Nonlinear propagation of quasiplanar shear wave beams in soft elastic media with transverse isotropy

Abstract

Model equations are developed for shear wave propagation in a soft elastic material that include effects of nonlinearity, diffraction, and transverse isotropy. A theory for plane wave propagation by Cormack [J. Acoust. Soc. Am. 150, 2566 (2021)] is extended to include leading order effects of wavefront curvature by assuming that the motion is quasiplanar, which is consistent with other paraxial model equations in nonlinear acoustics. The material is modeled using a general expansion of the strain energy density to fourth order in strain that comprises thirteen terms defining the elastic moduli. Equations of motion for the transverse displacement components are obtained using Hamilton's principle. The coupled equations of motion describe diffraction, anisotropy of the wave speeds, quadratic and cubic plane wave nonlinearity, and quadratic nonlinearity associated with wavefront curvature. Two illustrative special cases are investigated. Spatially varying shear vertical wave motion in the fiber direction excites a quadratic nonlinear interaction unique to transversely isotropic soft solids that results in axial second harmonic motion with longitudinal polarization. Shear horizontal wave motion in the fiber plane reveals effects of anisotropy on third harmonic generation, such as beam steering and dependence of harmonic generation efficiency on the propagation and fiber directions.

I. INTRODUCTION

The shear stiffness of muscle tissue is a useful biomarker for characterization of muscle health and function.^1,2 Muscular tissue may be considered as a nearly incompressible elastic material, and due to its structure of mainly parallel-running fibers, it must also be modeled as transversely isotropic (TI). Of particular interest in current diagnostic practice is the change in small-strain stiffness with stretching or applied load, which results from the acoustoelastic effect of nonlinear elasticity.³ Recently, theoretical and experimental effort has focused on quantitative evaluation of the nonlinear shear stiffness of muscle tissue using ultrasound shear wave elastography techniques based on acoustoelastography principles.^4–6

In addition to the acoustoelastic effect, nonlinear elasticity in soft tissues may manifest as waveform steepening or harmonic generation in propagating shear waves. Finite-amplitude propagation effects in plane shear waves in an isotropic gelatin were first observed by Catheline et al.,⁷ and Zabolotskaya et al.⁸ then presented and analyzed a material model for plane nonlinear shear wave propagation in isotropic soft solids.⁹ In the isotropic case, nonlinear effects in plane waves appear at cubic order due to the symmetry of the deformation and material response. A similar constitutive model accounting for nonlinearity and transverse isotropy in soft solids⁶ was recently employed to describe nonlinear effects in plane shear wave propagation in those materials.¹⁰ It was found that nonlinear effects in plane wave propagation may appear at quadratic order in TI materials because fiber elasticity breaks the symmetry of the material response found in isotropic materials.^10,11 Quadratic nonlinear effects in plane shear wave propagation in TI soft solids are due to stretching and compression along the fiber direction.

For application to diagnostic elastography, propagating shear waves with finite amplitude must be excited externally at the tissue surface using a source with finite size (e.g., those employed by Gennisson et al.^12,13), thereby generating a nonplanar shear wave field subject to diffraction effects. Nonlinear shear wave beam propagation has been described in isotropic elastic solids in the paraxial, or quasiplanar, approximation,^14,15 which is similar to the method employed to obtain the KZK (Khokhlov-Zabolotskaya-Kuznetsov) equation in nonlinear acoustics. Wavefront curvature results in quadratic nonlinear effects in shear wave beam propagation in isotropic media.¹⁵ Quasiplanar nonlinear sound waves in anisotropic elastic media were modeled in the same approximation by Zabolotskaya.¹⁶

The present work provides an extension of the theory for plane nonlinear shear wave propagation in a TI soft solid¹⁰ to include effects of diffraction in the quasiplanar (paraxial) approximation. The approach employs the same scaling laws as the corresponding model for isotropic solids.^15,17 The material model, Lagrangian formalism, and ordering scheme employed in the analysis are presented in Sec. II. Two coupled nonlinear equations of motion are obtained in Sec. III for the transverse displacement components that account for elastic anisotropy along with leading order nonlinear and diffraction effects. Two special cases of the model equations are analyzed in Secs. IV and V that illustrate the combined effects of anisotropy, diffraction, and nonlinearity on shear wave propagation. Of particular interest in diagnostic elastography is the special case considered in Sec. IV, in which a configuration similar to that employed in Refs. 12 and 13 is proposed for measurement of muscle fiber nonlinear elasticity.

II. BASIC EQUATIONS

A. Transverse isotropy

The material model used in the present work to describe the nonlinear elasticity of an incompressible elastic solid with transverse isotropy is a general expansion of the strain energy density $\mathcal{E}$ up to fourth order in strain,^6,18 similar to expansions employed by Zabolotskaya et al.^8,14 for investigations of nonlinear wave motion in isotropic elastic solids:

$$\begin{array}{c}\mathcal{E}=\mu I_2+{\alpha }_1{I}_4^2+{\alpha }_2{I}_5\\ +\frac{1}{3}A{I}_3+{\alpha }_3{I}_2{I}_4+{\alpha }_4{I}_4^3+{\alpha }_5{I}_4{I}_5\\ +D{I}_2^2+{\alpha }_7{I}_2{I}_4^2+{\alpha }_8{I}_2{I}_5+{\alpha }_9{I}_4^4+{\alpha }_{10}{I}_5^2+{\alpha }_{11}{I}_3{I}_4,\end{array}$$ (1)

where

$$I_2=\text{tr}\hspace{0.17em}{\mathbf{E}}^2,\hspace{1em}{I}_3=\text{tr}\hspace{0.17em}{\mathbf{E}}^3,\hspace{1em}{I}_4=\mathbf{m}·\mathbf{Em},\hspace{1em}{I}_5=\mathbf{m}·{\mathbf{E}}^{\mathbf{2}}\mathbf{m},$$ (2)

are invariants of the Green-Lagrange finite strain tensor,

$$\mathbf{E}=\frac{1}{2}({\nabla }_0{\mathbf{u}}_0+{\nabla }_0{\mathbf{u}}_0^{\mathrm{T}}+{\nabla }_0{\mathbf{u}}_0^{\mathrm{T}}{\nabla }_0{\mathbf{u}}_0),$$ (3)

and m is the fiber direction in the undeformed configuration. The quantity ${\nabla }_0{\mathbf{u}}_0$ is the gradient of the displacement vector ${\mathbf{u}}_0$ with respect to the coordinates ${\mathbf{x}}_0={[x_0,\hspace{0.17em}{y}_0,\hspace{0.17em}{z}_0]}^{\mathrm{T}}$ of the undeformed configuration. Zero subscripts are used to distinguish the displacement ${\mathbf{u}}_0$ and coordinate ${\mathbf{x}}_0$ vectors from their scaled counterparts, u and x, to be introduced in Sec. II C. Elastic moduli μ, A, and D are the second, third, and fourth order moduli associated with isotropic soft solids,⁹ and moduli α_k account for anisotropy.

It is assumed that the material is incompressible, meaning that it is subject to the constraint that no local volume change occurs during any deformation:

$$\det \hspace{0.17em}\mathbf{F}-1=0,$$ (4)

where $\mathbf{F}=\mathbf{I}+{\nabla }_0{\mathbf{u}}_0$ is the deformation gradient tensor.

Two plane wave modes of propagation are supported in a soft TI material: SH (shear horizontal) waves with particle motion perpendicular to the fibers, and SV (shear vertical) waves with a component of particle motion in the fiber direction.¹⁹ By virtue of incompressibility, particle motion in both plane wave modes is perpendicular to the propagation direction. Thus, we define the frame of reference to have SH and SV motion polarized in the x₀ and y₀ directions, respectively, with propagation in the z₀ direction, and the fiber direction oriented in the y₀-z₀ plane,

$$\mathbf{m}=\text{sin}\hspace{0.17em}\theta \hspace{0.17em}{\mathbf{e}}_{y_0}+\text{cos}\hspace{0.17em}\theta \hspace{0.17em}{\mathbf{e}}_{z_0},$$ (5)

where θ is the angle between the propagation and fiber directions (Fig. 1). A reference frame in which the fiber direction possesses an x₀ component may be transformed by simple rotation about the z₀ axis to eliminate the x₀ component. SH and SV waves will emerge relative to the fiber direction as described by Chadwick¹⁹ regardless of the chosen reference frame. Thus, there is no loss of generality by expressing m as in Eq. (5).

FIG. 1.

(Color online) Schematic diagram of the reference frame. Quasiplanar SH and SV waves are polarized at leading order in the x₀ (red) and y₀ (blue) directions, respectively, and the fiber direction (green) lies in the y₀-z₀ plane according to Eq. (5). Wave motion is generated by a source at $z_0=0$ that has finite extent in the x₀-y₀ plane.

B. Lagrangian formalism

In similar investigations of nonlinear elastic wave mechanics,^6,10,15,17 the equations of motion are obtained by appealing to the relation for conservation of momentum as expressed in terms of a balance of material acceleration with the divergence of stress. Here, we prefer the energy-based approach of Hamilton's principle that minimizes variations in the action, or Lagrangian density. The Lagrangian density is

$$\mathcal{L}=\mathcal{T}-\mathcal{E}+p_0\left(\det \hspace{0.17em}\mathbf{F}-1\right),$$ (6)

where

$$\mathcal{T}=\frac{1}{2}\rho {\left|\frac{\partial {\mathbf{u}}_0}{\partial t}\right|}^2$$ (7)

is the kinetic energy density, ρ is the density of the material, the potential energy density $\mathcal{E}$ is defined in Eq. (1), and the hydrostatic pressure p₀ is a Lagrange multiplier of the third term of Eq. (6) that enforces the constraint of incompressibility.

For each $i=1,2,$ and 3, the $i^{\text{th}}$ equation of motion is found by varying the Lagrangian according to²⁰

$$\frac{\partial }{\partial t}\frac{\partial \mathcal{L}}{\partial (\partial u_{0,i}/\partial t)}+\frac{\partial }{\partial x_0}\frac{\partial \mathcal{L}}{\partial (\partial u_{0,i}/\partial x_0)}+\frac{\partial }{\partial y_0}\frac{\partial \mathcal{L}}{\partial (\partial u_{0,i}/\partial y_0)}+\frac{\partial }{\partial z_0}\frac{\partial \mathcal{L}}{\partial (\partial u_{0,i}/\partial z_0)}=0,\hspace{1em}i=1,2,3.$$ (8)

Together with the incompressibility condition, Eq. (4), Eqs. (8) yield four relations for the four unknown quantities $u_{0,i}$ and p₀.

C. Ordering scheme

Evaluation of Eq. (8) for a Lagrangian that accounts exactly for finite, three-dimensional deformation is not tractable. Instead, we turn to an approximation of the problem with the aim of describing leading order effects of elastic nonlinearity and wavefront curvature on shear wave propagation in a TI soft solid. As with other models of nonlinear diffracting beam propagation, we assume that deviations from plane wave propagation due to nonlinearity and diffraction are small. Further, following Wochner et al.¹⁵ and Berjamin and Destrade,¹⁷ we assume that those effects are of comparable smallness. Thus, we take the leading order problem as the linear plane wave case¹⁹ and regard the plane shear strain, $\partial u_{0,1}/\partial z$ and $\partial u_{0,2}/\partial z$, to be $O(ϵ)$ with $ϵ\ll 1$. Our aim is to obtain a set of equations of motion that are valid at $O({ϵ}^3)$. The resulting equations will describe leading order effects of nonlinearity, diffraction, and their interaction in shear wave motion in a TI soft solid.

For clarity, we introduce explicitly the small parameter ϵ to characterize the relative magnitudes of terms. While ϵ appears explicitly in the following, the same scaling strategy is employed implicitly in Refs. 8, 10, and 21, where the smallness of terms is identified easily. With the parameter ϵ included, we introduce the scaled displacement components u_i and coordinates (x, y, z) of the undeformed configuration:^15,17

$${\mathbf{u}}_0={\left[ϵu_1\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}ϵu_2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{ϵ}^2{u}_3\right]}^{\mathrm{T}},$$ (9)

$$\mathbf{x}={[ϵx_0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}ϵy_0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{z}_0]}^{\mathrm{T}}$$ (10)

$$={[x\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}y\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}z]}^{\mathrm{T}},$$ (11)

where the displacements u_i are considered functions of x.¹⁷ Leading order displacements are u₁ and u₂ for the SH and SV motions, respectively. The longitudinal component u₃ is identified as $O({ϵ}^2)$ because it arises from wavefront curvature via the condition of material incompressibility. Equation (11) denotes that gradients perpendicular to the propagation direction are much smaller than those along the propagation direction, which are on the order of the characteristic wavenumber. Thus, the displacement gradient becomes

$${\nabla }_0{\mathbf{u}}_0=\left(\begin{array}{ccccc}{ϵ}^2\frac{\partial u_1}{\partial x}& & {ϵ}^2\frac{\partial u_1}{\partial y}& & ϵ\frac{\partial u_1}{\partial z}\\ {ϵ}^2\frac{\partial u_2}{\partial x}& & {ϵ}^2\frac{\partial u_2}{\partial y}& & ϵ\frac{\partial u_2}{\partial z}\\ {ϵ}^3\frac{\partial u_3}{\partial x}& & {ϵ}^3\frac{\partial u_3}{\partial y}& & {ϵ}^2\frac{\partial u_3}{\partial z}\end{array}\right),$$ (12)

and the incompressibility condition reduces to

$$\det \hspace{0.17em}\mathbf{F}-1={ϵ}^2\left(\frac{\partial u_1}{\partial x}+\frac{\partial u_2}{\partial y}+\frac{\partial u_3}{\partial z}\right)+O\left({ϵ}^4\right)=0.$$ (13)

In the present approximation, the incompressibility condition reduces to the familiar relation $\nabla ·\mathbf{u}=0$ for linear elasticity. Note that as a result of the O(1) difference between SH and SV wave speeds, a retarded time frame is not employed here because such a transformation would yield significant error regardless of the chosen translation speed of the reference frame.

In order to obtain equations of motion consistent to $O({ϵ}^3)$ after evaluation of Eq. (8), terms in $\mathcal{L}$ that are $O({ϵ}^n)$ with $n\le 4$ are retained, and those with n > 4 discarded as negligible in the present approximation. Thus, the strain energy density is calculated according to Eqs. (1)–(3) using ${\nabla }_0{\mathbf{u}}_0$ given by Eq. (12), the kinetic energy expressed as

$$\mathcal{T}=\frac{1}{2}\rho \left[{ϵ}^2{\left(\frac{\partial u_1}{\partial t}\right)}^2+{ϵ}^2{\left(\frac{\partial u_2}{\partial t}\right)}^2+{ϵ}^4{\left(\frac{\partial u_3}{\partial t}\right)}^2\right],$$ (14)

and the pressure p₀ identified¹⁹ as $O(ϵ)$ in order to construct the Lagrangian according to Eq. (6). Care must be taken to preserve the ordering scheme in the evaluation of Eq. (8). Note that the derivative of $g=O({ϵ}^n)$ with respect to $h=O({ϵ}^m)$ is $\partial g/\partial h=O({ϵ}^{n-m})$, and that $\partial /\partial x_0$ and $\partial /\partial y_0$ are both $O(ϵ)$.

III. EQUATIONS OF MOTION

Here we derive equations of motion, consistent to $O({ϵ}^3)$, for which both transverse displacement components vary in three dimensions (3D) and the fiber direction is taken in an arbitrary direction θ in the y-z plane [Eq. (5)]. An expression is obtained for the hydrostatic pressure p₀ along with two coupled equations of motion for the transverse displacement components u₁ and u₂. The equations of motion contain terms at $O(ϵ)$ (linear plane wave terms), $O({ϵ}^2)$ (beam steering and quadratic plane wave nonlinearity terms), and $O({ϵ}^3)$ (diffraction, nonlinearity associated with wavefront curvature, and cubic plane wave nonlinearity terms). In the isotropic case there are only terms at $O(ϵ)$ and $O({ϵ}^3)$.¹⁵ It is discussed following Eq. (21) how terms that are $O({ϵ}^2)$ may be considered to be $O({ϵ}^3)$ in most cases of interest.

First, the pressure p₀ is found by evaluating Eq. (8) with i = 3. The longitudinal displacement u₃ is eliminated using the incompressibility constraint given by Eq. (13), and time derivatives are eliminated using the leading order plane wave relations:^10,19

$$ϵ\rho \frac{{\partial }^2{u}_1}{\partial t^2}=ϵ{\mu }_1\frac{{\partial }^2{u}_1}{\partial z^2}+O({ϵ}^2),$$ (15)

$$ϵ\rho \frac{{\partial }^2{u}_2}{\partial t^2}=ϵ{\mu }_2\frac{{\partial }^2{u}_2}{\partial z^2}+O({ϵ}^2),$$ (16)

where

$${\mu }_1=\mu +\frac{1}{2}{\alpha }_2\hspace{0.17em}{\text{cos}}^2\theta ,$$ (17)

$${\mu }_2=\mu +\frac{1}{2}{\alpha }_2+\frac{1}{2}{\alpha }_1\hspace{0.17em}{\text{sin}}^{2}2\theta$$ (18)

define the SH and SV plane wave propagation speeds according to ${\mu }_1=\rho c_1^2$ and ${\mu }_2=\rho c_2^2$. The resulting expression for the pressure is

$$p_0=ϵp_1\frac{\partial u_2}{\partial z}+{ϵ}^2[p_2\frac{\partial u_1}{\partial x}+p_3\frac{\partial u_2}{\partial y}+p_4{\left(\frac{\partial u_1}{\partial z}\right)}^2+p_5{\left(\frac{\partial u_2}{\partial z}\right)}^2]+O({ϵ}^3),$$ (19)

where the coefficients p_k are given in the Appendix. Errors of approximation that result from using the plane wave relations are $O({ϵ}^3)$ in the expression for pressure and thus comprise $O({ϵ}^5)$ contributions to the Lagrangian, which may be neglected. The term that is $O(ϵ)$ in Eq. (19) is associated with the linear SV plane wave. Note that the corresponding coefficient that appears in Chadwick,¹⁹ denoted by q_a in their Eqs. (3.11), differs from p₁ in Eq. (19) here by a negative sign; this is because in Ref. 19, the SV displacement is defined positive in the −y direction. There are two terms of $O({ϵ}^2)$ that are linear in strain and associated with curvature of the wavefront. Finally, nonlinear terms appear at $O({ϵ}^2)$ and are functionally the same as those in the isotropic case,¹⁷ but the coefficients are augmented to account for anisotropy.

The equation of motion for the x displacement u₁, corresponding at leading order to the SH mode, is found by evaluating Eq. (8) with i = 1 and eliminating u₃ and p₀ with Eqs. (13) and (19), respectively:

$$\begin{array}{c}ϵ\left(\rho \frac{{\partial }^2{u}_1}{\partial t^2}-{\mu }_1\frac{{\partial }^2{u}_1}{\partial z^2}\right)={ϵ}^2\left[a_1\frac{{\partial }^2{u}_1}{\partial y\partial z}+a_2\frac{{\partial }^2{u}_2}{\partial x\partial z}+{\mathcal{A}}_1\frac{\partial }{\partial z}\left(\frac{\partial u_1}{\partial z}\frac{\partial u_2}{\partial z}\right)\right]\\ +\hspace{0.17em}{ϵ}^3\{d_1\frac{{\partial }^2{u}_1}{\partial x^2}+d_2\frac{{\partial }^2{u}_1}{\partial y^2}+d_3\frac{{\partial }^2{u}_2}{\partial x\partial y}+b_1\left(\frac{\partial u_1}{\partial y}\frac{{\partial }^2{u}_2}{\partial z^2}+2\frac{{\partial }^2{u}_1}{\partial y\partial z}\frac{\partial u_2}{\partial z}\right)-2b_2\frac{{\partial }^2{u}_1}{\partial z^2}\frac{\partial u_2}{\partial y}-b_3\frac{\partial u_1}{\partial z}\frac{{\partial }^2{u}_2}{\partial y\partial z}\\ +\hspace{0.17em}{b}_4\frac{\partial u_2}{\partial x}\frac{{\partial }^2{u}_2}{\partial z^2}-b_5\frac{\partial u_2}{\partial z}\frac{{\partial }^2{u}_2}{\partial x\partial z}-b_6\left(\frac{\partial u_1}{\partial x}\frac{{\partial }^2{u}_1}{\partial z^2}+2\frac{\partial u_1}{\partial z}\frac{{\partial }^2{u}_1}{\partial x\partial z}\right)+\frac{\partial }{\partial z}[{\mathcal{B}}_1{\left(\frac{\partial u_1}{\partial z}\right)}^3+{\mathcal{B}}_{12}\frac{\partial u_1}{\partial z}{\left(\frac{\partial u_2}{\partial z}\right)}^2]\},\end{array}$$ (20)

where all coefficients except for the density ρ are combinations of elastic moduli and propagation direction and are given in the Appendix. Similarly for the y displacement u₂, which corresponds at leading order to the SV mode, Eq. (8) is evaluated with i = 2 before elimination of u₃ and p₀:

$$\begin{array}{c}ϵ\left(\rho \frac{{\partial }^2{u}_2}{\partial t^2}-{\mu }_2\frac{{\partial }^2{u}_2}{\partial z^2}\right)={ϵ}^2\left[a_3\frac{{\partial }^2{u}_1}{\partial x\partial z}+a_4\frac{{\partial }^2{u}_2}{\partial y\partial z}+\frac{1}{2}{\mathcal{A}}_1\frac{\partial }{\partial z}{\left(\frac{\partial u_1}{\partial z}\right)}^2+{\mathcal{A}}_2\frac{\partial }{\partial z}{\left(\frac{\partial u_2}{\partial z}\right)}^2\right]\\ +\hspace{0.17em}{ϵ}^3\{d_4\frac{{\partial }^2{u}_2}{\partial x^2}+d_5\frac{{\partial }^2{u}_2}{\partial y^2}+d_6\frac{{\partial }^2{u}_1}{\partial x\partial y}+b_7\left(\frac{{\partial }^2{u}_1}{\partial z^2}\frac{\partial u_2}{\partial x}+2\frac{\partial u_1}{\partial z}\frac{{\partial }^2{u}_2}{\partial x\partial z}\right)-2b_8\frac{\partial u_1}{\partial x}\frac{{\partial }^2{u}_2}{\partial z^2}-b_9\frac{{\partial }^2{u}_1}{\partial x\partial z}\frac{\partial u_2}{\partial z}\\ +\hspace{0.17em}{b}_{10}\frac{\partial u_1}{\partial y}\frac{{\partial }^2{u}_1}{\partial z^2}-b_{11}\frac{\partial u_1}{\partial z}\frac{{\partial }^2{u}_1}{\partial y\partial z}-b_{12}\left(\frac{\partial u_2}{\partial y}\frac{{\partial }^2{u}_2}{\partial z^2}+2\frac{\partial u_2}{\partial z}\frac{{\partial }^2{u}_2}{\partial y\partial z}\right)+\frac{\partial }{\partial z}[{\mathcal{B}}_{12}{\left(\frac{\partial u_1}{\partial z}\right)}^2\frac{\partial u_2}{\partial z}+{\mathcal{B}}_2{\left(\frac{\partial u_2}{\partial z}\right)}^3]\},\end{array}$$ (21)

where the expressions for all coefficients in terms of shear moduli and propagation direction appear in the Appendix.

Equations (20) and (21) are the main results of the present work. They are consistent $O({ϵ}^3)$ equations of motion for the transverse displacement components in a directional shear wave beam in a TI soft solid. The left-hand sides of each equation correspond to the leading order problem of linear plane wave propagation at an angle θ with respect to the fiber direction. The right-hand sides, which would be identically zero for linear plane wave propagation, specify the leading order effects of diffraction and nonlinearity.

Equations (20) and (21) describe the propagation of two quasiplanar shear wave modes that are coupled by nonlinearity. At leading order, the modes are polarized in the x and y directions and correspond exactly to plane SH and SV modes. Effects of diffraction and nonlinearity result in slight deviations of the polarization of each mode from the x and y directions.¹⁶

In the isotropic limit ( ${\alpha }_i\to 0$) the coefficients reduce to ${\mu }_{1,2}=\mu$, $a_{1-4}={\mathcal{A}}_{1,2}=d_{3,6}=b_{6,12}=0,\hspace{0.17em}{d}_{1,2,4,5}=\mu ,\hspace{0.17em}{b}_{1-5,7-11}=\mu +A/4$, and ${\mathcal{B}}_1={\mathcal{B}}_{12}={\mathcal{B}}_2=\mu +A/2+D$, and subsequent transformation to a retarded time frame $\tau =t-z/\sqrt{\mu /\rho }$ yields the equations derived first by Zabolotskaya¹⁴ and expressed in the limit of incompressibility.^15,22 In the plane wave limit, $\partial /\partial x=\partial /\partial y=0$, Eqs. (20) and (21) reduce to Eqs. (49) and (50) in Ref. 10.

Coefficients multiplying terms in Eqs. (20) and (21) that are $O({ϵ}^2)$ are significant only in cases where θ is not close to either $0°$ or $90°$. In these cases, the terms that are $O({ϵ}^2)$ and are linear in strain result in significant beam steering that directs propagation away from the z axis and renders the quasiplanar approximation invalid. For cases where the quasiplanar approximation is valid, i.e., θ close to $0°$ or $90°$, the coefficients $a_{1-4}$ and ${\mathcal{A}}_{1,2}$ may be considered small of $O(ϵ)$ (see the Appendix) such that all terms on the right-hand sides of Eqs. (20) and (21) are approximately of the same order.

Presented in Secs. IV and V are special cases that illustrate the effects of anisotropy and diffraction on nonlinear propagation, and for which analytical progress with Eqs. (20) and (21) is feasible.

IV. SV PROPAGATION PERPENDICULAR TO FIBERS

We take inspiration from the apparatus of Gennisson et al.^12,13 developed for transient elastography of skeletal muscle. Their apparatus comprised a thin bar connected to an electromechanical vibrator and had a single-element ultrasonic sensor embedded in the center of the bar. Vibration of the bar perpendicular to the skin generated a two-dimensional shear wave field with transverse motion polarized perpendicular to the bar, and longitudinally polarized motion in the direction of propagation that accompanied the transverse component, arising from the low compressibility of the tissue. The ultrasonic sensor measured the longitudinal component of the wave as it propagated into the tissue, permitting calculation of the shear wave speed and shear modulus. In the present section, we extend the theoretical description of such a configuration to include nonlinear effects and suggest the possibility of calculating a combination of nonlinear elastic fiber moduli from measurements of harmonic generation in the longitudinal shear wave field on axis.

The bar is approximated as infinitely long in the x direction and of finite width 2a in y, such that the resulting transverse displacement is an SV wave polarized in the y direction (Fig. 2). In muscular tissue the fibers are oriented approximately parallel to the skin, such that in this special case

$$\mathbf{m}={\mathbf{e}}_y\hspace{1em}(\theta =90°).$$ (22)

The resulting two-dimensional wave field has only SV particle motion with no variation in the x coordinate, and therefore $u_1=0\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\partial /\partial x=0$ in this case. Each term in Eq. (20) is zero, and Eq. (21) reduces to a tractable form for the SV field. The SV plane wave speed c₂ is given by [recall Eq. (18)]

$$\rho c_2^2=\mu +\frac{1}{2}{\alpha }_2\hspace{1em}(\text{SV}\hspace{0.17em}\text{mode},\hspace{0.17em}\theta =90°).$$ (23)

The expression for the pressure is, following reduction of Eq. (19),

$$p_0={ϵ}^2{\alpha }_2\frac{\partial u_2}{\partial y}+{ϵ}^2\left(\mu +\frac{1}{4}A\right){\left(\frac{\partial u_2}{\partial z}\right)}^2.$$ (24)

Simplification of Eq. (21) for this case yields, with the coefficients expanded in terms of shear moduli,

$$\begin{array}{c}ϵ\left[\rho \frac{{\partial }^2{u}_2}{\partial t^2}-\left(\mu +\frac{1}{2}{\alpha }_2\right)\frac{{\partial }^2{u}_2}{\partial z^2}\right]={ϵ}^3[\left(\mu +\frac{1}{2}{\alpha }_2+2{\alpha }_1\right)\frac{{\partial }^2{u}_2}{\partial y^2}\hspace{1em}+\left({\alpha }_2+{\alpha }_3+\frac{{\alpha }_5}{2}\right)\left(\frac{\partial u_2}{\partial y}\frac{{\partial }^2{u}_2}{\partial z^2}+2\frac{\partial u_2}{\partial z}\frac{\partial u_2}{\partial y\partial z}\right)+\hspace{0.17em}\left(\mu +\frac{A}{2}+D+\frac{{\alpha }_8}{2}+\frac{{\alpha }_{10}}{4}\right)\frac{\partial }{\partial z}{\left(\frac{\partial u_2}{\partial z}\right)}^3].\end{array}$$ (25)

FIG. 2.

(Color online) Schematic of the source motion for the special case analyzed in Sec. IV.

To elucidate anisotropy of the wave speed for this special case, ignore nonlinear terms in Eq. (25) and let $u_2\propto \hspace{0.17em}\text{exp}\hspace{0.17em}[i\omega (\mathbf{n}·\mathbf{r}/c-t)]$, where $\mathbf{n}=[\text{cos}\hspace{0.17em}\varphi ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{sin}\hspace{0.17em}\varphi ],\hspace{0.17em}\mathbf{r}=[z,\hspace{0.17em}\hspace{0.17em}y]$, and c is the propagation speed of a plane wave in the direction that differs from ${\mathbf{e}}_z$ by the angle $\varphi$. The resulting wave speed is given by

$$\rho c^2=\mu +\frac{1}{2}{\alpha }_2+2{\alpha }_1\hspace{0.17em}{\text{sin}}^2\varphi ,$$ (26)

which leads to an error proportional to $2\hspace{0.17em}{\text{sin}}^4\varphi$ compared to the exact value

$$\rho c_2^2=\mu +\frac{1}{2}{\alpha }_2+\frac{1}{2}{\alpha }_1\hspace{0.17em}{\text{sin}}^{2}2\varphi .$$ (27)

For example, the error in predicted propagation speed at $\varphi =20°$ is about 10%, which is consistent with the paraxial approximation employed here.

In the absence of SH motion, there is no longer asynchronous interaction between the SH and SV fields, and we are able to express Eq. (25) in retarded time with negligible loss of accuracy. Introduction of the retarded time ${\tau }_2=t-{ϵ}^{2}z/c_2$ and neglecting terms of $O({ϵ}^4)$ yields

$$\frac{{\partial }^2{u}_2}{\partial z\partial {\tau }_2}=\frac{c_2\eta }{2}\frac{{\partial }^2{u}_2}{\partial y^2}+\frac{\gamma }{2c_2}\left(\frac{{\partial }^2{u}_2}{\partial {\tau }_2^2}\frac{\partial u_2}{\partial y}+2\frac{\partial u_2}{\partial {\tau }_2}\frac{{\partial }^2{u}_2}{\partial y\partial {\tau }_2}\right)+\frac{{\beta }_2}{3c_2^3}\frac{\partial }{\partial {\tau }_2}{\left(\frac{\partial u_2}{\partial {\tau }_2}\right)}^3,$$ (28)

in which all terms are $O({ϵ}^3)$ and

$$\eta =1+\frac{2{\alpha }_1}{\rho c_2^2},$$ (29)

$$\gamma =\frac{{\alpha }_2+{\alpha }_3+\frac{1}{2}{\alpha }_5}{\rho c_2^2},$$ (30)

$${\beta }_2=\frac{3}{2}\frac{\mu +\frac{1}{2}A+D+\frac{1}{2}{\alpha }_8+\frac{1}{4}{\alpha }_{10}}{\rho c_2^2}.$$ (31)

There are several unique features of Eq. (28) compared to the corresponding isotropic case. The coefficient of the first term on the right-hand side contains the dimensionless factor η, which accounts for anisotropy of the wave speed in the diffracting field. In an isotropic material η = 1, but recent measurements²³ of human thigh muscle suggest $\eta \approx 8$ for that tissue. Consequently, the beam spreads more rapidly in the anisotropic case due to the perpendicular orientation of the fibers. See also further discussion after Eq. (37) regarding the diffraction length.

Second, the quadratic nonlinear term in Eq. (28) is absent in the corresponding isotropic case.¹⁵ Indeed, the coefficient γ is proportional only to fiber properties α₂, α₃, and α₅. Quadratic nonlinearity in this case is due to extension and contraction of the fibers, similar to the origins of quadratic nonlinearity in plane shear wave propagation.¹⁰ Finally, the last term in Eq. (28) represents cubic nonlinearity, which is functionally equivalent to that found in the plane wave case.¹⁰

A. Harmonic generation

Harmonic generation according to Eq. (28) is investigated via successive approximations by letting

$$u_2=\frac{1}{2}{U}_1(y,z)e^{-i\omega {\tau }_2}+\frac{1}{2}{U}_2(y,z)e^{-i2\omega {\tau }_2}+\frac{1}{2}{U}_3(y,z)e^{-i3\omega {\tau }_2}+\mathrm{c}.\mathrm{c}.,$$ (32)

where $\left|U_3\right|\ll \left|U_2\right|\ll \left|U_1\right|$ is assumed, which is validated numerically in Sec. IV B. In this section, $i=\sqrt{-1}$ is the imaginary unit, not to be confused with the subscript index employed in Secs. II and III. Separately collect terms proportional to $e^{-in\omega {\tau }_2}$, n = 1, 2, and 3, and neglect terms much smaller than the leading order (linear) terms in U_n:

$$\frac{\partial U_1}{\partial z}+\frac{\eta }{i2k}\frac{{\partial }^2{U}_1}{\partial y^2}=0,$$ (33)

$$\frac{\partial U_2}{\partial z}+\frac{\eta }{i4k}\frac{{\partial }^2{U}_2}{\partial y^2}=-\frac{i3\gamma k}{8}{U}_1\frac{\partial U_1}{\partial y},$$ (34)

$$\frac{\partial U_3}{\partial z}+\frac{\eta }{i6k}\frac{{\partial }^2{U}_3}{\partial y^2}=-i\gamma k\left(\frac{2}{3}\frac{\partial U_1}{\partial y}{U}_2-\frac{5}{12}{U}_1\frac{\partial U_2}{\partial y}\right)+\frac{i{\beta }_2{k}^3}{12}{U}_1^3,$$ (35)

where $k=\omega /c_2$. Equation (33) is solved first for a given source condition for the fundamental field U₁, which is then substituted into right-hand side of Eq. (34) in order to determine the second harmonic field U₂. Solutions for U₁ and U₂ may likewise be substituted into Eq. (35), but that analytical calculation is unwieldy and not reported here. Numerical results are presented in Sec. IV B for all three harmonics.

The source condition for a thin push-bar that oscillates perpendicular to the source plane is approximated by a Gaussian-derivative distribution of the SV displacement,

$$U_1(y,z=0)=u_0\sqrt{2e}\hspace{0.17em}\frac{y}{a}{e}^{-y^2/a^2},$$ (36)

where a is a characteristic source half-width and the factor $\sqrt{2e}$ is introduced so that the peak source amplitude is equal to u₀. This source condition is motivated by antisymmetric transverse motion that occurs near the push-bar source as a result of the low compressibility of the medium. The distribution in Eq. (36) is the derivative with respect to y of the Gaussian shaded source condition commonly used in the analysis of acoustic beam propagation. More exact source conditions related to reported experiments may be obtained using Fourier decomposition techniques.^24,25

The Gaussian-derivative source leads to analytical solutions for the fundamental and second harmonic components of the transverse displacement field. The fundamental field is

$$U_1=u_0\sqrt{2e}\frac{y/a}{{\left(1+iz/z_0\right)}^{3/2}}\text{exp}\hspace{0.17em}\left(-\frac{y^2/a^2}{1+iz/z_0}\right),$$ (37)

where $z_0=k{a}^2/2\eta$ is a characteristic diffraction length. The fundamental field U₁ exhibits near and far field behavior typical of sound beams, where the extent of the near field is characterized by the diffraction length z₀. The isotropic case²² has a corresponding diffraction length equal to $k{a}^2/2$, thus the effect of anisotropy on the fundamental field is a decrease in the extent of the near field by a factor of $\eta >1$, which results from the increased propagation speed for components of the wave field directed away from the beam axis.

The solution of Eq. (34) with U₁ given by Eq. (37) is

$$\begin{array}{c}{U}_2=\frac{3\gamma k{z}_{0}e{u}_0^2}{4a}\text{exp}\hspace{0.17em}\left(-\frac{2y^2/a^2}{1+iz/z_0}\right)[4\left(\frac{y^3}{a^3}\right)\frac{1+iz/z_0-\sqrt{1+iz/z_0}}{{(1+iz/z_0)}^4}\\ -\left(\frac{y}{a}\right)\frac{4(1-\sqrt{1+iz/z_0})+(iz/z_0)(4-\sqrt{1+iz/z_0})}{{(1+iz/z_0)}^3}].\end{array}$$ (38)

The second harmonic field contains terms within the square brackets proportional to y and to y³. Only the term proportional to y contributes to the axial longitudinal field.

The second harmonic field has anomalous far field behavior. Substituting $y=r\hspace{0.17em}\text{sin}\hspace{0.17em}\varphi$ and $z=r\hspace{0.17em}\text{cos}\hspace{0.17em}\varphi$ and taking the limit $r\to \infty$ one finds that to leading order, the magnitude is

$$\left|U_2\right|\simeq \frac{3\gamma k{u}_0^2{z}_0^4}{2a^4}\hspace{0.17em}{\text{tan}}^3\varphi \hspace{0.17em}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-2\frac{z_0^2}{a^2}\hspace{0.17em}{\text{tan}}^2\varphi \right),\hspace{1em}r\to \infty .$$ (39)

The amplitude far from the source is thus independent of propagation distance r; it is an example of perfect balance between nonlinear energy transfer and geometrical spreading, which also occurs for second harmonic generation in cylindrical sound beams in ideal fluids and in surface acoustic waves.^26,27 This feature is not significant in reality, as attenuation and the finite size of real sources influence the motion well before the steady state amplitude is reached.

In a transient elastography application, the longitudinally polarized field u₃ is measured along the z axis (y = 0) with an ultrasonic detector embedded in the center of the push bar. The longitudinal displacement is computed here from the transverse component u₂ by locally enforcing the constraint of incompressibility. In the retarded time frame and with $u_1=0$, Eq. (13) becomes

$$\frac{\partial u_2}{\partial y}-\frac{1}{c_2}\frac{\partial u_3}{\partial {\tau }_2}=0.$$ (40)

Expand the longitudinal field as

$$u_3=\frac{1}{2}{W}_1(y,z)e^{-i\omega {\tau }_2}+\frac{1}{2}{W}_2(y,z)e^{-i2\omega {\tau }_2}+\frac{1}{2}{W}_3(y,z)e^{-i3\omega {\tau }_2}+\mathrm{c}.\mathrm{c}.,$$ (41)

and then from Eq. (40) one has

$$W_n=-\frac{1}{\mathit{ink}}\frac{\partial U_n}{\partial y},\hspace{1em}n=1,2,3.$$ (42)

The expressions for the fundamental and second harmonic components along the beam axis (y = 0) are

$$W_1=\frac{i{u}_0\sqrt{2e}}{ka{(1+iz/z_0)}^{3/2}},$$ (43)

$$W_2=-\frac{3ie\gamma u_0^2{z}_0}{8a^2}\frac{4\left(1-\sqrt{1+iz/z_0}\right)+(iz/z_0)\left(4-\sqrt{1+iz/z_0}\right)}{{(1+iz/z_0)}^3}.$$ (44)

Thus, simple push-bar excitation of a nonlinear shear wave beam in this special case yields a longitudinally polarized second harmonic component that may be measured using an ultrasonic sensor embedded in the push bar.^12,13 Further, the mechanism for second harmonic generation is identified as nonlinear elasticity of the fibers and is quantified by the coefficient of nonlinearity γ given in Eq. (30). Measurements of second harmonic generation can enable quantification of γ, and Eq. (44) can be used to design future experiments. For example, near the source, the second harmonic amplitude increases linearly according to

$$W_2\approx \frac{3\gamma e{u}_0^2}{8a^2}z,\hspace{1em}z\ll z_0.$$ (45)

Initial second harmonic growth is notably independent of frequency and inversely proportional to the square of the source width. The second harmonic amplitude may be increased by decreasing the source size, although this must be balanced by the requirement $ka\gg 1$ for the quasiplanar approximation upon which Eq. (44) is based.

B. Numerical results

Numerical results for second and third harmonic generation are presented here, obtained from Eqs. (33)–(35). Equation (28) is recast in the dimensionless variables $\mathcal{U}=u_2/u_0,\hspace{0.17em}Z=z/z_0,\hspace{0.17em}Y=y/a$, and $T=\omega {\tau }_2$, where u₀ is a characteristic wave amplitude, $z_0=k{a}^2/2\eta$ the characteristic diffraction length, a a characteristic source half-width, and ω a characteristic angular frequency, yielding

$$\frac{{\partial }^2\mathcal{U}}{\partial Z\partial T}=\frac{1}{4}\frac{{\partial }^2\mathcal{U}}{\partial Y^2}+\frac{N_2}{2}\left(\frac{{\partial }^2\mathcal{U}}{\partial T^2}\frac{\partial \mathcal{U}}{\partial Y}+2\frac{\partial \mathcal{U}}{\partial T}\frac{{\partial }^2\mathcal{U}}{\partial Y\partial T}\right)+\frac{N_3}{3}\frac{\partial }{\partial T}{\left(\frac{\partial \mathcal{U}}{\partial T}\right)}^3,$$ (46)

where

$$N_2=z_0/z_2,\hspace{1em}{N}_3=z_0/z_3,$$ (47)

are dimensionless coefficients indicating the relative importance of quadratic and cubic nonlinearity, respectively, to that of diffraction. The coefficients are written in terms of the characteristic nonlinear length scales

$$z_2=\frac{a}{\gamma k{u}_0},\hspace{1em}{z}_3=\frac{1}{{\beta }_2{k}^3{u}_0^2}.$$ (48)

Substitution of $ϵ=k{u}_0$ for the strain amplitude reveals that the cubic nonlinear length scale z₃ is identical to the plane wave shock formation distance for an initially sinusoidal waveform.⁸

The dimensionless coefficients N₂ may be employed to estimate the efficiency of second harmonic generation. The second harmonic amplitude in Eq. (38) exhibits the dependence $\left|U_2\right|/u_0\propto N_2$, and the corresponding dependence of the longitudinal component in Eq. (44) is

$$\frac{\left|W_2\right|}{u_0}\propto \frac{N_2}{ka}=\frac{\gamma k{u}_0}{2\eta }.$$ (49)

Longitudinal motion results from the local requirement of volume conservation, resulting in dependence of the longitudinal harmonic amplitudes along the z axis on an additional factor of ${(ka)}^{-1}$, and therefore the longitudinal second harmonic amplitude is proportional to strain amplitude ku₀ but independent of source size ka. Additionally, the initial growth on axis of the longitudinal second harmonic, given by Eq. (45), is characterized by $\left|W_2\right|/u_0\propto z/z_2$.

The dimensionless forms of Eqs. (33)–(35) are

$$\frac{\partial {\mathcal{U}}_1}{\partial Z}+\frac{1}{4i}\frac{{\partial }^2{\mathcal{U}}_1}{\partial Y^2}=0,$$ (50)

$$\frac{\partial {\mathcal{U}}_2}{\partial Z}+\frac{1}{8i}\frac{{\partial }^2{\mathcal{U}}_2}{\partial Y^2}=-\frac{i3N_2}{8}{\mathcal{U}}_1\frac{\partial {\mathcal{U}}_1}{\partial Y},$$ (51)

$$\frac{\partial {\mathcal{U}}_3}{\partial Z}+\frac{1}{12i}\frac{{\partial }^2{\mathcal{U}}_3}{\partial Y^2}=-i{N}_2\left(\frac{2}{3}\frac{\partial {\mathcal{U}}_1}{\partial Y}{\mathcal{U}}_2-\frac{5}{12}{\mathcal{U}}_1\frac{\partial {\mathcal{U}}_2}{\partial Y}\right)+\frac{i{N}_3}{12}{\mathcal{U}}_1^3,$$ (52)

where ${\mathcal{U}}_n=U_n/u_0$ are the dimensionless harmonic amplitudes. Equations (50)–(52) are solved by operator splitting,²⁸ with the contribution from diffraction calculated in the spatial frequency domain.²⁹ The formula for the longitudinal component in dimensionless notation,

$$\frac{W_n}{u_0}=-\frac{1}{\mathit{inka}}\frac{\partial {\mathcal{U}}_n}{\partial Y},$$ (53)

is then evaluated numerically.

Harmonic amplitude curves are presented for two source conditions, the Gaussian-derivative given by Eq. (36), and a perhaps more realistic source condition based on a super-Gaussian radial weighting,

$${\mathcal{U}}_1(Y,Z=0)={(2me)}^{1/2m}Y{e}^{-Y^{2m}},$$ (54)

where m is a positive integer, not to be confused with the fiber direction vector m, and ${(2me)}^{1/2m}$ is a normalization factor that yields $|{\mathcal{U}}_1{|}_{\text{max}}=1$ at Z = 0. Equation (54) is equivalent to Eq. (36) for m = 1. For m > 1, the source condition given by Eq. (54) has width approximately equal to 2, and becomes increasingly N-shaped with increasing m.

Numerical results for harmonic generation in the longitudinally polarized component along the beam axis are presented in Fig. 3. Source parameters used for simulation are $\omega =2\pi \times 200$ s^{− 1}, $u_0=100$ μm, a = 25 mm; linear material parameters for human thigh²³ are $c_2=2.4$ m/s and η = 8; nonlinear parameters are simply chosen to be O(1) in lieu of measured values: γ = 2 and ${\beta }_2=2$. For this case, the diffraction length is $z_0\approx 30$ mm, with $N_2=0.2,\hspace{0.17em}{N}_3=0.3$, and ka = 20. The first column of Fig. 3 shows the source condition used to generate the curves in each row, and the second through fourth columns show the amplitude of the longitudinally polarized harmonic components $W_n/u_0$ for each case. The upper and lower rows have m = 1 and m = 4 in the source condition given by Eq. (54), respectively. The curves in Figs. 3(b) and 3(c) correspond to Eqs. (43) and (44), respectively.

FIG. 3.

(Color online) Amplitudes of the three harmonic components in the longitudinally polarized field along the beam axis, obtained numerically using Eqs. (50)–(53) with $N_2=0.2,\hspace{0.17em}{N}_3=0.3$, and ka = 20, and for the source condition given by Eq. (54) with m = 1 (upper row) and m = 4 (lower row).

While the harmonic amplitude curves with m = 1 (upper row of Fig. 3) vary smoothly with propagation distance, those with m = 4 (lower row) exhibit rapid oscillations in the near field ( $Z\ll 1$). Rapid oscillations near the source for m > 1 result from the interference of the direct and edge waves (no distinct edge wave exists for m = 1), and this phenomenon is also common to the acoustic field resulting from ultrasonic piezoelectric transducers. The impact on the peak amplitudes of the second and third harmonic components is twofold: they are increased as both the amplitude of the fundamental and the transverse (Y) gradients are increased in the near field, and they occur closer to the source as the fundamental field achieves its maximum amplitude at approximately Z = 0.27. For the parameters used to generate Figs. 3(g) and 3(h), the peak second and third harmonic amplitudes occur at approximately z = 10 mm and z = 11 mm, respectively. As mentioned following Eq. (28), note that for the case at hand $\left|W_2\right|$ is less than $\left|W_1\right|$ by an order of magnitude, and similarly for $\left|W_3\right|$ relative to $\left|W_2\right|$.

V. SH BEAM PROPAGATION IN THE y-z PLANE

The special case analyzed in the present section illustrates the combined effects of nonlinearity, diffraction, and anisotropy on the propagation of SH waves by considering source motion in only the x direction and with no variation in the x direction [Fig. 4(a)]. If nonlinear generation of SV waves by SH waves is further neglected due to the asynchronicity of that interaction,¹⁰ one obtains $u_2=0$, and Eq. (20) reduces to

$$\frac{{\partial }^2{u}_1}{\partial t^2}-c_1^2\frac{{\partial }^2{u}_1}{\partial z^2}=2c_1^2\nu \frac{{\partial }^2{u}_1}{\partial y\partial z}+c_1{c}_{\perp }\frac{{\partial }^2{u}_1}{\partial y^2}+\frac{1}{2}{\beta }_1{c}_1^2\frac{\partial }{\partial z}{\left(\frac{\partial u_1}{\partial z}\right)}^3,$$ (55)

where $\rho c_1^2=\mu +({\alpha }_2/2)\hspace{0.17em}{\text{cos}}^2\theta ,$

$$\nu =\frac{{\alpha }_2\hspace{0.17em}\text{sin}\hspace{0.17em}2\theta }{4\rho c_1^2},$$ (56)

$$c_{\perp }=\frac{\mu +({\alpha }_2/2)\hspace{0.17em}{\text{sin}}^2\theta }{\rho c_1},$$ (57)

and ${\beta }_1=3{\mathcal{B}}_1/2\rho c_1^2$ is the coefficient of nonlinearity for plane SH waves.¹⁰

FIG. 4.

(Color online) (a) Schematic representation of the example investigated in Sec. V, showing source motion (red arrows) and fiber direction (green arrow). (b) Beam steering parameter ν versus fiber direction θ. (c) Ratio of anisotropic diffraction length to the isotropic diffraction length. Grey regions in (b) and (c) indicate fiber directions for which beam steering renders the paraxial approximation invalid. Parameters used in (b) and (c) are for in vivo human thigh: (Ref. 23) $\mu =1.9$ kPa, ${\alpha }_1=19.7$ kPa, and ${\alpha }_2=7.8$ kPa. (d) Third harmonic amplitude ${\widehat{V}}_3=|V_3(y=\nu z)|/(k_1{z}_R{\beta }_1{v}_0^3/4c_1^2)$ along the beam axis $y=\nu z$.

Anisotropy of the wave speed is examined by substituting into the linear terms of Eq. (55) $u_1\propto \hspace{0.17em}\text{exp}\hspace{0.17em}[i\omega (\mathbf{n}·\mathbf{r}/c-t)]$ as before. The resulting dispersion relation is

$$\rho c^2=\mu +\frac{1}{2}{\alpha }_2(\hspace{0.17em}{\text{cos}}^2\theta \hspace{0.17em}{\text{cos}}^2\varphi +\text{sin}\hspace{0.17em}2\theta \hspace{0.17em}\text{sin}\hspace{0.17em}\varphi \hspace{0.17em}\text{cos}\hspace{0.17em}\varphi +{\text{sin}}^2\theta \hspace{0.17em}{\text{sin}}^2\varphi ),$$ (58)

which leads to an error proportional to $\text{sin}\hspace{0.17em}2\theta \hspace{0.17em}\text{sin}\hspace{0.17em}2\varphi$ compared to the exact directional dependence $\rho c^2=\mu +({\alpha }_2/2){\text{cos}}^2(\theta -\varphi )$.

Equation (55) may be recast in a moving frame with ${\tau }_1=t-{ϵ}^{2}z/c_1$:

$$\frac{{\partial }^2{v}_1}{\partial z\partial {\tau }_1}=-\nu \frac{{\partial }^2{v}_1}{\partial y\partial {\tau }_1}+\frac{c_{\perp }}{2}\frac{{\partial }^2{v}_1}{\partial y^2}+\frac{{\beta }_1}{3c_1^3}\frac{{\partial }^2{v}_1^3}{\partial {\tau }_1^2},$$ (59)

where $v_1=\partial u_1/\partial {\tau }_1$. Note that the first term on the right-hand side is $O({ϵ}^2)$ while all other terms in the equation are $O({ϵ}^3)$. As shown in Sec. V A, Eq. (59) is valid in the quasiplanar approximation only if $\nu =O(ϵ)$, and therefore all terms in the equation are comparable in magnitude for meaningful cases.

A. Harmonic generation

An analytical solution is available for third harmonic generation from a time harmonic source with Gaussian amplitude shading. The solution is assumed to have the form

$$v_1=\frac{1}{2}{V}_1{e}^{-i\omega {\tau }_1}+\frac{1}{2}{V}_3{e}^{-i3\omega {\tau }_1}+\mathrm{c}.\mathrm{c}.,$$ (60)

with $\left|V_3\right|\ll \left|V_1\right|$. The coupled differential equations to be solved sequentially are

$$\frac{\partial V_1}{\partial z}+\nu \frac{\partial V_1}{\partial y}+\frac{1}{i2k_{\perp }}\frac{{\partial }^2{V}_1}{\partial y^2}=0,$$ (61)

$$\frac{\partial V_3}{\partial z}+\nu \frac{\partial V_3}{\partial y}+\frac{1}{i6k_{\perp }}\frac{{\partial }^2{V}_3}{\partial y^2}=\frac{i\omega {\beta }_1}{4c_1^3}{V}_1^3,$$ (62)

where $k_{\perp }=\omega /c_{\perp }$. For the Gaussian-shaded source condition $V_1(0)=v_0\hspace{0.17em}\text{exp}\hspace{0.17em}(-y^2/a^2)$ and $V_3(0)=0$, the solutions are

$$V_1=\frac{v_0}{\sqrt{1+iz/z_R}}\text{exp}\hspace{0.17em}\left[-\frac{{(y-\nu z)}^2/a^2}{1+iz/z_R}\right],$$ (63)

$$V_3=\frac{k_1{z}_R{\beta }_1{v}_0^3}{4c_1^2}\frac{\hspace{0.17em}\text{log}\hspace{0.17em}(1+iz/z_R)}{\sqrt{1+iz/z_R}}\text{exp}\hspace{0.17em}\left[-\frac{3{(y-\nu z)}^2/a^2}{1+iz/z_R}\right],$$ (64)

where $z_R=k_{\perp }{a}^2/2$ and $k_1=\omega /c_1$.

The solution differs from the isotropic case in two ways. The first is the combination of terms $(y-\nu z)$ in the exponentials of each solution, which corresponds to steering of the beam away from the z axis during propagation that results from anisotropy of the wave speed. Note that $\nu =\text{tan}\hspace{0.17em}\psi$ where ψ is the angle between the z axis and the beam axis defined by the maximum of $\left|V_1\right|$. In accordance with the quasiplanar approximation, which is consistent with the paraxial approximation, one must have $\left|\psi \right|\lesssim 15°$– $20°$, or equivalently $\left|\nu \right|\lesssim 0.26$–0.33. The parameter ν is plotted versus fiber angle θ in Fig. 4(b) for parameters corresponding to measurements of in vivo human thigh muscle.²³ The gray regions indicate fiber angles θ that result in extreme deviations of the propagation direction from the z direction such that the quasiplanar approximation is violated.

The second difference is the form of the diffraction length z_R, which differs from the isotropic value $z_R^{\text{iso}}$ by the factor

$$\frac{z_R}{z_R^{\text{iso}}}=\frac{\mu +({\alpha }_2/2)\hspace{0.17em}{\text{cos}}^2\theta }{\mu +({\alpha }_2/2)\hspace{0.17em}{\text{sin}}^2\theta },$$ (65)

where $z_R^{\text{iso}}=k_1{a}^2/2$ is the diffraction length if variation of the wave speed across a wavefront is ignored. Thus anisotropy can lead to defocusing ( $z_R<z_R^{\text{iso}}$, θ close to $90°$) or something like self-focusing ( $z_R>z_R^{\text{iso}}$, θ close to $0°$). The latter case is not truly self-focusing, because the amplitude of the fundamental remains a monotonically decreasing function of z, but the rate of decrease with propagation distance is slower than in the isotropic case. The quantity $z_R/z_R^{\text{iso}}$ is presented as a function of θ in Fig. 4(c). The gray regions correspond to those in Fig. 4(b), and indicate where anisotropy renders the quasiplanar assumption invalid. For fiber directions $-30°\lesssim \theta \lesssim 30°$, one has $z_R>z_R^{\text{iso}}$ such that anisotropy increases the extent of the near field. For $\theta \approx \pm 90°$, one has $z_R<z_R^{\text{iso}}$ such that anisotropy causes the beam to spread rapidly, thus decreasing the extent of the near field.

The third harmonic propagates in the same direction as the fundamental field, and the dependence on z is likewise scaled everywhere by the factor z_R, which depends on fiber direction θ. Near the source and along the beam axis ( $y=\nu z$) the third harmonic grows as $V_3\approx -i{k}_1{\beta }_1{v}_0^{3}z/4c_1^2\propto c_1^{-3}$. The peak amplitude of V₃, which occurs at $z/z_R\approx 3.47$, is $\left|V_{3,\text{max}}\right|\approx 0.96\hspace{0.17em}{k}_1{z}_R{\beta }_1{v}_0^3/4c_1^2\propto c_1^{-2}$. Similarly, a characteristic nonlinear length scale corresponds to the shock formation distance for plane SH waves, ${\overline{z}}_1=c_1^2/{\beta }_1{k}_1{v}_0^2\propto c_1^3$, which may be compared to characteristic length scales of attenuation and diffraction to assess the relative significance of each effect in experiments, where a shorter characteristic length indicates that nonlinearity is of greater importance during propagation. Thus, nonlinearity is most efficient in this case, in terms of initial growth rate and eventual peak amplitude of the third harmonic, when the wave speed c₁ is a minimum at $\theta =90°$.

Numerical solutions of Eq. (59) are presented in Fig. 5. The solutions are obtained in the frequency domain with an algorithm based on the approach employed by Yuldashev and Khokhlova,²⁹ altered to include effects of anisotropy and cubic nonlinearity. The spatial variation of the fundamental and third harmonic amplitudes resulting from a Gaussian shaded source with frequency 200 Hz, width a = 13 mm, and amplitude $v_0=250$ mm/s are shown, along with time waveforms along the beam axis $y=\nu z$. Linear material parameters are for human thigh,²³ and the coefficient of nonlinearity is taken to be ${\beta }_1=2$. The number of harmonics used in each calculation is 250. A small amount of viscous attenuation is included for stabilization of shocks by including the term ${\alpha }_{\text{SH}}{\partial }^2{v}_1/\partial {\tau }_1^2$, with ${\alpha }_{\text{SH}}=0.00014$ mm⁻¹, appended to the right-hand side of Eq. (59).

FIG. 5.

(Color online) Numerical solution of Eq. (59) for propagation direction at 0°, 20°, and 90° relative to the fiber direction m. Linear material parameters are for human thigh (Ref. 23) and the coefficient of nonlinearity is ${\beta }_1=2$. Source parameters used are frequency 200 Hz, width a = 13 mm, and amplitude $v_0=250$ mm/s. Time waveforms in the right-most column are located along the beam axis at $y=\nu z$.

VI. SUMMARY AND DISCUSSION

A model for plane nonlinear shear wave propagation in soft elastic media with transverse isotropy¹⁰ is extended to include leading order effects of wavefront curvature. The material model is a general expansion of the strain energy density up to fourth order in the strain with material incompressibility assumed.⁶ The ordering scheme employed is common to other quasiplanar approximations used in the study of nonlinear sound beams in fluids and solids,¹⁶ and nonlinear shear wave beams in isotropic solids.^15,17 Coupled nonlinear equations of motion derived using Hamilton's principle describe the influence of nonlinearity, anisotropy, and wavefront curvature on wave propagation. It is found that fiber elasticity results in quadratic nonlinear terms that are unique to the case accounting for anisotropy and diffraction. Two special cases are considered: one involving only SV waves that has potential utility in nonlinear elastography applications, and one involving only SH waves that is illustrative of the combined effects of nonlinearity, anisotropy, and diffraction.

Not considered in this work, but inherently significant in elastography applications in soft tissues, is material viscoelasticity. Viscoelastic effects result in the frequency dependence of attenuation and shear wave speed in soft tissues, in addition to directional dependence induced by material anisotropy. Furthermore, due to the ratchet-type mechanism of muscle fiber contraction, it is thought that viscoelasticity in muscle tissue may itself be nonlinear.³⁰ Viscoelastic effects such as stress relaxation must be included in the presented framework by considering a three-dimensional approach.³¹

The material model used in this work was preferred due to its generality and because it extends previous analyses of nonlinear shear waves in isotropic solids.^8,15 The model may be compared to other approaches, such as continuum models that allow for less general dependence of the strain energy on strain¹¹ or composite models that consider the medium to be composed of separate fiber and background components.³²

Finally, future applications of the present work may be related to the special case detailed in Sec. IV of harmonic generation of SV waves with $\theta =90°$. This analysis is the nonlinear extension of that of Gennisson et al.,^12,13 which employed transient elastography in muscle with a thin push bar ( $\omega a/c_2\approx 0.1$). Notable features of the nonlinear case include a coefficient of quadratic nonlinearity proportional only to fiber elastic properties, and frequency independence of the growth rate of the second harmonic component of the longitudinal displacement along the beam axis. The use of sources with larger values of $\omega a/c_2$ along with focusing may be employed to improve the signal-to-noise ratio and enhance the measurement. Thus, second harmonic generation in SV beams may be employed in nonlinear elastography applications, with γ [Eq. (30)] a specific biomarker for nonlinear fiber elasticity.

Future investigations can involve further analytical treatments and new numerical approaches for solving Eqs. (20) and (21) for cases of interest. Analysis of focusing effects, similar to ongoing investigations in isotropic media,²⁴ are of interest for increasing robustness of measurements. Numerical approaches may be used to investigate deviations from the special conditions assumed in Sec. IV, for example, cases in which the fiber angle is $\theta \lesssim 90°$ as may be encountered in elastography applications.²³ Incorporation of viscoelastic effects may be performed using the technique detailed by Saccomandi and Vianello.³¹

APPENDIX: COEFFICIENTS IN TERMS OF MODULI AND FIBER DIRECTION

Listed here are expressions for the coefficients appearing in Eqs. (19), (20), and (21) in terms of the elastic moduli defined in Eq. (1) and the angle θ between the z axis and the fiber direction as defined in Eq. (5).

Coefficients in the expression for hydrostatic pressure, Eq. (19):

$$p_1=\text{sin}\hspace{0.17em}2\theta \left({\alpha }_1\hspace{0.17em}{\text{cos}}^2\theta +\frac{1}{2}{\alpha }_2\right),$$ (A1)

$$p_2=-{\text{cos}}^2\theta \left(2{\alpha }_1\hspace{0.17em}{\text{cos}}^2\theta +{\alpha }_2\right),$$ (A2)

$$p_3=-({\alpha }_1+{\alpha }_2)\hspace{0.17em}\text{cos}\hspace{0.17em}2\theta -{\alpha }_1\hspace{0.17em}\text{cos}\hspace{0.17em}4\theta ,$$ (A3)

$$p_4=\mu +\frac{A}{4}+\left({\alpha }_2+\frac{1}{2}{\alpha }_3\right)\hspace{0.17em}{\text{cos}}^2\theta +\left({\alpha }_1+\frac{1}{4}{\alpha }_5\right)\hspace{0.17em}{\text{cos}}^4\theta ,$$ (A4)

$$p_5=\mu +\frac{A}{4}+\left({\alpha }_2+\frac{1}{2}{\alpha }_3+\frac{5}{4}{\alpha }_5\hspace{0.17em}{\text{sin}}^2\theta \right)\hspace{0.17em}{\text{cos}}^2\theta +\left({\alpha }_1+3{\alpha }_4\hspace{0.17em}{\text{sin}}^2\theta +\frac{1}{4}{\alpha }_5\right)\hspace{0.17em}{\text{cos}}^4\theta .$$ (A5)

Coefficients in the equation of motion for SH waves, Eq. (20):

$${\mu }_1=\mu +\frac{1}{2}{\alpha }_2\hspace{0.17em}{\text{cos}}^2\theta ,$$ (A6)

$${\mathcal{A}}_1=\frac{1}{2}\text{sin}\hspace{0.17em}2\theta \left[\left(2{\alpha }_1+\frac{1}{2}{\alpha }_5\right)\hspace{0.17em}{\text{cos}}^2\theta +{\alpha }_2+{\alpha }_3\right],$$ (A7)

$$a_1=\frac{1}{2}{\alpha }_2\hspace{0.17em}\text{sin}\hspace{0.17em}2\theta ,$$ (A8)

$$a_2=-\text{sin}\hspace{0.17em}2\theta \left({\alpha }_1\hspace{0.17em}{\text{cos}}^2\theta +\frac{1}{4}{\alpha }_2\right),$$ (A9)

$$d_1=\mu +\frac{1}{2}{\alpha }_2\hspace{0.17em}{\text{cos}}^2\theta +2{\alpha }_1\hspace{0.17em}{\text{cos}}^4\theta ,$$ (A10)

$$d_2=\mu +\frac{1}{2}{\alpha }_2\hspace{0.17em}{\text{sin}}^2\theta ,$$ (A11)

$$d_3=\left({\alpha }_1+\frac{1}{2}{\alpha }_2\right)\text{cos}\hspace{0.17em}2\theta +{\alpha }_1\hspace{0.17em}\text{cos}\hspace{0.17em}4\theta ,$$ (A12)

$$b_1=\mu +\frac{A}{4}+\frac{1}{2}{\alpha }_2+\left(2{\alpha }_1+\frac{1}{2}{\alpha }_5\right)\hspace{0.17em}{\text{cos}}^2\theta \hspace{0.17em}{\text{sin}}^2\theta ,$$ (A13)

$$b_2=\mu +\frac{A}{4}-\frac{1}{2}{\alpha }_3\hspace{0.17em}{\text{sin}}^2\theta +\left[{\alpha }_2+\frac{1}{2}{\alpha }_3-\left({\alpha }_1+\frac{1}{4}{\alpha }_5\right)\hspace{0.17em}{\text{sin}}^2\theta \right]\hspace{0.17em}{\text{cos}}^2\theta +\left({\alpha }_1+\frac{1}{4}{\alpha }_5\right)\hspace{0.17em}{\text{cos}}^4\theta ,$$ (A14)

$$b_3=\mu +\frac{A}{4}-\left(\frac{1}{2}{\alpha }_2+{\alpha }_3\right)\hspace{0.17em}{\text{sin}}^2\theta +\left[\frac{3}{2}{\alpha }_2+{\alpha }_3-(4{\alpha }_1+{\alpha }_5)\hspace{0.17em}{\text{sin}}^2\theta \right]\hspace{0.17em}{\text{cos}}^2\theta +2\left({\alpha }_1+\frac{1}{4}{\alpha }_5\right)\hspace{0.17em}{\text{cos}}^4\theta ,$$ (A15)

$$b_4=\mu +\frac{A}{4}+\frac{1}{2}({\alpha }_2+{\alpha }_5\hspace{0.17em}{\text{sin}}^2\theta )\hspace{0.17em}{\text{cos}}^2\theta ,$$ (A16)

$$b_5=\mu +\frac{A}{4}+\left(\frac{3}{2}{\alpha }_2+{\alpha }_3+2{\alpha }_5\hspace{0.17em}{\text{sin}}^2\theta \right)\hspace{0.17em}{\text{cos}}^2\theta +\left(2{\alpha }_1+\frac{1}{2}{\alpha }_5+6{\alpha }_4\hspace{0.17em}{\text{sin}}^2\theta \right)\hspace{0.17em}{\text{cos}}^4\theta ,$$ (A17)

$$b_6=({\alpha }_2+{\alpha }_3)\hspace{0.17em}{\text{cos}}^2\theta +\left(2{\alpha }_1+\frac{1}{2}{\alpha }_5\right)\hspace{0.17em}{\text{cos}}^4\theta ,$$ (A18)

$${\mathcal{B}}_1=\mu +\frac{1}{2}A+D+\left({\alpha }_2+{\alpha }_3+\frac{1}{2}{\alpha }_8\right)\hspace{0.17em}{\text{cos}}^2\theta +\left({\alpha }_1+\frac{1}{2}{\alpha }_5+\frac{1}{4}{\alpha }_{10}\right)\hspace{0.17em}{\text{cos}}^4\theta ,$$ (A19)

$$\begin{array}{c}{\mathcal{B}}_{12}=\mu +\frac{1}{2}A+D+{\alpha }_1{\hspace{0.17em}\text{cos}\hspace{0.17em}}^4\theta +\left({\alpha }_2+{\alpha }_3+\frac{1}{4}{\alpha }_{10}\right){\hspace{0.17em}\text{cos}\hspace{0.17em}}^2\theta \\ +3{\alpha }_4{\hspace{0.17em}\text{cos}\hspace{0.17em}}^4\theta {\hspace{0.17em}\text{sin}\hspace{0.17em}}^2\theta +\frac{1}{8}{\alpha }_5{\hspace{0.17em}\text{cos}\hspace{0.17em}}^2\theta (7-3\hspace{0.17em}\text{cos}\hspace{0.17em}2\theta )+{\alpha }_7{\hspace{0.17em}\text{cos}\hspace{0.17em}}^2\theta {\hspace{0.17em}\text{sin}\hspace{0.17em}}^2\theta +\frac{1}{8}{\alpha }_8(3+\text{cos}\hspace{0.17em}2\theta ).\end{array}$$ (A20)

Coefficients in the equation of motion for SV waves, Eq. (21):

$${\mu }_2=\mu +\frac{1}{2}{\alpha }_2+\frac{1}{2}{\alpha }_1\hspace{0.17em}{\text{sin}}^{2}2\theta ,$$ (A21)

$${\mathcal{A}}_2=\frac{3}{4}\text{sin}\hspace{0.17em}2\theta \left(2{\alpha }_1\hspace{0.17em}{\text{cos}}^2\theta +{\alpha }_2+{\alpha }_3+\frac{1}{2}{\alpha }_4\hspace{0.17em}{\text{sin}}^{2}2\theta +\frac{1}{2}{\alpha }_5\right),$$ (A22)

$$a_3=a_2=-\text{sin}\hspace{0.17em}2\theta \left({\alpha }_1\hspace{0.17em}{\text{cos}}^2\theta +\frac{1}{4}{\alpha }_2\right),$$ (A23)

$$a_4=-{\alpha }_1\hspace{0.17em}\text{sin}\hspace{0.17em}4\theta ,$$ (A24)

$$d_5=\mu +\frac{1}{4}{\alpha }_1\left(1+7\hspace{0.17em}\text{cos}\hspace{0.17em}4\theta \right)+\frac{1}{2}{\alpha }_2,$$ (A25)

$$d_4=d_2=\mu +\frac{1}{2}{\alpha }_2\hspace{0.17em}{\text{sin}}^2\theta ,$$ (A26)

$$d_6={\text{cos}}^2\theta \left[-{\alpha }_1\left(1-3\hspace{0.17em}\text{cos}\hspace{0.17em}2\theta \right)+\frac{1}{2}{\alpha }_2\right],$$ (A27)

$$b_7=\mu +\frac{A}{4}+\frac{1}{2}\left({\alpha }_2+{\alpha }_5\hspace{0.17em}{\text{sin}}^2\theta \right)\hspace{0.17em}{\text{cos}}^2\theta ,$$ (A28)

$$b_8=\mu +\frac{A}{4}+{\text{cos}}^2\theta \left({\alpha }_2+\frac{1}{2}{\alpha }_3+\frac{5}{4}{\alpha }_5\hspace{0.17em}{\text{sin}}^2\theta \right)+{\text{cos}}^4\theta \left({\alpha }_1+\frac{1}{4}{\alpha }_5+3{\alpha }_4\hspace{0.17em}{\text{sin}}^2\theta \right),$$ (A29)

$$b_9=\mu +\frac{A}{4}+\left(\frac{3}{2}{\alpha }_2+{\alpha }_3\right)\hspace{0.17em}{\text{cos}}^2\theta +2\left({\alpha }_1+\frac{1}{4}{\alpha }_5+3{\alpha }_4\hspace{0.17em}{\text{sin}}^2\theta \right)\hspace{0.17em}{\text{cos}}^4\theta +\frac{1}{2}{\alpha }_5\hspace{0.17em}{\text{sin}}^{2}2\theta ,$$ (A30)

$$b_{10}=\mu +\frac{A}{4}+\frac{1}{2}{\alpha }_2+\frac{1}{2}\left({\alpha }_1+\frac{1}{4}{\alpha }_5\right)\hspace{0.17em}{\text{sin}}^{2}2\theta ,$$ (A31)

$$b_{11}=\mu +\frac{A}{4}+\frac{1}{2}{\alpha }_2+\frac{1}{4}\left({\alpha }_1+\frac{1}{4}{\alpha }_5\right)\left(1+\frac{1}{4}\text{cos}\hspace{0.17em}2\theta +3\hspace{0.17em}\text{cos}\hspace{0.17em}4\theta \right)+({\alpha }_2+{\alpha }_3)\hspace{0.17em}\text{cos}\hspace{0.17em}2\theta ,$$ (A32)

$$b_{12}=\left({\alpha }_1+{\alpha }_2+{\alpha }_3+\frac{3}{8}{\alpha }_4+\frac{1}{2}{\alpha }_5\right)\text{cos}\hspace{0.17em}2\theta +{\alpha }_1\hspace{0.17em}\text{cos}\hspace{0.17em}4\theta -\frac{3}{8}{\alpha }_4\hspace{0.17em}\text{cos}\hspace{0.17em}6\theta ,$$ (A33)

$$\begin{array}{c}{\mathcal{B}}_2=\mu +\frac{1}{2}A+D+{\alpha }_1{\hspace{0.17em}\text{cos}\hspace{0.17em}}^4\theta +\left({\alpha }_2+{\alpha }_3\right){\hspace{0.17em}\text{cos}\hspace{0.17em}}^2\theta +6{\alpha }_4{\hspace{0.17em}\text{cos}\hspace{0.17em}}^4\theta {\hspace{0.17em}\text{sin}\hspace{0.17em}}^2\theta +\frac{1}{2}{\alpha }_5{\hspace{0.17em}\text{cos}\hspace{0.17em}}^2\theta \left(3-2\hspace{0.17em}\text{cos}\hspace{0.17em}2\theta \right)\\ +2{\alpha }_7{\hspace{0.17em}\text{cos}\hspace{0.17em}}^2\theta {\hspace{0.17em}\text{sin}\hspace{0.17em}}^2\theta +\frac{1}{2}{\alpha }_8+4{\alpha }_9{\hspace{0.17em}\text{cos}\hspace{0.17em}}^4\theta {\hspace{0.17em}\text{sin}\hspace{0.17em}}^4\theta +\frac{1}{4}{\alpha }_{10}.\end{array}$$ (A34)