrasound, and (ii) accounting for the effect of ultrasound modulation.The key advancement in the proposed framework over previous analyses revolves around the dual-time-scale treatment ofthe temporal variable, which allows one to parse out the contribution of ultrasound and its modulation in the nonlinearsolution. In this setting the mean tissue motion and affiliated (linear) field equations are effectively extracted from thosedescribing the nonlinear, modulated ultrasound problem through the introduction of a ‘‘fast’’ time average of the germanestate variables. For completeness the problem is tackled from both Eulerian and Lagrangian viewpoints, which are shownto coincide provided that the formulation is written in terms of the mean tissue displacement and mean specific entropy.The results are compared with the existing formula for the ARF that is derived assuming (i) Newtonian fluid as thepropagation medium, (ii) non-modulated ultrasound field, and (iii) heuristic modification to account for the effects ofmodulation. The comparison reveals a number of key features of the ARF that are brought to light by the new formulation.In particular, it is found that:

�

The effective body force for the mean balance of linear momentum is shown to be comprised of a potential term (thatincludes the effects of thermal expansion due to heating), and a directional term that is typically referred to as the ARF.In this setting, the analysis of the mean tissue motion via the concept of the ARF requires suitable spatial filtering of theobserved displacement field. To the authors’ knowledge, this study provides the first complete expression for thepotential component of the radiation body force; � Even in the case of no modulation the featured formula for the ARF, as derived for tissue-like solids, has an additional

multiplier relative to its predecessor computed for Newtonian fluids. From the limited experimental data on tissue-mimicking phantoms it appears that this multiplier, that bears linear relationship with the Landau’s third-order elasticmoduli, may be significantly larger than unity and thus explain reported discrepancies between the (indirect)experimental observations of the ARF in soft tissues and its estimates based on the premise of a Newtonian fluid;

� The newly obtained formula for the ARF contains an additional term that is proportional to the time rate of ultrasound

intensity. This term, that carries (in part) the effect of modulation on the ARF and affiliated mean tissue motion, opensthe possibilities toward enhanced spatiotemporal control of the radiation body force within the focal region, and thusmore effective tissue interrogation;

� By way of a systematic asymptotic treatment, the existing analytical framework which postulates ‘‘narrow’’ ultrasound

beam (generated by transducers with focal numbers Z5) is generalized to permit significantly wider class ofultrasound beams (with focal numbers Z1) that are more representative of biomedical applications entailing theuse of the ARF.

Assuming a set of ultrasound and soft-tissue parameters that are relevant to germane biomedical applications, theabove findings are illustrated through the example of a modulated ultrasound beam generated by the axisymmetricGaussian source.

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–1813 1811

Appendix A

A.1. Expansion of the thermoelastic response function

To demonstrate (25) one may first make use of (10) and (22), and expand the second of (9) as

~R ¼e2

~poIþð1þe ~rÞF � 1

e@U

@I1

!o

þ@2U

@I21

!o

I1ð~EÞþ

e2

@3U

@I31

!o

I21ð~EÞþe @2U

@I1@I2

!o

I2ð~EÞ

( )I

"

þ@U

@I2

!o

þe @2U

@I1@I2

!o

I1ð~EÞ

( )ðI1ð

~EÞI� ~EÞþe @U

@I3

!o

ðI2ð~EÞI�I1ð

~EÞ ~Eþ ~E2Þ�e bT

cp

~SI

#� FT, ðA:1Þ

where ~R ¼ ðR�roÞ=e, and ro ¼�e ~poI is the initial hydrostatic stress field as postulated before. By virtue of the facts that (i)~R is homogeneous in ~u, and (ii) the terms that are linear in ~u are associated with the Hooke’s law for an isotropic solid, onefinds that

@U

@I1

!o

¼�e ~po,@2U

@I21

!o

¼ 1þe ~po,@U

@I2

!o

¼�2eð ~Gþ ~poÞ, ðA:2Þ

which relates the internal energy derivatives to the initial pressure and linear elastic moduli. To keep the accuracy ofapproximation in (A.1) within Oðe2Þ, it follows from (A.2) that ~E and I2ð

~EÞ can be approximated with their Oð1Þ terms, whileI1ð~EÞ requires expansion up to the linear i.e. OðeÞ term. Accordingly, one finds from (8) and (22) that

~E ¼erS ~u, I1ð

~EÞ ¼e2

r � ~uþe r ~u : rT ~uþ12r ~u : r ~u

� , I2ð

~EÞ ¼e 1

2½ðr � ~uÞ2�rS ~u : rS ~u�,

where rS ~u ¼ 1=2ðr ~uþrT ~uÞ. By virtue of (25) and (A.2), these approximations allow one to rewrite (A.1) as

~R ¼e2

ð1þe ~rÞF � r � ~uþe r ~u : rT ~uþ1

2r ~u : r ~u

� �þ

1

2eBðr � ~uÞ2þ 1

2eC½ðr � ~uÞ2�rS ~u : rS ~u��e bT

cp

~S

� �I

�

þf2e ~G�eCr � ~ugðrS ~u�r � ~u IÞþeD 1

2½ðr � ~uÞ2�rS ~u : rS ~u� I�r � ~u rS ~uþðrs ~uÞ2

� ��� FT, ðA:3Þ

where B,C and D are constants given by (26). With the aid of (21a), (22) and relationship detðFÞ ¼e2

1þer � ~u, the latterexpression for ~R can be further expanded as

~R ¼e2

ð1�2e ~GÞr � ~u Iþ2e ~GrS ~uþe 1

2ðBþ3CþD�2Þðr � ~uÞ2�

1

2ðCþDÞrS ~u : rS ~u

�

þr ~u : rT ~uþ1

2r ~u : r ~u�

bT

cp

~S

�I�e½ðCþD�2Þr � ~u I�Drs ~u� � rs ~u, ðA:4Þ

which recovers claim (25).To compute the thermoelastic part of the first Piola–Kirchhoff stress tensor, ~P ¼ detðF ÞF

�1� ~R , one may note that

r ~u ¼ F�T� r ~u which, together with (A.4), yields

~P ¼e2

ð1�2e ~GÞr � ~uIþ2e ~GrS ~uþe 1

2ðBþ3CþDÞðr � ~u Þ2�

1

2ðCþDÞrS ~u : rS ~u :

�

þ1

2r ~u : r ~u�

bT

cp

~S

�I�e½ðCþDÞr � ~uI�Drs ~u � � rs ~uþer � ~ur ~u : ðA:5Þ

A.2. Intensity relationship (81)

To compute the ‘‘fast’’ time average of (80), it is necessary to evaluate the average of each term in the equation. To thisend, the Lagrangian counterpart of (68), namely

~v z ¼@ ~u z

@t¼eW�@ ~u z

@zo¼eW ~r , ðA:6Þ

allows one to deal with the acceleration term in (80) as

@ ~u z

@zo

@2 ~u z

@t2

* +L

¼ e @@t

@ ~u z

@zo

@ ~u z

@t

* +L

�1

2eW @

@zo

@ ~u z

@t

!2* +L

¼e2W

�e@~IL

@t �1

2eW@

~IL

@zo: ðA:7Þ

Similarly, the first term on the right-hand side of (80) can be computed via (A.6) as

@ ~u z

@zo

@2 ~u z

@z2o

* +L

¼1

2eW @

@zo

@ ~u z

@zo

!2* +L

¼e2W 1

2eW@

~IL

@zo, ðA:8Þ

E.V. Dontsov, B.B. Guzina / J. Mech. Phys. Solids 60 (2012) 1791–18131812

while (21b), (57b) and (A.6) can be used to show that

@ ~u z

@zo

@2 ~u i

@zo@wio

* +L

¼eW @

@wio

@ ~u z

@t

@ ~u i

@t

* +L

�@2 ~u z

@zo@wio

@ ~u i

@zo

* +L

¼eW @/ ~v z ~v iSL

@wio

þ@2 ~u i

@t2

@ ~u i

@t

* +L

¼eW @/ ~v z ~v iSL

@wio

: ðA:9Þ

The term in (80) multiplying ðBþ3Þ=2 is negligible because

@ ~u z

@zo

@

@zo

@ ~u z

@zo

!2* +L

¼2

3eW @

@zo

@ ~u z

@zo

!3* +L

¼eW

0: ðA:10Þ

On the other hand, the entropy term can be shown via (32), the Lagrangian statement of (73) and (A.6) to read

@ ~u z

@zo

@ ~S

@zo

* +L

¼eW� ~k @ ~u z

@zo

@2 ~T

@z2o

* +L

¼eW ~k bT

cp

@ ~u z

@zo

@3 ~u z

@z3o

* +L

¼eW� ~k bT

cp

~In

L : ðA:11Þ

Finally, the viscosity term can be written using (A.6) as

@ ~u z

@zo

@2 ~v z

@z2o

* +L

¼eW� ~r@

2 ~r@t2

* +L

¼eW ~I

n

L : ðA:12Þ

On substituting (A.7)–(A.12) into the Lagrangian ‘‘fast’’ time average of (80), one immediately recovers (81).

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