Function for the horizontal layered seismic field is generally derived by
Function for the horizontal layered seismic field is normally derived by reflectivity technique, which was proposed by Fuchs and M ler [1] and extended to many other types [2], like reflection and transmission coefficient matrix system [3], discrete wavenumber approach [4], discrete wavenumber finite element approach [5], and generalized reflection transmission coefficient matrix process [6]. Inside the frequency domain, the Green function, derived by the reflectivity system, could be written in the Sommerfeld integral form in cylindrical coordinate method for symmetrical media. It really is well known that the numerical evaluation of Sommerfeld integral (SI) is computationally costly because of the oscillatory and slow convergence with the integrands. To overcome this issue, a number of approaches happen to be proposed, which may be divided into two principal categories: one could be the approximation from the spatial domain Green functions within a closed kind where no numerical integration is needed, along with the other may be the numerical integration of SI in conjunction with some acceleration methods [7]. Inside the initial category, discrete complex image approach (DCIM), which approximates the integrand of Sommerfeld integral by a series of complex exponential functions, is commonly utilised for the positive Hydroxyflutamide site aspects of high computational efficiency, however it requires to deal with the surface wave poles contributions, which not only makes the computation complex but additionally brings singularity towards the near area [8], as well as the calculation accuracy and helpful variety are tough to be accurately estimated. For the latter category, the popular practice is dividing the entire Sommerfeld integral into two parts: the very first element is definitely the path to bypass the singularity; the second aspect, the path to infinity, is the Sommerfeld tail integral. The finite-range integrals may possibly readily be evaluated by the Gauss acobi quadrature [9] or byPublisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Copyright: 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is definitely an open access article distributed below the terms and situations of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).Symmetry 2021, 13, 1969. https://doi.org/10.3390/symhttps://www.mdpi.com/journal/symmetrySymmetry 2021, 13,two ofthe double-exponential (DE) rule [10,11]. Mosig 1st utilised DE guidelines to calculate Sommerfeld integrals in [11,12] which indicated its validity of suppressing endpoint singularities. The calculation of tail integral is tough to converge as a result of Bessel’s oscillation and slow attenuation characteristics, so this sort of process normally requires extrapolation to accelerate convergence. The WA strategy has shown greater levels of convergence amongst a variety of extrapolation solutions [135]. This sort of method will not should strictly find the position of singularity in Sommerfeld integral but only requires to make sure that the very first integral path avoids each of the singularities. It has good adaptability and BI-0115 medchemexpress controllable numerical accuracy; on the other hand, this is dependent upon the number of intervals (n) that are selected to evaluate the tail region. The computational time also quickly increases because the value of (n) increases [16]. Offered the benefits and disadvantages of these two solutions, we propose a system by combining DE guidelines and DCIM to calculate Sommerfeld integral. This paper very first presents the Green function of a point supply in.