Wave propagation is one of essential interests in several fields of electro-magnetism and acoustics

Wave propagation is one of essential interests in several fields of electro-magnetism and acoustics. The scattering of elastic acoustic waves by dielectric spherical obstacle has been studied by many researchers for last years. There has been a continuous attention on the concentrating properties of scattered waves by dielectric spheres of several wave-lengths in diameter. The medium that supports the sound wave is a squeezable inviscid fluid are of less complex nature (J. H. Lopes(a), 2016). In the great sphere scattering, there is outstanding feature to overcome the diffraction-limit in which width of the beam must be greater than the wavelength. Moreover, the optical ?eld near to the surface of the sphere provides high intensity. These striking properties have enhanced the concept of photonic jet (Zhigang Chen, 2004). Which emerges close to ?eld scattering of laser irradiation by means of a dielectric sphere of several wavelength in diameter. Much interest has been clearly concentrated on improving photonic jets in utilized research (Dholakia & Reece, 2006). The wonderful physical characteristics of photonic jets, such as concentrate of subwavelength and high intensity has encouraged us to study the acoustic analog of this phenomenon theoretically (Minin, 2017).
The initial experimental researches of the characteristics of acoustic jets were described in acoustic wave physics years ago (E. P. Mednikov and B. G. Novitskiœ, 1975), (I. V. Lebedeva, 1980). In one investigate (E. P. Mednikov and B. G. Novitskiœ, 1975), the jet was generated by a low frequency resource of a vibration-resonance kind. The apparatus was prepared and studied in relation with the practical mission of powder spraying. In the other investigate (I. V. Lebedeva, 1980), the jets were formed as a product of the propagation of intensive sound through a hole in a screen placed in the cross section of the waveguide. This study was achieved in relation with the watching of the nonlinear absorption of intensive sound by resonant systems (S. P. Dragan and I. V. Lebedeva, 1998).It is possible to formulate the acoustic plan wave scattering by using the spectral element method (SEM).
Spectral element method used to compute the fluid dynamic by Patera (PATERA, 1984). He suggested that SEM merges the precision of spectral element method (the status where P-type method is used for a one element domain) with the pliability of the finite element method (FEM). In the SEM, he applied high order for lagrangian polynomial interpolant on Chebyshev collocation points to represent the speed of all elements in the computational domain.
Spectral element method has been newly applied to find a solution to electromagnetic scattering problems governed by Maxwell’s equations because of its high accuracy (I. Mahariq H. I., 2014). They presented that the maximum proportional error is calculated by SEM with Legendre polynomials being the interpolation functions. They applied Green’s function to study the degree of accuracy on single elements at dissimilar aspect ratios. And then solve the problem of real electromagnetic scattering. They discovered in the elements having straight side, the error was more than that of elements with one side being curved, as the aspect ratio increase. But, the degree of accuracy is highly decreased when the aspect ratio increases because there are mixed elements of straight and curved sides in the electromagnetic scattering problem.
SEM are supposed as a family of approximation schemes according to Galerkin method. Discretization the computational domain is common characteristic between SEM and FEM, and this gives the reason why they can be offered as p- or h-versions of finite element method. That is, when offered as h-version, a Lagrangian interpolation equation on the parent element exists in both, in addition to the shape function has local support. On the other hand, SEM utilizes high degree polynomials on a steady geometric mesh in order to enhanced accuracy, and this is the reality characterizing the p-version of the FEM. (Deville, 2002).
As a comparison between SEM and other numerical methods FEM and FDM, these numerical methods used to solve specified problems and the identical error calculations are subjected to a specified error measure. From the results, it is obvious that SEM is much accuracy than the other numerical methods. (I. Mahariq, 7 july 2015)

For simulations associated with wave propagation in unlimited domains, truncation of the domain is required, to limited computational resources. In the last years the ‘Perfectly Matched Layers’ (PML) have become the optimum absorbing boundaries for the elastic wave equation, due to its flexibility and good efficiency as a comparison with other conditions. The PML was presented by Berenger as a boundary condition for wave propagation problems in electromagnetism. (BERENGER, 1994). The mechanism comprise of surrounding the material domain of attention with an absorbing layer of limited width such that all leaving waves are damped out regardless of their direction of propagation and frequency. The most important property of the PML is that, before discretization, it does not enter reflections at the interface between the physical material and the layer that means perfectly matched in spite of the fact that a small reflection is constantly present after discretization. Being a material absorbing state, the PML leads surely to scattered systems, it is well suited for accomplishment in parallel computers, and therefore, very appealing for numerical simulations. (Papageorgiou, june 2012)