Acoustic radiation pressure in laterally unconfined plane wave beams

The acoustic radiation pressure in laterally unconfined, plane wave beams in inviscid fluids is derived via the direct application of finite deformation theory for which an analytical accounting is made ab initio that the radiation pressure is established under static, laterally unconstrained conditions, while the acoustic wave that generates the radiation pressure propagates under dynamic (sinusoidal), laterally constrained conditions. The derivation reveals that the acoustic radiation pressure for laterally unconfined, plane waves along the propagation direction is equal to (3/4)〈2K〉, where 〈 K 〉 is the mean kinetic energy density of the wave, and zero in directions normal to the propagation direction. The results hold for both Lagrangian and Eulerian coordinates. The value ( 3 / 4 ) 〈 2 K 〉 differs from the value 〈 2 K 〉 , traditionally used in the assessment of acoustic radiation pressure, obtained from the Langevin theory or from the momentum flux density in the Brillouin stress tensor. Errors in traditional derivations leading to the Brillouin stress tensor and the Langevin radiation pressure are pointed out and a long-standing misunderstanding of the relationship between Lagrangian and Eulerian quantities is corrected. The present theory predicts a power output from the transducer that is 4/3 times larger than that predicted from the Langevin theory. Tentative evidence for the validity of the present theory is provided from measurements previously reported in the literature, revealing the need for more accurate and precise measurements for experimental confirmation of the present theory.

It has long been known that radiation pressure in fluids is highly dependent on whether motion of fluid normal to the wave propagation direction is allowed-i.e., on whether the acoustic beam is laterally confined or laterally unconfined. The focus of the present work is to understand acoustic radiation pressure in a progressive, laterally unconfined, plane wave, acoustic beam via the direct application of finite deformation theory. The derivation shows that for both Lagrangian and Eulerian coordinates the radiation pressure for plane waves along the propagation direction is equal to á ñ ( ) / K 3 4 2 , where á ñ K is the mean kinetic energy density of the acoustic Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
wave. This result differs considerably from the value á ñ K 2 for the radiation pressure obtained from the Langevin theory [29] or from the Brillouin stress tensor [26,27]. In the directions normal to the propagation direction, the radiation pressure is zero for both Lagrangian and Eulerian coordinates.
The difference between the present assessment of radiation pressure and that obtained from the Langevin theory or from the Brillouin stress tensor is significant, since the assessment is used to link the radiation force on a target with the power generated by acoustic sources. As pointed out by Beissner [21], the 'measured radiation force must be converted to the ultrasonic power value and this is carried out with the help of theory.' It is generally assumed that for laterally unconfined, plane wave beams the relationship between the acoustic radiation pressure and the energy density for plane waves in the direction of wave propagation is that obtained from the Langevin theory [29] or from the Brillouin stress tensor [26,27]. The present model predicts a transducer output power 4/3 times larger than that predicted from the Langevin theory or from the Brillouin stress tensor. This has considerable implications regarding safety issues for medical transducers, calibrated using radiation pressure.
Issenmann et al [52] point out that 'despite the long-lasting theoretical controversies K the Langevin radiation pressure K has been the subject of very few experimental studies.' Indeed, absolute measurements obtained independently in the same experiment of the acoustic power generated by an acoustic source and the radiation force incident on a target to assess the radiation pressure-energy density relationship are relatively rare. If the transducer acoustic output power is calibrated from the measured radiation force and the Langevin theory is used as the radiation pressure-energy density relationship in the calibration (directly or indirectly via a secondary standard), then measurements taken with the transducer necessarily reflect (and, hence, by default 'confirm') the Langevin theory. Thus, radiation pressure measurements taken with 'Langevin-calibrated' transducers cannot in turn be used to validate the Langevin theory. Indeed, radiation pressure measurements taken with a transducer calibrated against a measured radiation force using any assumed radiation pressureenergy density relationship will necessarily reflect, and by default 'confirm,' the assumed pressure-energy density relationship. The correct radiation pressure-energy density relationship can only be validated from absolute, independent measurements of the radiation pressure and energy density in a single experiment under identical conditions. Such an experiment has been reported by Breazeale and Dunn [23] of the absolute pressure amplitude of a progressive, plane wave obtained using a force balance, three different optical techniques, a thermoelectric probe, and a direct assessment of the transducer output utilizing the piezoelectric constant. Experiments by Haran et al [24], have also been reported employing Raman-Nath diffraction measurements of the intensity of a progressive, plane wave impinging on a calibrated force balance. The measurements of Breazeale and Dunn [23] and Haran et al [24] are shown to provide tentative evidence for the validity of the present model, but reveal that more accurate and precise measurements are necessary for model confirmation.
A critical analysis of Lagrangian and Eulerian coordinates and quantities is presented in section 2. The analysis is necessary to understand errors made in previous derivations of the acoustic radiation pressure and to provide the framework for a radical (though mathematically punctilious and physically realistic) departure from previous approaches to resolve the long-lasting confusion surrounding the subject. Various widespread misconceptions concerning Lagrangian and Eulerian quantities and the relationships between them are pointed out-in particular, those related to Lagrangian and Eulerian pressures. A derivation via the direct application of finite deformation theory of the acoustic radiation pressure for laterally unconfined, plane wave beams is given in section 3.1 where, in contrast to previous derivations, an analytical accounting is made ab initio that the radiation pressure is established under static, laterally unconstrained conditions, while the acoustic wave that generates the radiation pressure propagates under dynamic (sinusoidal), laterally constrained conditions. Section 3.2 presents an assessment of the radiation pressure using the approach of Brillouin [26,27], who employed the Boltzmann-Ehrenfest Principle of Adiabatic Invariance in his derivation. It is shown that the terms in the Brillouin stress tensor are incorrectly referred to Eulerian coordinates and do not apply to laterally unconfined beams. Section 4 addresses various derivations of the Langevin relation for the radiation pressure. It is shown that the derivations improperly assume that the pressure in question is the Eulerian pressure rather than, correctly, the thermodynamic pressure (thermodynamic tensions or second Piola-Kirchhoff stress). The present theory is shown in section 5 to predict a power output from the transducer that is 4/3 times larger than that predicted from the Langevin theory. Tentative experimental evidence for the validity of the present model is presented.

Elements of finite deformation theory
In contrast to previous derivations, the present derivation of the acoustic radiation pressure for laterally unconfined, plane wave beams analytically accounts ab initio and a priori that the radiation pressure is established under static, laterally unconstrained conditions, while the acoustic wave that generates the radiation pressure propagates under dynamic (sinusoidal), laterally constrained conditions. The present model is based on the application of finite deformation theory to derive the relevant relationships between Lagrangian and Eulerian quantities. The derivation of the relationships via finite deformation theory is presented in some detail (a) to avoid the misinterpretation of mathematical operations and terms that has led to much of the confusion and controversy in the literature, (b) to provide the analytical underpinning critical to the derivation of the radiation pressure-energy density relationship, and (c) to allow the identification of errors made in previous derivations of the acoustic radiation pressure. The relationships between Lagrangian and Eulerian quantities are central to the theory of finite deformations, which was originally developed by Murnaghan [57], codified as a field theory by Truesdell and Toupin [58], Truesdell and Noll [59], and applied to acoustic wave propagation by Truesdell [60], Thurston [61], Thurston and Brugger [62], Thurston and Shapiro [63], and Wallace [64]. Finite deformation theory applies to any material of arbitrary crystalline symmetry including ideal fluids, which can be viewed as an isotropic material with zero shear modulus.

Lagrangian and Eulerian coordinates
Consider a material for which the initial (rest) configuration of particles comprising the material body is denoted by the set of position vectors {X}={X 1 , X 2 , X 3 } in a three-dimensional Cartesian reference frame having unit vectors e 1 , e 2, and e 3 along the coordinate axes. The (X 1 , X 2 , X 3 ) coordinates are known as Lagrangian or material (initial or rest) coordinates. Under an impressed stress the positions of the material particles will move from the initial (rest) set of vectors {X} to new positions described by the set of position vectors {x}= {x 1 , x 2 , x 3 } in the same three-dimensional Cartesian reference frame. The (x 1 , x 2 , x 3 ) coordinates are known as Eulerian or spatial (present) coordinates in the Cartesian reference frame. It is assumed that x and X are functionally related as where t is time. The present configuration of particles {x} is then related to the initial configuration {X} by means of an elastic deformation defined by the set of transformation (deformation) coefficients α ij = ∂x i /∂X j , where x i and X j , respectively, are the Cartesian components of the vectors x and X. The indices i and j take the values 1, 2, 3 representing the three mutually orthogonal Cartesian axes. An elemental length dX in the Lagrangian coordinates is transformed to an elemental length dx in the Eulerian coordinates as dx i = α ij dX j . The Einstein convention of summation over repeated indices is used in the present work. The inverse deformation is described by the set of transformation coefficients g ij defined such that g a d = , ij jk ik where δ ik is the Kronecker delta. If the deformation is non-uniform (i.e., varies with spatial position), the deformation is considered to be local in X and time t.
The deformation is defined by following the motion of a given particle originally at rest in the Lagrangian position X, which during deformation is displaced to the Eulerian position x. The particle displacement u is defined by u=x − X. The transformation coefficients α ij are related to the displacement gradients u ij = ∂u i /∂X j as For finite deformations Murnaghan [57] pointed out that the Lagrangian strains η ij defined as are rotationally invariant and provide an alternative to the displacement gradients u ij as a strain measure. Equations (1) and (2) hold for any material system having arbitrary crystalline symmetry -solid or fluid.

Lagrangian and Eulerian quantities
A physical quantity q in the deformed state but referred to the Lagrangian (initial, rest, or un-deformed state) coordinates at time t is defined as the Lagrangian quantity ( ) X q t , .

L
The same quantity referred to the Eulerian (present or deformed state) coordinates at the same time t is defined as the Eulerian quantity ( ) represent the same physical quantity q in the deformed state at the same position and same time t in Cartesian space, the relationship between the Lagrangian and Eulerian expressions of that quantity must necessarily be , 3 x X u L E t or, inversely, as = = - , It is generally assumed in traditional derivations of the radiation pressure that in equation (3), for example, the Lagrangian quantity is ( ) X q t , L and that the relevant Eulerian quantity is not L 0 in the un-deformed state at the initial (Lagrangian) position X at the initial time t 0 . Equation (3) states that both the Lagrangian quantity and the Eulerian quantity involve the same particle that initially is in the un-deformed position X at time t 0 in Cartesian space but has moved at time t from the undeformed position X to the deformed position x(t)=X+u(t) in Cartesian space [61].
A Eulerian quantity represents the value of a quantity associated with a particle in the present (deformed) position. Equation (4) states that the value of a Eulerian quantity at position x at time t corresponds to that of a particle whose present position at x originates from some un-deformed (Lagrangian) position X, but at a different time ¢ t the particle that appears at x is a different particle, originating from a different un-deformed (Lagrangian) position ¢ X . More importantly, both equations (3) and (4) state that the Lagrangian and Eulerian quantities corresponding to the same position in Cartesian space are exactly equal at all times t. A more complete analysis of the relationship between Lagrangian and Eulerian quantities is given in [65], where it is shown that the correct transformation between Lagrangian and Eulerian quantities must be obtained via the transformation coefficients a ij -not the displacement u. It is also important to note that the quantity q in equations (3) and (4) is assumed to be singly defined and is generally treated as a scalar (such as mass density being singly defined as a mass per unit volume). For tensor quantities, such as stress or pressure, the relationship between Lagrangian and Eulerian quantities can be more complicated, as shown in section 2.4.

Mass density in Lagrangian and Eulerian coordinates
A direct application of the transformation coefficients given in equation (1) for an initially un-deformed volume of material results in the well-known relationship [ where ρ 0 is the mass density in the initial (un-deformed) state, ρ is the mass density in the deformed state, and J is the Jacobian of the transformation defined as the determinant of the transformation coefficients α ij . It is important to recognize that ρ 0 is the mass density in the un-deformed state for both the Lagrangian and Eulerian coordinates and that ρ is the mass density in the deformed state for both the Lagrangian and Eulerian coordinates. This is apparent from equation (4), which states that the mass density r , , , , Thus, both equations (3) and (4) state that the mass density ρ has the same value at the same point and time in Cartesian space whether referred to Lagrangian or Eulerian coordinates, since for either coordinates the mass density refers to the same state of deformation at a given point and time t. Different values for the mass density in the two coordinates are obtained, when, as often occurs in the acoustics literature, the first term in a power series expansion of equations (3) or (4) in terms of the displacement u is assumed to represent the relevant conjugate density. In view of the equalities in equations (3) and (4), this assumption is clearly incorrect. Although equation (5) provides an expression of ρ as a function of the displacement gradients, which are referred to the Lagrangian coordinates, this does not mean that ρ in equation (5) now becomes exclusively the Lagrangian mass density, as often assumed. It is still the mass density in the deformed state for both Lagrangian and Eulerian coordinates in accordance with equations (3) and (4).

Stress in Lagrangian and Eulerian coordinates
Stress is defined in terms of the derivative of the internal energy per unit volume with respect to the relevant strain measure (a second rank tensor), which leads to the stress-strain relationships. Stress is thus a second rank tensor -not a scalar as is the mass density. The internal energy per unit mass U(x, S m ) of material depends on the relative positions of the particles comprising the material and the entropy per unit mass S . m This means that the internal energy per unit volume f=ρ 0 U(x, S m ), from which the stress is obtained by differentiation, can be expressed as a function of the displacement gradients u ij or as a function of the Lagrangian strains η ij as [64] , , , where A ij , A ijkl , and A ijklpq , respectively, are the first, second, and third-order Huang coefficients and C ij , C ijkl , and C ijklpq , respectively, are the first, second, and third-order Brugger elastic constants [61,64]. Substituting equation (2) in equation (6) and comparing the coefficients of like powers of the displacement gradients yield the relations [64] The first-order constants A ij are the initial stresses at position x=X in the material and are denoted in various alternative ways in equation (7) that will become apparent below. A stress is a force per unit area obtained by differentiating equation (6) with respect to the appropriate strain measure, h ij or u ij . It is noted that while the strain is defined with respect to the initial state of the material (i.e., with respect to the Lagrangian coordinates), the force F i is usually defined with respect to a unit area of deformed material (i.e., with respect to the Eulerian coordinates) [58,61]. An exception is the thermodynamic tensions (second Piola-Kirchhoff stress) for which both the strain and the force are referred to the initial state [58,61]. The stresses most relevant to acoustic wave propagation are the Eulerian (Cauchy) stresses and Lagrangian (first Piola-Kirchhoff) stresses. The Eulerian or Cauchy stress T ij is the force per unit area referred to the present configuration. It is a force per unit area for which both the force and the area are referred to the deformed state x [58][59][60][61][62][63][64]. The Cauchy stresses, evaluated in the present (perturbed or deformed) configuration x, are given in terms of the derivatives of the internal energy per unit volume with respect to the Lagrangian strains h ij as The Lagrangian or first Piola-Kirchhoff stress σ ij is a stress for which the force is referred to the deformed state x but the area is referred to the initial state X of the material [58][59][60][61][62][63][64]. The Cauchy stresses T ij are related to the first Piola-Kirchhoff stresses σ ij as [58][59][60][61][62][63][64][65] Equations (11) and (12) reveal that the relationship between the Cauchy (Eulerian) and first Piola-Kirchhoff (Lagrangian) stresses is more complicated than that of the Lagrangian and Eulerian mass densities, given by equations (3) and (4). The complication results from the differing definitions of the Eulerian and Lagrangian stresses, in contrast to the single definition of the mass density as simply a mass per unit volume. Note that when evaluated at x=X (the initial or un-deformed state) equation (11) yields Brillouin [26,27] preferred to use the Boussinesq stress tensor B ij , which is defined directly in terms of the derivatives of the internal energy per unit volume with respect to the displacement gradients u ij . The Boussinesq stress tensor is related to the first Piola-Kirchhoff stress tensor as [66] f From equations (6) and (13) It is extremely important to note that for purely longitudinal, plane wave propagation along the Cartesian direction e 1 , the shear strains and equation (12) simplifies without approximation to s = ( ) T . 1 5 11 11 Equation ( Equation (16) shows, in contrast to previous derivations in the literature, that the Eulerian pressure p E 1 is exactly equal to the Lagrangian pressure p L 1 for longitudinal, plane wave propagation along e 1 in materials. The subscript '1' in equation (16) denotes that the pressure corresponding to longitudinal, plane wave propagation along e 1 is the i=j=1 component of the second rank tensors T ij and s . ij More generally, from equation (5) J −1 can be approximated as [61,64] » - The equations-of-state for fluids are generally defined as functions of pressure in terms of the mass density ρ. As shown in section 2.3, ρ refers to the mass density in the present state of deformation and has the same value whether referred to Lagrangian or Eulerian coordinates. The pressure, in contrast to the mass density, is generally different in Lagrangian and Eulerian coordinates, except for the case of purely longitudinal wave propagation. Since ρ refers to the mass density in the present state of deformation, the pressure in the equationsof-state for fluids is quite naturally referred to Eulerian coordinates. Thus, the equations-of-state for fluids, when expressed as functions of the mass density, are Eulerian equations. For liquids, the equation-of-state is given as an expansion of the Eulerian pressure p E in terms of the mass density ρ as [34] r r r where the relation = p p E L 1 1 follows from equation (16) for plane waves. The Brugger elastic coefficients are related to the Fox-Wallace-Beyer coefficients as   3  2 5   123  132  213  231  312  321  0 where p 0 is the initial hydrostatic pressure and the Voigt contraction of indices (11→1, 22→2, 33→3, 23=32→4, 13=31→5, 12=21→6) has been used in equations (20)- (25) for the Brugger coefficients. The Huang elastic coefficients A ij are assessed from equations (7)-(9), (20)- (25) in terms of the Fox-Wallace-Beyer coefficients as (using Voigt contraction of indices)  The equation-of-state for ideal gases is given as where γ is the ratio of specific heats. The relationships between the Huang coefficients and the corresponding elastic parameters for ideal gases are obtained by setting

Time-averaging of Lagrangian and Eulerian quantities
Since the acoustic radiation pressure is a time-averaged, steady-state property of the wave, it is useful to define the time-average of a continuous periodic function ( ) f t under steady-state conditions by the operation where the angular bracket denotes time-averaging of the function enclosed in the bracket. It is often assumed that time-averaging a Lagrangian quantity yields values different from that of time-averaging the corresponding Eulerian quantity, since time-averaging the Lagrangian quantity á ñ ( ) X q t , L occurs while holding the Lagrangian coordinates fixed, but time-averaging the Eulerian quantity á ñ ( ) x q t , E occurs while holding the Eulerian coordinates constant. For fixed Lagrangian coordinates X, , , results from the fact that for sinusoidal waves u(t) averages to zero and x(t) averages to X. For fixed Eulerian coordinates x, á ñ = á ñ = á ñ = - L results from the fact that for sinusoidal waves u(t) averages to zero and X(t) averages to x. When x and X correspond to the same point Y in Cartesian space, then x=X=Y and á , .

L E
It is often assumed in the acoustics literature for fluids that the Eulerian coordinates correspond to surfaces fixed in Cartesian space and that the Lagrangian coordinates correspond to surfaces that oscillate in space under an impressed sinusoidal wave [28,34,35,41,48]. An oscillating material surface is defined by a set of n contiguous, particle displacements u n (t) (n=1, 2, 3, ···) that vary sinusoidally in time t. Relative to fixed Lagrangian coordinates X n , the time-dependent particle displacements u n (t) are defined by u n (t)=x n (t)−X n (n=1, 2, 3, ···). The Eulerian coordinates x in this case are time-dependent. Relative to fixed Eulerian coordinates x n the particle displacements u n (t) are defined by u n (t)=x n −X n (t), where it is the Lagrangian coordinates that are now dependent on time t. Since it is the displacements that define the motion of the surface, it is apparent that the displacements can occur with respect to either fixed Lagrangian or fixed Eulerian coordinates -a consequence of the relativistic principle that for coordinate systems moving relative to each other it does not matter in regard to the relative displacement which system is regarded as moving and which is considered fixed.
It is noted that the quantity q in equations (3) and (4) is assumed to be a singly defined scalar quantity. The relationship between the time-averaged Cauchy (Eulerian) stress and the first Piola-Kirchhoff (Lagrangian) stress is more complicated, since stress is not a scalar but, rather, a second rank tensor, defined as a force per unit area for which the area is defined differently for the two stresses. The force in the definition of both stresses refers to Eulerian coordinates (present or deformed state) but the area in the first Piola-Kirchhoff stress refers to Lagrangian coordinates (initial or un-deformed state) and the area in the Cauchy stress refers to Eulerian coordinates. The relationship between the two stresses is thus governed by the transformation between the Lagrangian and Eulerian areas and the time-averaging must be assessed from the equation, obtained from finite deformation theory, linking the quantities [64]. For plane, longitudinal acoustic stresses such that σ 11 =T 11 at a given point in Cartesian space, s á ñ = á ñ T , 11 11 exactly, resulting from the fact that the areas in the two stresses transform such that the areas are equal in magnitude. In other cases, equations (11), (12) or (17) must be used to assess the time-averaged relationship between Eulerian and Lagrangian stresses.
The time-averaged displacement gradient á ñ u 11 is the radiation-induced static strain. Since σ ij is the force per unit area referred to the Lagrangian coordinates, s á ñ ij is the Lagrangian radiation stress (also known as the first Piola-Kirchhoff radiation stress). Since T ij is the force per unit area referred to the Eulerian coordinates, á ñ T ij is quite properly the Eulerian radiation stress (or Cauchy radiation stress). It is generally assumed in the acoustics literature that for plane wave propagation the radiation stress in Eulerian coordinates is not á ñ T ij but, rather, the momentum flux density r á ñ v v .
i j It is shown in section 3.2 that this assumption is based on an incorrect interpretation, originally proposed by Brillouin [26,27], of the terms in what is now known as the Brillouin stress tensor. It is shown that the term in the Brillouin stress tensor that Brillouin assumed to be the momentum flux density is actually the contribution to the fractional variation in the period of the acoustic wave resulting from a change in the sample length from slow, virtual variations in the strain parameter. Equally important, it is shown that the Brillouin stress tensor is not a Eulerian tensor at all, as generally assumed, but a Lagrangian tensor. The incorrect identification of the momentum flux density with the radiation stress for plane waves has greatly contributed to the considerable misunderstanding of acoustic radiation stress in the literature.

Acoustic radiation pressure in laterally unconfined, plane waves
It has been known since the work of Hertz and Mende [28] that for fluids the radiation pressure in an acoustic beam is highly dependent on whether motion of fluid normal to the wave propagation direction is allowed-i.e., on whether the acoustic beam is laterally confined or laterally unconfined. Brillouin [26,27], Hertz and Mende [28], and Beyer [34,49] assess the radiation pressure by assuming a longitudinal, plane wave acoustic beam of cylindrical cross-section incident on a target in laterally confined and laterally unconfined volumes. Their derivation is questionable for several reasons given in [65], including an incorrect assessment of the relationship between Lagrangian and Eulerian quantities. Moreover, as pointed out by Beissner [50,51], a beam of finite cross-section is three-dimensional and, thus, diffracted, which leads to additional issues in assessing the radiation pressure. The seminal papers [25][26][27][28][29][30][40][41][42] on acoustic radiation pressure, however, assume idealized, one-dimensional, plane wave propagation. Since these papers are responsible for much of the confusion and misunderstanding surrounding acoustic radiation pressure, it is appropriate to focus on the derivation of the radiation pressure for plane wave propagation, beginning with a derivation based on the direct application of finite deformation theory to laterally unconfined beams.

Derivation of radiation pressure via direct application of finite deformation theory
In assessing the acoustic radiation pressure for laterally unconfined, plane waves propagating along e 1 in inviscid fluids, it is crucial to recognize that the time-averaged energy density á ñ K 2 (K =kinetic energy density) that drives the radiation pressure is produced by a sinusoidally oscillating plane wave of finite beam cross-section (usually cylindrical) propagating under laterally constrained conditions. That is, the dynamic (sinusoidal) plane wave propagation is defined such that ¹ u 0, 11 = = u u 0. 22 33 In contrast, the radiation pressure itself is governed by static (time-averaged, steady-state) conditions associated with a laterally unconstrained volume. This is quite unlike the case for laterally confined, plane wave beams where both the dynamical wave and the radiation (static) pressure are subject to the same lateral constraints, ¹ u 0, 11 = = u u 0. 22 33 Strictly, for laterally unconfined conditions a cylindrical acoustic beam of finite cross-section is not planar because of diffraction but becomes increasingly planar in an area around the center of the beam as the ratio of the acoustic wavelength to the beam radius r approaches zero. Further, as pointed out by Lee and Wang [48] the amplitude of the wave does not abruptly decrease to zero beyond the beam radius but does so smoothly in a manner approximated by the zeroth order Bessel function a where a c is a constant corresponding to the reciprocal of some characteristic beam radius. For present purposes, there is no loss in generality for onedimensional wave propagation to assume an idealized plane wave beam of cylindrical cross-section with a 'tophat' amplitude profile. More importantly, as shown below, lateral unconfinement modifies the assessment of the elastic coefficients relevant to static conditions.
The radiation-induced static strain generated by the acoustic plane wave statically deforms the volume of material through which the wave propagates. For a laterally unconfined beam, the reaction of the statically deformed volume, however, is governed not by the dynamic, laterally constrained conditions associated with plane wave propagation but by the static (time-averaged, steady state) conditions governing a laterally unconstrained volume. To emphasize that the elastic properties associated with static, laterally unconstrained conditions are distinct from those of the laterally constrained conditions associated with dynamic, acoustic, plane wave propagation, the functions, parameters, and variables associated with static, laterally unconstrained conditions are designated by the superscript 'S'. Thus, under static, laterally unconstrained conditions It is convenient to begin with the derivation of acoustic radiation pressure in Lagrangian coordinates. Performing the summation in the last equality in equation (6), time-averaging, and substituting within the timeaverage expressions the equations (26)-(31) for the Huang coefficients (in Voigt notation) and the relation 11 corresponding to laterally unconstrained conditions lead to an assessment of the mean internal energy density f S as  33 for static conditions. Taking the time derivative of equation (37) yields . Taking the derivative of equation (37) with respect to Integrating equation (41) with respect to u 11 yields   and writing r = á ñ ¶ ¶ Equation (46) shows that the mean Eulerian and Lagrangian excess pressures for laterally unconfined, plane waves are exactly equal with magnitude á ñ ( ) / K 3 4 2 along the direction of wave propagation. Equation (46) is quite different from the Langevin expression for laterally unconfined plane waves, which posits that the mean Lagrangian pressure á ñ p L 1 along e 1 is obtained from Langevin's first relation f á ñ = á ñ + á ñ + p K C, L 1 where C=constant [29]. Assuming C=p 0 leads to Langevin's result for the acoustic radiation pressure in laterally unconfined, plane wave beams as where, as shown in section 3.2, for plane waves á ñ K 2 is approximately equal to the energy density á ñ E . It is relevant to point out that for laterally constrained conditions, u 22 =u 33 =0 and = ( ) f f u . 11 Applying these conditions and following the derivation leading to equation (43) now results in the compatibility relation Equation (50) has been experimentally confirmed along the three, independent, pure mode propagation directions in monocrystalline silicon [45,55] and in isotropic vitreous silica [45,56]. The experimental confirmation attests the validity of the derivation leading to equations (48)- (50) and lends support to the analogous derivation leading to equation (46). (It is noted that equation is mistakenly used in reference [65] to assess the acoustic radiation pressure for laterally unconfined, plane wave beams in fluids. The appropriate relationship for fluids is clearly given by equation (46).) The static stresses s á ñ  This also means, from equation (17) and the null value under static conditions of the excess Lagrangian pressures in directions e 2 and e 3 , that the excess Eulerian pressures normal to the wave propagation direction are given as Thus, for laterally unconfined, plane wave propagation the acoustic radiation pressure along the direction of propagation is á ñ K 2 3 4 and zero in directions normal to the propagation direction. This result holds for both Lagrangian and Eulerian coordinates for laterally unconfined, plane wave beams.

Acoustic radiation pressure and the Boltzmann-Ehrenfest Adiabatic Principle
Brillouin [26,27] approached the problem of acoustic radiation stress (pressure) by applying the Boltzmann-Ehrenfest Principle of Adiabatic Invariance to longitudinal, plane wave propagation. His result, as shown below, differs considerably from equation (46). Since Brillouin's theory has played such a pivotal role in assessing the radiation pressure for fluids, it is instructive to reconsider the Boltzmann-Ehrenfest approach in detail.
The Boltzmann-Ehrenfest (B-E) Adiabatic Principle [68,69] states that if the constraints of a periodic system are allowed to vary sufficiently slowly, then the product of the mean (time-averaged, steady-state) kinetic energy * á ñ K and the period T of the system is an adiabatic invariant or constant of the motion such that the virtual variation According to the B-E Principle, a slow virtual variation δq * in a constraint q * (generalized displacement) of a conservative, oscillatory system leads to a change in the system configuration that results in a change * d á ñ E in the mean total energy * á ñ E of the system. The change in the mean total energy is quantified by the product of the generalized reaction force Q * and virtual constraint variation δq * such that To understand Brillouin's results, it is instructive to consider first the derivation in Lagrangian coordinates for laterally confined, longitudinal, plane wave propagation along e 1 . In Lagrangian coordinates, the virial theorem states that [70,71]  where for longitudinal plane waves, the potential energy density corresponding to the excess stress s s -[ ( ) ] 11 11 0 is obtained from equation (6) by letting f f - ¢ ( ) A u , 1 11 dropping the prime on f¢, and writing The relationship between the mean kinetic energy density and the mean internal (potential) energy density for plane waves can be established by substituting equations (55) in (54) to obtain a power series expansion of equation (54), and then solving equation (55) for A u 11 11 2 and iteratively substituting for u 11 2 in the terms of the expanded equation (54) to obtain where the constant term r ( ) U X S , 0, 0 has been dropped, since it makes no contribution to the kinetic energy. The mean total energy density á ñ E for nonlinear plane waves is then where the last equality follows from equation (56). It is interesting to note from equation (57) that for nonlinear waves the total average energy density á ñ E is not exactly equal to á ñ K 2 . According to the B-E Adiabatic Principle [70,71], a slow virtual variation δq * in a constraint q * (generalized displacement) of a conservative, oscillatory system leads to a change in the system configuration that results in a change * d á ñ E in the mean (time-averaged, steady-state) total energy * á ñ E of the system. The change in the mean total energy is quantified by the product of the generalized reaction force Q * and virtual constraint variation δq * such that * * * d d á ñ = E Q q . For longitudinal, acoustic plane wave propagation, the generalized reaction force in Lagrangian coordinates is the mean excess radiation stress s s s s á -ñ=á ñ-( ) ( ) , 11 11 0 11 11 0 the constraint (generalized displacement) is the displacement gradient u 11 , the mean kinetic energy * á ñ K corresponds to the mean kinetic energy density á ñ K , and the mean total energy * á ñ E corresponds to the mean total energy density á ñ E . Thus, for plane wave propagation the relation where the last equality in equation (58) follows from equation (57).
in equation (53) and substituting in equation (58) lead, to first order in the nonlinearity, to the relation for the acoustic radiation stress where the subscripted '0' denotes evaluation at u 11 =0. The factor df d u u 11 11 in equation (59) is evaluated as 2 . 60 11 11 11 11 It is noted that Brillouin omitted in his derivation the nonlinear contribution corresponding to the last term in equation (59). It is extremely important to note that the radiation stress given by equation (59) is the Lagrangian radiation stress. The fractional change in the oscillation period T −1 δT/δu 11 with respect to the virtual variation δu 11 can be easily assessed from the fractional change in the natural velocity W. The natural velocity is the velocity defined as the ratio of the length of the sound path in the un-deformed state to the propagation time in the deformed state [61][62][63]. Since the path length in the un-deformed state is constant, only the propagation time in the deformed state plays a role in assessing the fractional variation in the system period when using the natural velocity for the assessment. The natural velocity is the velocity referred to the Lagrangian coordinates and is obtained from equation (14)  The excess acoustic radiation stress in Lagrangian coordinates s s á ñ -( ) 11 11 0 is generally known as the Rayleigh radiation stress s á ñ Rayleigh 11 and is evaluated from equations (59), (60), and (62)  For laterally confined beams in liquids with initial (hydrostatic) pressure p 0 , equation (16) yields that the excess Eulerian pressure and the excess Lagrangian radiation pressure are equal along the wave propagation direction e 1 and equations (27) and (29)

11
Thus, for laterally confined beams in liquids equation (63) gives is in agreement with the results of Brillouin (except for the contribution, omitted by Brillouin, of the nonlinear factor corresponding to the last term in equation (59)). However, Brillouin incorrectly assumed that the equations he used, leading to equation (64), refer to Eulerian coordinates rather than to Lagrangian coordinates, as shown in the present derivation. It is crucial to understand how Brillouin came to such an assumption, since the assumption has led to a deep foundational misunderstanding of acoustic radiation stress and pressure. It is noted that the equality of the Lagrangian and Eulerian stresses (pressures), obtained from finite deformation theory and given by equation (16), for longitudinal, plane wave propagation was not known to Brillouin. The equations that Brillouin interpreted as Eulerian equations can be obtained from a consideration of the relationship between the natural velocity W and the true velocity c. The true velocity c is the velocity defined as the ratio of the length of the sound path in the deformed state to the propagation time in the deformed state. The true velocity is related to the natural velocity W as = where ℓ 0 is the length of material in the un-deformed state and ℓ is the length of material in the deformed state [61][62][63]66]. It is shown Appendix B of Brillouin largely ignored the 'mean excess Eulerian pressure' term in applications of the Brillouin stress tensor, assuming the term to be irrelevant in assessing the radiation pressure. Several attempts to justify Brillouin's assumption have appeared in the literature. The approach to establishing a null 'mean excess Eulerian pressure' has been to utilize the relationship between pressure and enthalpy. The derivations of Lee and Wang [48]. Beissner [50], Beissner and Makarov [51], and Hasegawa et al [47] are representative of such approaches.

Consider the wave equation in Eulerian coordinates
For irrotational sound waves j =  u , SP where j SP is the scalar potential. Equation (70) can thus be re-written as It is generally assumed that equation (71) can be solved using the thermodynamic relationship Th m is the enthalpy per unit mass, S m is the entropy per unit mass, and p Th is the thermodynamic pressure [58,61,64]. It is critically important to recognize that the thermodynamic pressure p Th is not the Eulerian pressure p . E The thermodynamic pressure is obtained from the thermodynamic tensions (second Piola-Kirchhoff stress) [58,61,64]. For longitudinal, plane wave propagation along direction e 1 , [58,61]. It is generally overlooked that the pressure in the thermodynamic relationship for enthalpy is the thermodynamic pressure p Th and it is incorrectly assumed in equation (71) that for adiabatic motion It is then assumed that the pressure p E can be expanded in a power series in the enthalpy H [48] or that can be expanded in terms of the pressure p E [47,50,51]. The power series expansion is a key operation in the derivations to obtain the relationship for propagation along e 1 (see [47,48,50,51] for details) where C is a constant and f á ñ and á ñ K are, respectively, the time-averaged potential and kinetic energies of the wave.
Equation (72) is known as Langevin's second relation [30,50] and is generally regarded as the expression defining the mean Eulerian pressure in a wave. For plane, progressive waves it is traditionally assumed that f á ñ = á ñ K . If C is assumed to be the initial pressure p 0 , equation (72) predicts that the mean excess Eulerian pressure is zero. This means, in regard to the traditional (incorrect) interpretation of the Brillouin stress tensor, that the acoustic radiation pressure depends only on the momentum flux density r á ñ = á ñ v v E 1 1 for laterally unconfined, plane wave propagation along e 1 .
The problem with the derivations leading to equation (72) is that the Eulerian pressure p E is not the thermodynamic pressure p .
Th Substituting H for r p E in equation (71) does not affect the terms on the left-hand side of the equation but it changes the right-hand side of the equation from a dependence on p E to a dependence on p .
Th By substituting p Th for p E in the relevant equations, the arguments of [47,48,50,51] lead to the relation Th 1 for propagation along direction e 1 , rather than to equation (71). For longitudinal wave propagation along e 1 [58,61,64], The last equality in equation (74) follows from the virial theorem, equation (54), where s = -=p p .
L E 11 1 1 Substituting equation (74) in equation (73) leads to f á ñ = á ñ = á ñ -á ñ + ( ) Equation (75) does not yield Langevin's second relation, equation (72). It is concluded that Langevin's second relation, equation (72), is incorrect and does not provide an assessment of the mean Eulerian pressure, as traditionally assumed. Indeed, the results of equations (16) and (46) already suggest that since á ñ = á ñ p p E L 1 1 for plane wave propagation along e 1 , Langevin's second relation [30], f á ñ = á ñ -á ñ + p K C , and Langevin's first relation [29], f á ñ = á ñ + á ñ + p K C , L 1 cannot both be correct. Equally important, it is seen from equation (46) that Langevin's first relation is also incorrect, as the equation does not account analytically a priori for the difference between the elastic properties under laterally unconfined, static conditions and that of the driving acoustic wave propagating under laterally confined, dynamic conditions.

Experimental evidence for the present theory
Issenmann et al [52] point out that 'despite the long-lasting theoretical controversies K the Langevin radiation pressure K has been the subject of very few experimental studies.' Indeed, few absolute measurements obtained independently in the same experiment of the acoustic power generated by an acoustic source and the radiation force incident on a target to assess the radiation pressure-energy density relationship have been reported. Typically, experimental assessments of the radiation pressure have either relied on the assumption of the Langevin relation a priori in evaluating the transducer power output [18][19][20][21][22] or have considered relative measurements without directly evaluating the transducer power output (cf, [8,9,38,39]). Beissner [21] points out that if acoustic radiation pressure is used to calibrate acoustic sources the 'measured radiation force must be converted to the ultrasonic power value and this is carried out with the help of theory.' The measured radiation force on a target is generally assumed to result from the Langevin relation, equation (47), between the radiation pressure generated by the acoustic source and the energy density of the wave [18][19][20][21][22]. It is appropriate to consider the implications of this assumption.
Consider a planar transducer of active area S A that emits an idealized plane wave. The average ultrasonic power á ñ W pw emitted by the transducer is related to the energy density á ñ » á ñ E K 2 of the plane wave as á ñ = á ñ W E S c, pw A where c is the sound velocity in the fluid. If Langevin's theory, equation (47), is assumed to be correct, then along the direction of plane wave propagation, the average force á ñ F generated over the area S A normal to the propagation direction is = ⟨ ⟩ ⟨ ⟩ F E S A and the average ultrasonic power is á ñ = á ñ W Fc .

pw Langevin
The present theory, however, predicts from equation (46) that for laterally unconfined plane waves along e 1 á ñ -= á ñ -= á ñ » á ñ Along the direction of plane wave propagation, the average force á ñ F generated over the area S A normal to the propagation direction is á ñ = á ñ - Since the measured pressure á ñp p L 1 0 must be the same whether using the present theory or the Langevin theory, equating the lefthand sides of equations (46) and (47) yields in agreement with equation (76).
Independent measurements of the absolute ultrasonic pressure amplitude in water have been reported by Breazeale and Dunn [23] using different methods subject to the same experimental conditions. They report a direct assessment of the transducer output, calculated from the piezoelectric constant and the voltage applied to the transducer, and measurements, referred to a force balance, of three different optical techniques and a thermoelectric probe. They find that 'the experimental results exhibit a total range of approximately ±10 percent about the mean and that this mean is approximately 27 percent below that calculated (from the piezoelectric constant) from the voltage applied to the transducer.' If it is assumed that the direct piezoelectric assessment represents the correct transducer output, then the measurements of Breazeale and Dunn [23] imply that the acoustical measurements, referred to the radiation force balance using the Langevin theory, are consistently well below the correct transducer output. This is consistent with the prediction of the present theory that the correct energy density is 33 percent greater than the value calculated from the Langevin equation.
In a similar experiment, Haran et al [24] report independent, simultaneous assessments of the acoustic power generated from a 1 MHz acoustic transducer from acousto-optic Raman-Nath diffraction measurements and measurements from a force balance in water using the Langevin equation. Critical to their assessment from the Raman-Nath measurements is the value 1.46 × 10 −10 Pa −1 assumed for the piezo-optic coefficient. There is considerable variation reported in the literature regarding the value of the piezo-optic coefficient in water, which is found to range from 1.32 × 10 -10 Pa −1 to 1.51 × 10 −10 Pa −1 [72][73][74][75]. Further, the active area of the transducer may be smaller than that of the stated physical size of the transducer. For a given measured value of the Raman-Nath parameter, the uncertainty in the magnitude of the piezo-optic coefficient and active transducer radius, as well as contributions from acoustic diffraction and attenuation, can lead to large changes in the calculated value of the intensity from the Raman-Nath measurements. For example, using in [24] the value 1.32 × 10 −10 Pa −1 for the piezo-optic coefficient and assuming that the active diameter of the transducer is 5 percent smaller than the stated physical diameter lead to an increase in the intensity assessed from the Raman-Nath measurements by 36 percent. This value is consistent with the prediction of the present theory that the correct energy density (and intensity) is 33 percent greater than the value calculated from the Langevin equation. It is concluded that although the measurements of Breazeale and Dunn [23] and Haran et al [24] provide tentative evidence for the validity of the present model, more accurate and precise measurements are needed for model confirmation.

Conclusion
Equation (46) shows that the mean excess Eulerian and Lagrangian pressures along the propagation direction for longitudinal waves in laterally unconfined, plane wave beams are exactly equal with magnitude á ñ ( ) / K 3 4 2 .This result is quite different from that derived from the Langevin expression for laterally unconfined, plane waves, which posits that the mean Lagrangian pressure is obtained as á ñ = á ñ » á ñ p K E 2 Langevin 1 [29]. A number of analytical efforts have been published [28,29,34,35,[40][41][42][47][48][49][50][51] in various attempts to validate Langevin's result, equation (47). The derivations do not distinguish analytically a priori, as done in section 3.1, that the elastic properties under static, laterally unconstrained conditions associated with the radiation pressure are quite different from those of the driving (dynamical) acoustic wave propagating under laterally constrained conditions. Recognition of the difference occurs a posteriori in [28,29,34,35,[40][41][42][47][48][49][50][51], which entail various erroneous and, in some cases, somewhat contrived arguments involving fluid flow to establish Langevin's result. Central to the arguments is the assumption that for laterally unconfined, plane wave propagation the Lagrangian and Eulerian radiation pressures are different, which, as shown in section 2.4, is not correct.
Other attempts to assess the acoustic radiation pressure along the propagation direction for laterally unconfined, plane waves are based on an application of the Brillouin stress tensor, which Brillouin assumed is an expression of the acoustic radiation stress in Eulerian coordinates [26,27]. Brillouin, who employed the Boltzmann-Ehrenfest Principle of Adiabatic Invariance in his derivation, obtained that the Brillouin stress tensor is composed of two contributions -a 'mean excess Eulerian stress' contribution and a momentum flux density contribution. It is shown in section 3.2 that the Brillouin stress tensor is not an expression in Eulerian coordinates, but rather an expression in Lagrangian coordinates. The 'mean excess Eulerian stress' in the Brillouin stress tensor is not a Eulerian stress at all, but rather the contribution to the fractional change in the system period (in terms of the B-E Principle) associated with the change in the true sound velocity (defined as the ratio of the length of the sound path in the deformed state to the propagation time in the deformed state). The contribution that Brillouin attributed to the momentum flux density is actually the contribution to the fractional variation in the period of acoustic oscillations resulting from a change in the sample length from slow, virtual variations in the strain parameter (generalized displacement in terms of the B-E Principle).
Efforts to assess the acoustic radiation pressure along the propagation direction for laterally unconfined, plane waves from the Brillouin stress tensor rely on establishing that the 'mean excess Eulerian pressure' (á ñp p E 1 0 ) along the propagation direction e 1 is zero, leaving the radiation pressure equal, incorrectly, to the momentum flux density r á ñ v v 1 1 The derivations of Langevin and Brillouin fail to recognize, as shown from finite deformation theory, that for laterally unconfined plane waves the Lagangian and Eulerian radiation pressures are exactly equal and that the Brillouin stress tensor is a Lagrangian tensor -not a Eulerian tensor. Equation (46) establishes that the radiation pressure along the propagation direction is (3/4)〈2K〉 and zero in the directions normal to the wave propagation direction. This result is consistent with the experimental studies of Herrey [52], who shows that the radiation pressure in laterally unconfined fluids is anisotropic, and of Rooney [53], who shows that the radiation pressure in such media is independent of the dynamic acoustic nonlinear parameter b = -( ) / A A 111 11 of the fluid -although equations (34) show that for laterally unconstrained beams the static nonlinearity parameter b = - for all inviscid fluids. The present model deviates from previous approaches in several, quite fundamental ways. The traditional derivations for laterally unconfined, plane waves in fluids do not account a priori for the difference between the elastic properties under static, laterally unconfined conditions (giving rise to free fluid flow) and that of the driving acoustic wave propagating under laterally confined conditions (that do not permit free flow). Rather than accounting a priori for the difference in elastic properties, a patchwork of a posteriori assumptions, definitions, and arguments has been used in various attempts to quantify the radiation pressure for laterally unconfined, plane waves. The previous derivations are typically based on a number of misconceptions that have permeated the acoustics literature including (a) a widespread misunderstanding of Lagrangian and Eulerian quantities and of the transformation between them (addressed in detail in section 2), (b) the misinterpretation by Brillouin of terms leading to the Brillouin stress tensor, discussed in section 3.2, and (c) the assumption that the pressure defined by the enthalpy in deriving Langevin's second relation is the Eulerian pressure rather than the thermodynamic pressure (second Piola-Kirchhoff pressure), as discussed in section 4. The present work corrects these misconceptions and provides a coherent, first principles examination of acoustic radiation pressure based on finite deformation theory. Equally important, as shown in section 5, a limited amount of experimental data have been reported that provide tentative evidence for the validity of the present theory. More precise and accurate measurements, however, are necessary for confirmation of the present theory.
The acoustic radiation pressure is used in a variety of applications , many of which depend on a reliable assessment of the force on a target generated by an acoustic source. It is appreciated that the measurements for diffracted and focused beams are not described by simple plane wave propagation, but because of the difference between the present value of á ñ ( ) / K 3 4 2 for the acoustic radiation pressure along the propagation direction for laterally unconfined, plane waves and the value á ñ » á ñ K E 2 from the Langevin theory or the momentum density r á ñ v v 1 1 from the Brillouin stress tensor, it would seem prudent to re-examine relevant applications in view of the present theoretical results. For example, a recent experimental study [76] suggests that acoustic radiation pressure may play a role in pulmonary capillary hemorrhage (PCH) but 'only a full understanding of PCH mechanisms will allow development of science-based safety assurance for pulmonary ultrasound.' The larger acoustic power levels predicted in the present manuscript could contribute to acquiring 'a full understanding.'