The exact solution of the Schrödinger equation with a polynomially spatially varying mass

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.

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Effect of inhomogeneous temperature fields on acoustic streaming structures in resonator

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.

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Description of waves in inhomogeneous domains using Heun’s equation

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.

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Acoustic bandpass filters employing shaped resonators

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.

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Variety of acoustic streaming in 2D resonant channels

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.

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Behavior of plane waves propagating through a temperature-inhomogeneous region

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.

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