To calculate the transmission coefficient of ultrasonic waves through a multi-layered medium, a new approach is proposed by expanding it into Debye’s series. Using this formalism, the transmission coefficient can be put in the form of resonance terms series. From this point of view, the relative amplitude of the transmitted wave can be considered as an infinite summation of terms taking into account all possible reflections and refractions on each interface. Our model is then used to investigate interaction between the ultrasonic plane wave and the N-plane-layer structure.
Obviously, the resulting infinite summation has to be reduced to a finite one, according to some level of accuracy. The numerical estimation of the transmission coefficient using the exact expression (Eq. (1)) is then compared to the one of our method in the case of two or three plane-layer structure. The effect of the order of the finite summation on the calculated value of the transmission coefficient is, as well, studied. Finally, our proposed method may be used, with the decomposition into Gaussian beams of a pressure field created by a circular source, to draw a 3D image of the pressure field transmitted through a multilayered structure.

Problems concerning structures dynamics are being one of most important subjects in recent investigations associated with railways constructions. The need of modelling of such structures and their behaviour prediction leads to necessity of seeking new approaches, mainly due to highly increasing speeds of vehicles and traffic intensity. Comparative studies carried out on experimental data, measurements and theoretical research show that models based on multi-layered approach supported by semi-analytical approximations of solutions can give new insight into undertaken analyses. More detailed consideration of roads components and their physical properties, along with application of effective estimations allowing to avoid numerical instabilities linked with extremal dynamic variations, can be important tools in obtaining new solutions both, theoretical and engineering. This paper briefly presents a number of multilayer railway track models, with special emphasis on nonlinear track properties. Existing analytical and semi-analytical solution methods are presented with main advantages of new approaches. The theoretical double-beam system with two nonlinear layers is solved and computational examples are presented along with possibility of their transition to other multilayer structures analysis.