Cervenka, M., Bednarik, M.: Optimized reactive silencers with narrow side-branch tubes , The Journal of the Acoustical Society of America 144(4), 2015-2021, 2018
Apstract: This paper presents a theoretical study of the sound propagation in a waveguide loaded by an array of flush-mounted narrow side-branch tubes, forming a simple low-frequency reactive silencer. The individual tube-lengths and the distances between the adjacent tubes are optimized in order to maximize the minimum transmission loss over a given frequency range. The transmission properties of the silencer are calculated using the transfer matrix method, heuristic evolutionary approach is employed for the determination of the optimal parameters. The numerical results are validated against the finite element method simulation. A comprehensive parametric study is performed to demonstrate the optimized silencer performance as a function of the number of side-branch tubes, and the frequency range. It is shown that for the given frequency range, the minimum transmission loss of the optimized silencer increases linearly with the number of the side-branch tubes.
Bednarik, M., Cervenka, M.: Propagation of electromagnetic waves through non-uniform dielectric layers, Journal of the Optical Society of America 35(10), 2541-2549, 2018.
The propagation of TE- and TM-polarized electromagnetic (EM) waves through a dielectric layer with spatial variation of the refractive index is reported in this paper. The relative permittivity of the layer is assumed to be graded along the thickness direction, and its spatial variation is described by a quartic polynomial. The corresponding mode equations are Helmholtz equations with variable coefficients that can be transformed to a triconfluent Heun equation, a local exact solution of which is expressed in terms of triconfluent Heun functions. The solutions cover many particular cases owing to a variability of four optional parameters (coefficients) of the quartic polynomial. The general local solution for a TE-polarized EM wave is employed for the calculation of transmission properties of a periodic one-dimensional photonic crystal using the Floquet theory.
Bednarik, M., Cervenka, M., Groby, J.-P., Lotton, P.: One-dimensional propagation of longitudinal elastic waves through functionally graded materials , International Journal of Solids and Structures 146, 43-54, 2018
Apstract: The one-dimensional propagation of longitudinal elastic waves along the thickness of a plate made of functionally graded materials excited by a harmonic force is reported in this article. The material properties of the plate are assumed to be graded along the thickness direction according to a trigonometric law distribution. This distribution smoothly connects the material properties of the upper and lower homogeneous materials that bounds the plate. The corresponding propagation equation is Ince-type equation that can be transformed to Heun’s equation a local exact solution of which is expressed in terms of local Heun functions. The general nature of these functions is demonstrated based on four degenerate cases of Heun’s equation. The transfer matrix method is used to study the elastic waves propagating in the inhomogeneous domain. The calculation of the transfer matrices requires the evaluation of the general solution in the interval containing two regular singular points. For this purpose, the modified Heun function is introduced. Based on the transfer matrices, the influence of both the asymmetry of the unit cell and various constituent materials on the transmission coefficient spectrum is studied. The transmission coefficient is also calculated for the locally periodic structures with the help of the Chebyshev polynomials.
Cervenka, M., Bednarik, M.: Numerical study of the influence of the convective heat transport on acoustic streaming in a standing wave , The Journal of the Acoustical Society of America 143(2), 727-734, 2018
Apstract: Within this work, acoustic streaming in an air-filled cylindrical resonator with walls supporting a temperature gradient is studied by means of numerical simulations. A set of equations based on successive approximations is derived from the Navier-Stokes equations. The equations take into account the acoustic-streaming-driven convective heat transport; as time-averaged secondary-field quantities are directly calculated, the equations are much easier to integrate than the original fluid-dynamics equations. The model equations are implemented and integrated employing commercial software COMSOL Multiphysics. Numerical calculations are conducted for the case of a resonator with a wall-temperature gradient corresponding to the action of a thermoacoustic effect. It is shown that due to the convective heat transport, the streaming profile is considerably distorted even in the case of weak wall-temperature gradients. The numerical results are consistent with available experimental data.
Bednarik, M., Cervenka, M.: The exact solution of the Schrödinger equation with a polynomially spatially varying mass, Journal of Mathematical Physics 58, 072103, 2017
Apstract: The Schrödinger equation with a position-dependent mass (SEPDM) is employed in many areas of quantum physics. Exact solutions for the SEPDM lie at the center of interest of the professional public because it helps us to understand the behavior of quantum particles in the cases in which their mass varies spatially. For this purpose, we used the mass function represented by a quartic polynomial and a quadratic potential function, which extends the current class of exact solutions of the SEPDM. The exact analytical solution of the problem is expressed as a linear combination of local Heun functions. Heun’s equation contains many parameters, resulting in its general nature. We studied how limit changes in some of these parameters will affect the solution of the SEPDM. The obtained solutions are particularly suitable for the transfer matrix method and solutions of scattering problems; this is demonstrated by the calculation of bound states.
Cervenka, M., Bednarik, M.: Effect of inhomogeneous temperature fields on acoustic streaming structures in resonators, The Journal of the Acoustical Society of America 141(6), 4418-4426, 2017
Apstract: Acoustic streaming in 2D rectangular resonant channels filled with a fluid with a spatial temperature distribution is studied within this work. An inertial force is assumed for driving the acoustic field; the temperature inhomogeneity is introduced by resonator walls with prescribed temperature distribution. The method of successive approximations is employed to derive linear equations for calculation of primary acoustic and time-averaged secondary fields including the streaming velocity. The model equations have a standard form which allows their numerical integration using a universal solver; in this case, COMSOL Multiphysics was employed. The numerical results show that fluid temperature variations in the direction perpendicular to the resonator axis influence strongly the streaming field if the ratio of the channel width and the viscous boundary layer thickness is big enough; the streaming in the Rayleigh vortices can be supported as well as opposed, which can ultimately lead to the appearance of additional vortices.
Bednarik, M., Cervenka, M.: Description of waves in inhomogeneous domains using Heun’s equation, Waves in Random and Complex Media 2017
Abstract: There are a number of model equations describing electromagnetic, acoustic or quantum waves in inhomogeneous domains and some of them are of the same type from the mathematical point of view. This isomorphism enables us to use a unified approach to solving the corresponding equations. In this paper, the inhomogeneity is represented by a trigonometric spatial distribution of a parameter determining the properties of an inhomogeneous domain. From the point of view of modeling, this trigonometric parameter function can be smoothly connected to neighboring constant-parameter regions. For this type of distribution, exact local solutions of the model equations are represented by the local Heun functions. As the interval for which the solution is sought includes two regular singular points. For this reason, a method is proposed which resolves this problem only based on the local Heun functions. Further, the transfer matrix for the considered inhomogeneous domain is determined by means of the proposed method. As an example of the applicability of the presented solutions the transmission coefficient is calculated for the locally periodic structure which is given by an array of asymmetric barriers.
Cervenka, M., Bednarik, M.: Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration 383, 76-88, 2016
Abstract: This work deals with acoustic bandpass filters realized by shaped waveguide-elements inserted between two parts of an acoustic transmission line with generally different characteristic impedance. It is shown that the formation of a wide passband is connected with the eigenfrequency spectrum of the filter element which acts as an acoustic resonator and that the required filter shape substantially depends on whether the filter characteristic impedance is higher or lower than the characteristic impedance of the waveguide. It is further shown that this class of filters can be realized even without the need of different characteristic impedance. A heuristic technique is proposed to design filter shapes with required transmission properties; it is employed for optimization of low-frequency bandpass filters as well as for design of bandpass filters with wide passband surrounded by wide stopbands as it is typical for phononic crystals, however, in this case the arrangement is much simpler as it consists of only one simple-shaped homogeneous element.
Cervenka, M., Bednarik, M.: Variety of acoustic streaming in 2D resonant channels, Wave Motion 66, 21-30, 2016.
Abstract: Acoustic streaming in 2D resonant channels with uniform or non-uniform cross-sections is studied within this work. An inertial force as well as a vibrating boundary are assumed for driving the acoustic field. The method of successive approximations is employed to derive linear equations for calculation of primary acoustic and time-averaged secondary fields including the radiation pressure and the mass transport velocity. The model equations have a standard form which allows their numerical integration using a universal solver; in this case, COMSOL Multiphysics was employed. As this software is based on the finite element method, it is simple and straightforward to perform the calculations with moderate computational costs even for complex geometries, which makes the proposed approach an operative tool for study of acoustic streaming. The numerical results are validated for the case of a rectangular channel by comparison with previously published analytical results; an excellent agreement is found. The numerical results show that the acoustic streaming can be quite complex even in rectangular channels and its structure depends on the manner of driving. Examples of acoustic streaming in wedged and elliptical channels are given to demonstrate a strong dependence of the acoustic streaming structure on the resonator shape.
Bednarik, M., Cervenka, M., Lotton, P., Penelet, G.: Behavior of plane waves propagating through a temperature-inhomogeneous region, Journal of Sound and Vibration 362, 292-304, 2016.
Abstract: Description and analysis of acoustic waves in ducts with a region containing temperature-inhomogeneous fluid represent a significant problem of scientific and practical interest. This interest is induced by the need of understanding how temperature fields affect acoustic processes which would lead to a more efficient design and control of systems involving thermoacoustic interactions. Most of the works addressing these problems limit themselves to the assumption of weak temperature profile gradients or to temperature profiles which do not connect neighboring temperature-homogeneous regions smoothly. In our work we investigate the behavior of plane acoustic waves that enter a region with an arbitrary temperature gradient. A polynomial character of the used temperature profile ensures smooth connection with constant-temperature regions. The one-dimensional wave equation for ducts with an axial mean temperature gradient is solved analytically. The derived solutions based on Heun functions extend the class of published exact analytical solutions of model wave equations taking into account the medium temperature gradient. Due to the property that our proposed polynomial temperature function has derivatives equal to zero at points which are connected with the surrounding temperature-homogeneous regions we can form more complex smooth temperature profiles for which it is possible to use the transfer matrix method.