Research paper
Scattering asymmetry parameters for a circular cylinder in arbitrary–shaped acoustical sheets

https://doi.org/10.1016/j.cnsns.2021.106022Get rights and content

Highlights

  • Quadratic scattering asymmetry parameters are derived in the context of the acoustic scattering.

  • An elastic cylinder material submerged in a non-viscous fluid is considered.

  • Acoustical sheets of arbitrary shape (or beam in 2D) demonstrate the results for Airy and Gaussian beams.

  • Longitudinal and transverse scattering asymmetry parameters and their resonance counterparts are defined.

  • The results are of particular importance in scattering applications.

Abstract

Quantitative quadratic physical observables pertaining to the longitudinal and transverse scattering asymmetry parameters are described in the context of the acoustic scattering theory, and their generalized mathematical expressions are derived for a cylinder material submerged in a non-viscous fluid and located arbitrarily in the field of an acoustical sheet of arbitrary shape (or beam in 2D). The derived partial-series expansions in cylindrical coordinates depend on the beam-shape coefficients of the acoustical sheet and the scattering coefficients of the cylinder. Depending on its sign, the longitudinal scattering asymmetry parameter is used to quantify the amount of energy scattered in either the forward (θ=0) or backward (θ=π) directions, while the sign of its transverse counterpart identifies the energy scattered in either the positive (θ=π/2) or negative (θ = –π/2) lateral directions, respectively. For symmetric scattering with respect to θ=±π/2, the longitudinal and transverse scattering asymmetry parameters vanish. Moreover, for a symmetric acoustical sheet centered on the cylinder, the transverse scattering asymmetry parameter is zero, while its longitudinal counterpart can alternate between negative, positive or neutral values depending on particle size and acoustical-sheet parameters. The analysis is extended to derive the expressions for the resonance scattering asymmetry parameters by removing a nonresonant smooth “background” coefficient corresponding to the totally rigid- or soft-body scattering. Numerical results for the longitudinal and transverse scattering asymmetry parameters and their resonance counterparts considering an elastic cylinder placed arbitrarily in nonparaxial symmetric Gaussian and asymmetric Airy acoustical sheets chosen as examples illustrate the analysis, with particular emphasis on the cylinder size and acoustical-sheet parameters. The results are of particular importance in scattering applications, imaging, particle characterization, negative (pulling) radiation force in tractor beams and acoustical tweezers, acoustic radiative transfer and remote sensing, and other related fields.

Introduction

The study of the scattering of acoustical waves by objects is of fundamental importance for numerous applications, encompassing imaging, acoustical tweezers, acoustic communication, particle characterization, remote sensing etc. An especially interesting quantitative physical characteristic is defined as the asymmetry parameter, originally introduced in the optical context [1] to calculate the time-averaged optical radiation force due to scattering by the particle. The (longitudinal) asymmetry parameter is the integral of the cosine-weighted intensity phase function [2], [3] and is a measure of the amount of the scattered energy along the axis of wave propagation. It was also introduced in acoustics [4], [5], [6] as a quantitative observable proportional to a force factorκ[4], where an explicit connection between the time-averaged acoustic radiation force and the scattering cross-section was established [4], [7], [8].

Accurate calculations of the asymmetry parameter are of particular importance in scattering and related phenomena, as shown in numerous investigations in optics [9], [10], [11], [12] to compute the albedo of snow and ice [13], the scattering of nonspherical particles [14], densely-packed grains [15], [16], particle aggregates [17], [18] and mineral dust aerosols [19] to name a few examples. The (longitudinal) asymmetry parameter characterizes the forward energetic scattering directivity of a scatterer, as well as the forward linear momentum carried by the scattered waves.

A specific particle geometry is the elongated cylinder, which is often considered for the modeling of the scattering and mechanical effects of sound waves from tubular structures, nanotubes [20], [21], and fibers [22], [23]. Earlier works have considered the induced time-averaged (quadratic/nonlinear) radiation force and torque of arbitrary-shaped acoustical sheets with a cylinder from the standpoint of particle manipulation and transport, such as the nonparaxial Gaussian [24], [25], [26], Hermite-Gaussian [27] and Airy [28] sheets. Nonetheless, a thorough physical analysis and exact formalism related to the quadratic acoustical scattering asymmetry parameters for cylindrical structures in finite acoustical-sheets are yet to be developed. Important questions that arise from the standpoint of acoustical scattering theory and related applications remain, especially those concerned with the numerical predictions related to the directionality of the scattering energy efficiency from the target under consideration, and the utility of the scattering asymmetry parameters.

This analysis addresses those challenges based upon the partial-wave series expansion method in cylindrical coordinates, and establishes a rigorous formulation stemming from the projections along the direction of wave propagation of the linear momentum transport associated with the scattering field, and the direction transverse to it. Although earlier works investigated the longitudinal scattering asymmetry parameter for circular [29] and elliptic [6] cylinders, a complete analysis accurately describing the transverse scattering asymmetry parameter as well as the resonance components is required. For a comprehensive analysis of the scattering, both longitudinal and transverse asymmetry parameters and their resonance counterparts must be calculated and used adequately in the design of experimental methods and optimization analyses, which are provided in the following.

Section snippets

Analysis of the linear acoustic scattering and quadratic scattering asymmetry parameters and their resonance counterparts – elastic cylinder example in a lossless fluid

Consider an acoustical-sheet of arbitrary wavefront in two-dimensions (2D) characterized by its dimensionless (complex) beam-shape coefficients (BSCs) bn, and propagating in a lossless fluid of density ρ and speed of sound c. A time-dependence in the form of eiωt is assumed, but omitted from the linear field equations for convenience. An elastic cylinder is located arbitrarily in space facing the acoustical-sheet, with its z-axis perpendicular to the plane in which the waves propagate (Fig. 1

Numerical computations and results

Eqs. (3)–(6) and their individual components given by Eqs. (8), (9) constitute the main contribution of this work. Notice that these equations depend on the dimensionless cylinder coefficients Cn as well as the BSCs that are related to the dimensionless Cartesian coordinates kx and ky, as shown subsequently in Eqs. (16), (17) for the nonparaxial Airy and Gaussian acoustical sheets. Numerical implementation and computations of those exact partial-wave series expressions for the longitudinal and

Conclusion

In this analysis, mathematical expressions for the quadratic asymmetry parameters related to the longitudinal and transverse energetic scattering of an elastic cylinder immersed in a non-viscous liquid have been derived analytically and computed numerically, using a generalized multipole solution applicable to any acoustical-sheet of arbitrary wavefront. The analysis of the scattering asymmetry parameters has been also examined from the standpoint of the resonance scattering theory, and

CRediT authorship contribution statement

F.G. Mitri: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Writing - review & editing, Validation, Writing – original draft.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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