Mathematical treatment of wave propagation in acoustic waveguides with n curved interfaces

There are some curved interfaces in ocean acoustic waveguides. To compute wave propagation along the range with some marching methods, a flattening of the internal interfaces and a transforming equation are needed. In this paper a local orthogonal coordinate transform and an equation transformation are constructed to flatten interfaces and change the Helmholtz equation as a solvable form. For a waveguide with a flat top, a fiat bottom and n curved interfaces, the coefficients of the transformed Helmholtz equation are given in a closed formulation which can be thought of as an extension of the formal work related to the equation transformation with two curved internal interfaces. In the transformed horizontally stratified waveguide, the one-way reformulation based on the Dirichlet-to-Neumann （DtN） map is then used to reduce the boundary value problem to an initial value problem. Numerical implementation of the resulting operator Riccati equation uses a large range step method to discretize the range variable and a truncated local eigenfunction expansion to approximate the operators. This method is particularly useful for solving long range wave propagation problems in slowly varying waveguides. Furthermore, the method can also be applied to wave propagation problems in acoustic waveguides associated with varied density.

LOCAL ORTHOGONAL TRANSFORMATION FOR ACOUSTIC WAVEGUIDE

In this paper, a local orthogonal transformation is created to transform the Helmholtz waveguide with curved interface to the one with a flat interface within the two layer medium, and the Helmholtz equation u xx +u zz +κ 2(x,z)u=0 is transformed to V +αV +β V +γV=0 . Numerical results demonstrate that the transformation is more feasible. This transformation is particularly useful for the research on wave propagation in acoustic waveguide.

GENERALIZED REVERBERATION MATRIX FORMULATION FOR WAVE PROPAGATION IN MULTILAYERED MEDIUM

Generalized reverberation matrix （GRM） formulation is presented to investigate elastic wave propagation in a complex multilayered solid by the combination of reverberation-ray matrix （RRM） method and stiffness matrix （SM） method. RRM method formulates a reverberation matrix, which reflects the reflection or refraction of the elastic waves in the multilayered solid. However, the dimension of RRM increases as the sublayer number increases, which may result in lower calculation efficiency of the generalized rays. SM formulation yields a system matrix of the constant dimension to promise higher calculation efficiency, but it is difficult to identify the generalized rays. In order to calculate the generalized rays in the complex multi-layered solid efficiently, the RRM formulation is applied to the interested sublayer for the evaluation of the generalized rays and SM formulation to the other sublayers, to construct a generalized reverberation matrix of the constant dimension, which is independent of the sublayer number. Numerical examples show that GRM formulation has higher calculation efficiency for the generalized rays in the complex multilayered-solid configuration compared with RRM formulation.

Sound scattering from underwater elastic spherical shell covered with multilayered medium approximated acoustic cloak

The anechoic performance and mechanism of underwater elastic spherical shell covered with coating are studied at low frequencies.The acoustic cloak is anisotropic material,which can be designed with homogeneous isotropic materials on the basis of effective medium approximation theory.The analytic expression of scattering acoustic field from the shell covered with multilayered medium is formulated and the scattering form function,resonance mode,acoustic field distribution are computed,the scattering characteristics and mechanism of transmission are analyzed.The results show that the direction of sound transmission inside the multilayered medium is changed,the acoustic field is deflected gradually,and the acoustic energy flux is guided around the target,which reduces the scattering intensity at low frequencies,the acoustic intensity of target＇s surface is very weak.Excepting the first resonance peak in spectrum produced by the zero order partial wave,the other resonance modes of elastic spherical shell are not excitated and the multilayered medium can suppress the resonance of the spherical shell effectively.

Axisymmetric loading on nanoscale multilayered media

Multilayered nanoscale structures are used in several applications.Because the effect of surface energy becomes nontrivial at such a small scale,a modified continuum theory is required to accurately predict their mechanical behaviors.A Gurtin–Murdoch continuum model of surface elasticity is implemented to establish a computational scheme for investigating an elastic multilayered system under axisymmetric loads with the incorporation of surface/interface energy.Each layer stiffness matrix is derived based on the general solutions of stresses and displacements obtained in the form of the Hankel integral transform.Numerical solutions to the global equation,which are formulated based on the continuity conditions of tractions and displacements across interfaces between layers,yield the displacements at each layer interface and on the top surface of the multilayered medium.The numerical solutions indicate that the elastic responses of multilayered structures are affected significantly by the surface material properties of both the top surface and interfaces,and that they become size dependent.In addition,the indentation problem of a multilayered nanoscale elastic medium under a rigid frictionless cylindrical punch is investigated to demonstrate the application of the proposed solution scheme.