Over the last 15 years, literature on nondestructive testing has shown that the generation of higher harmonics and nonlinear mixing of waves could be used to obtain the nonlinearity parameters of an elastic medium and thereby gather information about its state, e.g., aging and fatigue. To design ultrasound measurement setups based on these phenomena, efficient numerical modeling tools are needed. In this paper, the iterative nonlinear contrast source method for numerical modeling of nonlinear acoustic waves is extended to the one-dimensional elastic case. In particular, nonlinear mixing of two collinear bulk waves (one compressional, one shear) in a homogeneous, isotropic medium is considered, taking into account its third-order elastic constants ([Formula: see text]). The obtained results for nonlinear propagation are in good agreement with a benchmark solution based on the modified Burgers equation. The results for the resonant waves that are caused by the one-way and two-way mixing of primary waves are in quantitative agreement with the results in the literature [Chen, Tang, Zhao, Jacobs, and Qu, J. Acoust. Soc. Am. 136(5), 2389–2404 (2014)]. The contrast source approach allows the identification of the propagating and evanescent components of the scattered wavefield in the wavenumber-frequency domain, which provides physical insight into the mixing process and explains the propagation direction of the resonant wave.

In nondestructive testing, the generation of higher harmonics and the mixing of elastic waves are used to measure the nonlinearity parameters, which in turn are closely related to the microscopic state of the material. A 3-D numerical tool that can simulate large scale, four-dimensional, nonlinear elastic wavefields would be very useful for the reliable interpretation of experimental results. Here, a method is presented that can efficiently compute the nonlinear displacement fields in a homogeneous, isotropic elastic medium, taking into account its third-order elastic constants (A, B, C). The method is based on the Neumann iterative solution of an integral equation involving a Green's function of the linear `background' medium, and a contrast source representing the nonlinearity of the medium. The integral equation is solved iteratively, and the computations are based on Fast Fourier Transforms using a sampling rate close to the Nyquist limit, i.e., two grid points of the shortest wavelength. The displacement fields are evaluated using the scalar and vector potential functions representing the compressional and shear displacements. In this presentation, the suitability of the INCS method for modeling the harmonics of 3-D nonlinear elastic waves and its validation by comparison with an analytical benchmark solution, will be presented.

In nondestructive testing, nonlinear wave mixing could be used to obtain the nonlinearity parameters of an elastic medium and thereby get information about its state, e.g., aging and fatigue. To better understand the mixing mechanisms and optimize the design of measurement setups, a physics-oriented tool for the simulation of nonlinear elastic wave propagation would be valuable. In this presentation, we extend the iterative nonlinear contrast source method (INCS) to study the nonlinear mixing of two plane, collinear bulk waves (one compressional, one shear) in a homogeneous, isotropic, elastic medium with two independent coefficients of nonlinearity (βL and βT). The method successfully captured the resonant wave generated due to the mixing (one-way and two-way) of primary waves of different frequencies. The obtained results for the resonant wave were in good agreement with the results reported in the literature. In addition, the contrast source allowed the propagating and evanescent components of the scattered wave field to be studied in the wavenumber-frequency domain, which provides physical insight into the mixing process and explains the propagation direction of the scattered wave. Thus, the INCS method seems to be a useful tool to investigate and predict wave mixing in nonlinear elastic media.

The behavior of nature is usually modelled with Differential Equations in various forms. Depending on the constrains and the accuracy of a model, the connected equations may be more or less complicated. For simple models we may use Non Homogeneous Equations but in general, we have to deal with Homogeneous ones since from a physicists point of view nature seems to be Homogeneous. In many applications of sciences, for solving many of them, often appear equations of type nth order Linear differential equations, where the number of them is Euler-Cauchy differential equations. i.e. Euler-Cauchy differential equations often appear in analysis of computer algorithms, notably in analysis of quick sort and search trees; a number of physics and engineering applications. In this paper, the researcher aims to present the solutions of a homogeneous Euler-Cauchy differential equation from the roots of the characteristics equation related with this differential equation using MATLAB. It is hoped that this work can contribute to minimize the lag in teaching and learning of this important Ordinary Differential Equation.

In recent days, the data are transformed in the form of multimedia data such as images, graphics, audio and video. Multimedia data require a huge amount of storage capacity and transmission bandwidth. Consequently, data compression is used for reducing the data redundancy and serves more storage of data. In this paper, addresses the problem (demerits) of the lossy compression of images. This proposed method is deals on SVD Power Method that overcomes the demerits of Python SVD function. In our experimental result shows superiority of proposed compression method over those of Python SVD function and some various compression techniques. In addition, the proposed method also provides different degrees of error flexibility, which give minimum of execution of time and a better image compression.