Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 9471581, 7 pages

https://doi.org/10.1155/2017/9471581

## A Study on the Scattering Energy Properties of an Elastic Spherical Shell in Sandy Sediment Using an Improved Energy Method

College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin 150001, China

Correspondence should be addressed to GuangPing Zhu

Received 23 September 2016; Accepted 22 December 2016; Published 13 February 2017

Academic Editor: Andrea Crivellini

Copyright © 2017 WenKai Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An elastic wave is composed of compressional (longitudinal) waves and shear (transverse) waves which have different wave velocities in solids. The acoustic field presents complex interference patterns which means its phenomena and properties are difficult to reveal. Fortunately, the energy method is more accurate than the potential function approach in describing the physical properties of the acoustic field. However, the polarization state of particle vibration excited by an elastic wave is spatially periodic in the wave propagation direction. Therefore, the energy propagation direction is not consistent with the wave propagation direction using commonly used energy method. According to the polarization state of particle vibration, a time-space averaging method based on the spatial periodicity of energy flux in the solid is proposed. The method could eliminate the influence of the interference due to local energy exchange and retain the trend of energy propagation. Several conclusions are illustrated through the analysis of the scattering energy properties of a steel shell in sandy sediment. Sandy sediment can not be regarded as a fluid nor a general solid. Scattering energy excited by an incident shear wave mainly concentrates in the vicinity of the directions of backscattering and forward scattering. Especially, at low frequency, it plays an important role in the total scattering energy excited by an incident compressional and shear wave.

#### 1. Introduction

The acoustic field in solids presents a complex interference pattern because the compressional wave and the shear wave have different propagation velocities which both have effects on particle velocity, although the usual energy method is more accurate than the potential function approach in describing the physical properties of acoustic fields. As yet no such energy method is applicable to describe the energy propagation in solids because the energy propagation direction is not consistent with the wave propagation direction using usual energy methods.

The problem of the scattering field in solids has been studied in detail by scholars [1–3]. But there is rare research on the acoustic energy propagation. The energy density and energy flux vectors derived with the aid of Hamilton’s principle and Umov-Poynting’s theorem is obtained by Pierce [4]. The time-average energy density is obtained from the projection of the average power flow vector onto the propagation direction by Carcione and Cavallini [5]. The further study found that the relevant physical phenomena are related to the energy flow direction rather than to the propagation direction [6]. Time averaged Umov-Poynting vector is defined as acoustic intensity by Ainslie and Burns [7] when discussing the energy of reflected wave and transmitted wave at the interface of solid. An algorithm is used for calculating Umov-Poynting vector and the pressure force exerted by the nonparaxial cylindrical Gaussian wave on a circular microcylinder by Kotlyar et al. [8]. The acoustic power radiated by plates in bending vibration is estimated using Statistical Energy Analysis by Rumerman [9]. The instantaneous Umov-Poynting vector field is shown in both freeze-frame and animated versions in Dean and Braselton’s research [10–12]. The Umov-Poynting vector is applied to solve problems of thermal conductivity in any media specifying the medium and values of heat fluxes at the interface by Kuts [13]. The statements of the elasticity problem about the wave propagation in an elastic cylindrical waveguide under different radiation conditions at infinity are considered by Nazarov [14]. The sound radiation from elastically restrained plates covered by a decoupling layer is studied using the Spectrogeometric Method by Wang et al. [15].

Present researches are focusing on the time average of the instantaneous energy flux. But, the physical image of energy propagation is too complex to understand this way. This paper presents a time-space average method by which the physical image and the properties of acoustic energy propagation in the solid are accurately described. The elastic scattering energy of a spherical shell in sandy sediment especially is analyzed using this method.

#### 2. Instantaneous Scattering Energy Flux of Elastic Spherical Shell

Each of the physical quantities involved in the acoustic energy flux are derived from potential functions. The vector Helmholtz equation is introduced in order to obtain the analytical solution of the scattered acoustic field, where is the displacement vector in solids and it can be decomposed into the following form by field theory:where , , and satisfy the vector Helmholtz equation. The vector solutions are related to the scalar functions as follows:where satisfy the scalar Helmholtz equation.

The scattering displacement vector is given by

The undetermined coefficients are obtained using the boundary conditions for displacement and stress. The simplified scattering displacement in the far field is derived using an approximation for the Hankel function:where is the compressional wavenumber while is the shear wavenumber. From expressions (4)–(6), the radial displacement is the compressional component of the scattered wave, while the tangential and circumferential displacement is the shear component of the scattered wave. The scattered field has three displacement components and six stress components. It is difficult to reveal more scattering rules even if the expressions are very succinct. In this paper we introduce the acoustic Umov-Poynting vector:where** v** is particle velocity and** T** is stress tensor. The acoustic Umov-Poynting vector characterizes the combined action of the particle velocity and the stress tensor. It represents the spatial instantaneous acoustic energy flux in solids.

#### 3. Particle Polarization and the Time-Space Average Method

The acoustic energy flux vector is the instantaneous energy flux as a time-space function. The observation at different time or spatial locations differs. For example, acoustic energy flux vectors of the incident plane wave and the spherical scattered wave at time zero are shown in Figure 1.