Diffraction by a set of three parallel impedance half-planes with the one amidst located in the opposite direction

2002 ◽  
Vol 80 (8) ◽  
pp. 893-909 ◽  
Author(s):  
G Çinar ◽  
A Büyükaksoy
Keyword(s):  
Fourier Transform ◽  
Opposite Direction ◽  
Infinite System ◽  
Plane Waves ◽  
Mode Matching ◽  
Algebraic Equations ◽  
Matching Method ◽  
Linear Algebraic Equations ◽  
Expansion Coefficients ◽  
The One

The problem of diffraction of plane waves by a set of three parallel half-planes with different surface impedances on upper and lower faces where the one in the middle is placed in the opposite direction, is solved by the mode-matching method where available, and by Fourier-transform technique elsewhere. The solution includes two independent Wiener–Hopf equations each involving an infinite number of expansion coefficients that satisfy an infinite system of linear algebraic equations. PACS No.: 41.20J

Sound Radiation from the Perforated End of a Lined Duct

2019 ◽  
Vol 105 (4) ◽  
pp. 591-599 ◽  
Author(s):  
Burhan Tiryakioglu
Keyword(s):  
Infinite System ◽  
Normal Modes ◽  
Mode Matching ◽  
Algebraic Equations ◽  
Matching Method ◽  
Linear Algebraic Equations ◽  
Matching Technique ◽  
Specific Impedance ◽  
Expansion Coefficients ◽  
The Fourier Transform

Radiation of sound wave through a lined duct with perforated end is analyzed rigorously. The problem considered is axisymmetric. By using the Fourier transform technique in conjunction with the Mode Matching method, the related boundary value problem is formulated as a Wiener-Hopf (W-H) equation. The Mode-Matching technique allows us to express the field component defined in the waveguide region in terms of normal modes. The solution involves a set of infinitely many expansion coefficients satisfying an infinite system of linear algebraic equations. The numerical solution of this system is obtained for different parameters of the problem such as the surface impedances, specific impedance of the perforated screen and their effects on the radiation phenomenon are shown graphically.

A Diagonally Dominant Solution for the Cylinder End Problem

1981 ◽  
Vol 48 (4) ◽  
pp. 876-880 ◽  
Author(s):  
T. D. Gerhardt ◽  
Shun Cheng
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Infinite Cylinder ◽  
Algebraic Equations ◽  
Precise Calculation ◽  
Linear Algebraic Equations ◽  
Present Solution ◽  
Diagonally Dominant ◽  
Expansion Coefficients

An improved elasticity solution for the cylinder problem with axisymmetric torsionless end loading is presented. Consideration is given to the specification of arbitrary stresses on the end of a semi-infinite cylinder with a stress-free lateral surface. As is known from the literature, the solution to this problem is obtained in the form of a nonorthogonal eigenfunction expansion. Previous solutions have utilized functions biorthogonal to the eigenfunctions to generate an infinite system of linear algebraic equations for determination of the unknown expansion coefficients. However, this system of linear equations has matrices which are not diagonally dominant. Consequently, numerical instability of the calculated eigenfunction coefficients is observed when the number of equations kept before truncation is varied. This instability has an adverse effect on the convergence of the calculated end stresses. In the current paper, a new Galerkin formulation is presented which makes this system of equations diagonally dominant. This results in the precise calculation of the eigenfunction coefficients, regardless of how many equations are kept before truncation. By consideration of a numerical example, the present solution is shown to yield an accurate calculation of cylinder stresses and displacements.

scholarly journals Radiation analysis of sound waves from semi-infinite coated pipe

2018 ◽  
Vol 18 (1) ◽  
pp. 92-111 ◽  
Author(s):  
Burhan Tiryakioglu ◽  
Ahmet Demir
Keyword(s):  
Fourier Transform ◽  
Sound Waves ◽  
Algebraic Equations ◽  
Total Field ◽  
Hopf Equation ◽  
Waveguide Modes ◽  
Linear Algebraic Equations ◽  
Pipe Radius ◽  
Expansion Coefficients ◽  
The Fourier Transform

An analytical solution is presented for the problem of radiation of sound waves from a semi-infinite circular cylindrical coated pipe which is partially lined from inside. By stating the total field in duct region in terms of normal waveguide modes (Dini’s series) and using the Fourier transform technique elsewhere, we obtain a Wiener–Hopf equation whose solution involving three sets of infinitely many unknown expansion coefficients satisfying three systems of linear algebraic equations. This system is solved numerically and the influence of some parameters (pipe radius, impedances, extension, etc.) on the radiation phenomenon is displayed graphically.

scholarly journals Sound Transmission in a Duct with Sudden Area Expansion, Extended Inlet, and Lined Walls in Overlapping Region

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Ahmet Demir
Keyword(s):  
Fourier Transform ◽  
Numerical Solution ◽  
Sound Transmission ◽  
Algebraic Equations ◽  
Hopf Equation ◽  
Linear Algebraic Equations ◽  
Expansion Coefficients ◽  
The Fourier Transform ◽  
Area Expansion ◽  
Overlapping Region

The transmission of sound in a duct with sudden area expansion and extended inlet is investigated in the case where the walls of the duct lie in the finite overlapping region lined with acoustically absorbent materials. By using the series expansion in the overlap region and using the Fourier transform technique elsewhere we obtain a Wiener-Hopf equation whose solution involves a set of infinitely many unknown expansion coefficients satisfying a system of linear algebraic equations. Numerical solution of this system is obtained for various values of the problem parameters, whereby the effects of these parameters on the sound transmission are studied.

Plane wave diffraction by two parallel overlapped thick semiinfinite impedance plates

2000 ◽  
Vol 77 (11) ◽  
pp. 873-891
Author(s):  
F Birbir ◽  
A Büyükaksoy
Keyword(s):  
Fourier Transform ◽  
Plane Waves ◽  
Separation Distance ◽  
Mode Matching ◽  
Matching Method ◽  
New Approach ◽  
Time Harmonic ◽  
Plate Separation ◽  
The Fourier Transform ◽  
Plane Wave Diffraction

A new approach consisting of employing the mode matching method inconjunction with the Fourier transform technique is used to analyse thediffraction of time harmonic plane waves by two parallel overlapped thickimpedance half-planes. The problem is formulated as a pair of uncoupledmodified Wiener-Hopf equations and solved approximately. Numerical resultsillustrating the effects of various parameters such as wall thickness, plateto plate separation distance, wall impedance, etc. on the diffractionphenomenon are presented.PACS No.: 41.20jb

scholarly journals Diffraction of sound waves by a lined cylindrical cavity

2020 ◽  
Vol 19 (1-2) ◽  
pp. 38-56
Author(s):  
Burhan Tiryakioglu
Keyword(s):  
Mode Matching ◽  
Sound Waves ◽  
Algebraic Equations ◽  
Hopf Equation ◽  
Matching Method ◽  
Infinite Systems ◽  
Linear Algebraic Equations ◽  
The Fourier Transform ◽  
Sound Diffraction ◽  
Selection Of

In this paper, diffraction of sound waves through a lined cavity is analyzed rigorously. The inner–outer surfaces of the cavity and the base of the cavity are coated with three different absorbing linings. By using the Fourier transform technique in conjunction with the Mode-Matching method, the related boundary value problem is formulated as a Wiener–Hopf equation. In the solution, two infinite sets of unknown coefficients are involved that satisfy two infinite systems of linear algebraic equations. Numerical solution of this system is obtained for various values of the parameters of the problem. The graphical results are also presented which show that how efficiently the sound diffraction can be reduced by selection of problem parameters.

scholarly journals Two Contact Problems for Annular Rigid Stamp on an Elastic Half Space

2018 ◽  
Vol 17 (6) ◽  
pp. 458-464
Author(s):  
S. V. Bosakov
Keyword(s):  
Functional Equation ◽  
Half Space ◽  
Bessel Functions ◽  
Infinite System ◽  
Annular Plate ◽  
Contact Problems ◽  
Elastic Half Space ◽  
Contact Stresses ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

The paper presents solutions of two contact problems for the annular plate die on an elastic half-space under the action of axisymmetrically applied force and moment. Such problems usually arise in the calculation of rigid foundations with the sole of the annular shape in chimneys, cooling towers, water towers and other high-rise buildings on the wind load and the load from its own weight. Both problems are formulated in the form of triple integral equations, which are reduced to one integral equation by the method of substitution. In the case of the axisymmetric problem, the kernel of the integral equation depends on the product of three Bessel functions. Using the formula to represent two Bessel functions in the form of a double row on the works of hypergeometric functions Bessel function, the problem reduces to a functional equation that connects the movement of the stamp with the unknown coefficients of the distribution of contact stresses. The resulting functional equation is reduced to an infinite system of linear algebraic equations, which is solved by truncation. Under the action of a moment on the annular plate  die, the distribution of contact stresses is searched as a series by the products of the Legendre attached functions with a weight corresponding to the features in the contact stresses at the die edges. Using the spectral G. Ya. Popov ratio for the ring plate, the problem is again reduced to an infinite system of linear algebraic equations, which is also solved by the truncation method. Two examples of calculations for an annular plate die on an elastic half-space on the action of axisymmetrically applied force and moment are given. A comparison of the results of calculations on the proposed approach with the results for the round stamp and for the annular  stamp with the solutions of other authors is made.

scholarly journals Diffraction of the field of vertical electric dipole on the spiral conductive sphere in the presence of a cone

2018 ◽  
Keyword(s):  
Hilbert Space ◽  
Electric Dipole ◽  
Infinite System ◽  
Matrix Operator ◽  
Algebraic Equations ◽  
Problem Statement ◽  
Vertical Electric Dipole ◽  
Linear Algebraic Equations ◽  
Method Of Regularization ◽  
The Matrix

The problem of diffraction of a vertical electric dipole field on a spiral conductive sphere and a cone has been solved. By the method of regularization of the matrix operator of the problem, an infinite system of linear algebraic equations of the second kind with a compact matrix operator in Hilbert space $\ell_2$ is obtained. Some limiting variants of the problem statement are considered.

scholarly journals Numerical solution of a weakly singular integral equation by the method of orthogonal polynomials

2021 ◽  
pp. 98-103
Author(s):  
Sergei M. Sheshko
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Orthogonal Polynomials ◽  
Numerical Solution ◽  
Singular Integral ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Expansion Coefficients ◽  
Analytical Expressions

A scheme is constructed for the numerical solution of a singular integral equation with a logarithmic kernel by the method of orthogonal polynomials. The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.

The Problem of an Elastic Stiffener Bonded to a Half Plane

1971 ◽  
Vol 38 (4) ◽  
pp. 937-941 ◽  
Author(s):  
F. Erdogan ◽  
G. D. Gupta
Keyword(s):  
Mechanical Properties ◽  
Integral Equation ◽  
Contact Problem ◽  
Half Plane ◽  
Elastic Constants ◽  
Infinite System ◽  
Stress Singularity ◽  
Algebraic Equations ◽  
Numerical Example ◽  
Linear Algebraic Equations

The contact problem of an elastic stiffener bonded to an elastic half plane with different mechanical properties is considered. The governing integral equation is reduced to an infinite system of linear algebraic equations. It is shown that, depending on the value of a parameter which is a function of the elastic constants and the thickness of the stiffener, the system is either regular or quasi-regular. A complete numerical example is given for which the strength of the stress singularity and the contact stresses are tabulated.

Sign in / Sign up

Export Citation Format

Share Document