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Gerald J Diebold - One of the best experts on this subject based on the ideXlab platform.

  • Mathieu Function Solutions for the Photoacoustic Effect in Two- and Three-Dimensional Structures and Optical Traps
    International Journal of Thermophysics, 2012
    Co-Authors: Gerald J Diebold
    Abstract:

    The wave equation for the photoacoustic effect in a three-dimensional spherically symmetric, or two-dimensional structure where the compressibility or density varies sinusoidally in space reduces to an inhomogeneous Mathieu equation. As such, exact solutions for the photoacoustic pressure can be found in terms of either Mathieu Functions, integer order Mathieu Functions, or fractional order Mathieu Functions, the last of these being of importance for problems pertaining to structures of finite dimensions. Here, frequency domain solutions are given for a spherical structure with material properties varying radially, and a two-dimensional structure with material variations in one direction. Solutions for the acoustic pressure are found that give closed form expressions for the resonance frequencies. It is also shown that Mathieu Functions give solutions for the motion of an optically levitated sphere trapped in an intensity modulated, Gaussian laser beam. By determining the frequencies at which the motions of the sphere are largest, that is, where the Mathieu Functions become unstable, it is shown that the trap can act to determine the radiation force relative to the gravitational force on the sphere.

  • Mathieu Function solutions for photoacoustic waves in sinusoidal one dimensional structures
    Physical Review E, 2012
    Co-Authors: Gerald J Diebold
    Abstract:

    The photoacoustic effect for a one-dimensional structure, the sound speed of which varies sinusoidally in space, is shown to be governed by an inhomogeneous Mathieu equation with the forcing term dependent on the spatial and temporal properties of the exciting optical radiation. New orthogonality relations, traveling wave Mathieu Functions, and solutions to the inhomogeneous Mathieu equation are found, which are used to determine the character of photoacoustic waves in infinite and finite length phononic structures. Floquet solutions to the Mathieu equation give the positions of the band gaps, the damping of the acoustic waves within the band gaps, and the dispersion relation for photoacoustic waves. The solutions to the Mathieu equation give the photoacoustic response of the structure, show the space equivalent of subharmonic generation and acoustic confinement when waves are excited within band gaps.

Yoon Seok Choun - One of the best experts on this subject based on the ideXlab platform.

  • The power series expansion of Mathieu Function and its integral formalism
    International journal of differential equations and applications, 2015
    Co-Authors: Yoon Seok Choun
    Abstract:

    By applying three term recurrence formula (Choun, 2012 \cite{chou2012b}), power series expansions in closed forms of Mathieu equation for an infinite series and their integral forms are constructed.  One interesting observation resulting from the calculations is the fact that a modified Bessel Function recurs in each of sub-integral forms: the first sub-integral form contains zero term of $A_n's$, the second one contains one term of $A_n$'s, the third one contains two terms of $A_n$'s, etc. Section 5 contains two additional examples of Mathieu equation.  This paper is 5th out of 10 in series ``Special Functions and three term recurrence formula (3TRF).'' See section 6 for all the papers in the series.

  • the power series expansion of Mathieu Function and its integral formalism
    arXiv: Mathematical Physics, 2013
    Co-Authors: Yoon Seok Choun
    Abstract:

    Mathieu ordinary differential equation is of Fuchsian types with the two regular and one irregular singularities. In contrast, Heun equation of Fuchsian types has the four regular singularities. Heun equation has the four kind of confluent forms: (1) Confluent Heun (two regular and one irregular singularities), (2) Doubly confluent Heun (two irregular singularities), (3) Biconfluent Heun (one regular and one irregular singularities), (4) Triconfluent Heun equations (one irregular singularity). Mathieu equation in algebraic forms is also derived from the Confluent Heun equation by changing all coefficients. In this paper I apply three term recurrence formula [arXiv:1303.0806] to the power series expansion in closed forms of Mathieu equation for infinite series and its integral forms including all higher terms of A_n's. One interesting observation resulting from the calculations is the fact that a modified Bessel Function recurs in each of sub-integral forms: the first sub-integral form contains zero term of A_n's, the second one contains one term of A_n's, the third one contains two terms of A_n's, etc. Section 5 contains two additional examples of Mathieu Function. This paper is 5th out of 10 in series "Special Functions and three term recurrence formula (3TRF)". See section 6 for all the papers in the series. Previous paper in series deals with asymptotic behavior of Heun Function and its integral formalism [arXiv:1303.0876]. The next paper in the series describes the power series expansion in closed forms of Lame equation in the algebraic form and its integral forms [arXiv:1303.0873].

Ayton Lorna - One of the best experts on this subject based on the ideXlab platform.

  • Do we need non-linear corrections? On the boundary Forchheimer equation in acoustic scattering
    'Organisation for Economic Co-Operation and Development (OECD)', 2021
    Co-Authors: Colbrook Matthew, Ayton Lorna
    Abstract:

    This paper presents a rapid numerical method for predicting the aerodynamic noise generated by foam-like porous aerofoils. In such situations, particularly for high-frequency noise sources, Darcy's law may be unsuitable for describing the pressure jump across the aerofoil. Therefore, an inertial Forchheimer correction is introduced. This results in a non-linear boundary condition relating the pressure jump across the material to the fluid displacement. We aim to provide a quick, semi-analytical model that incorporates such non-linear effects without requiring a full turbulent simulation. The numerical scheme implemented is based on local Mathieu Function expansions, leading to a semi-analytical boundary spectral method that is well-suited to both linear and non-linear boundary conditions (including boundary conditions more general than the Forchheimer correction). In the latter case, Newton's method is employed to solve the resulting non-linear system of equations for the unknown coefficients. Whilst the physical model is simplified to consider just the scattering by a thin porous aerofoil with no background flow, when the non-linear inertial correction is included good agreement is seen between the model predictions and both experimental results and large eddy simulations. It is found that for sufficiently low-permeability materials, the effects of inertia can outweigh the noise attenuation effects of viscosity. This helps explain the discrepancy between experimental results and previous (linear) low-fidelity numerical simulations or analytical predictions, which typically overestimate the noise reduction capabilities of porous aerofoils

  • Reducing aerofoil–turbulence interaction noise through chordwise-varying porosity
    'Organisation for Economic Co-Operation and Development (OECD)', 2021
    Co-Authors: Ayton Lorna, Colbrook Matthew, Geyer, Thomas F, Chaitanya Paruchuri, Sarradj Ennes
    Abstract:

    This paper considers the effects of smoothly varying chordwise porosity of a finite perforated plate on turbulence-aerofoil interaction noise. The aeroacoustic model is made possible through the use of a novel Mathieu Function collocation method, rather than a traditional Wiener–Hopf approach which would be unable to deal with chordwise varying quantities. The main focus is on two bio-inspired porosity distributions, modelled from air flow resistance data obtained from the wings of barn owls (tyto alba) and common buzzards (buteo buteo). Trailing-edge noise is much reduced for the owl-like distribution, but, perhaps surprisingly, so too is leading-edge noise, despite both wings having similar porosity values at the leading edge. A general monotonic variation is then considered indicating that there may indeed be a significant acoustic impact of how the porosity is distributed along the whole chord of the plate, not just its values at the scattering edges. Through this investigation, it is found that a plate whose porosity continuously decreases from the trailing edge to a zero-porosity leading edge can, in fact, generate lower levels of trailing-edge noise than a plate whose porosity remains constant at the trailing-edge value

  • Reducing aerofoil-turbulence interaction noise through chordwise-varying porosity
    'Organisation for Economic Co-Operation and Development (OECD)', 2020
    Co-Authors: Ayton Lorna, Colbrook Matthew, Tf Geyer, Chaitanya P, Sarradj E
    Abstract:

    This paper considers the effects of smoothly varying chordwise porosity of a finite perforated plate on turbulence-aerofoil interaction noise. The aeroacoustic model is made possible through the use of a novel Mathieu Function collocation method, rather than a traditional Wiener–Hopf approach which would be unable to deal with chordwise varying quantities. The main focus is on two bio-inspired porosity distributions, modelled from air flow resistance data obtained from the wings of barn owls (tyto alba) and common buzzards (buteo buteo). Trailing-edge noise is much reduced for the owl-like distribution, but, perhaps surprisingly, so too is leading-edge noise, despite both wings having similar porosity values at the leading edge. A general monotonic variation is then considered indicating that there may indeed be a significant acoustic impact of how the porosity is distributed along the whole chord of the plate, not just its values at the scattering edges. Through this investigation, it is found that a plate whose porosity continuously decreases from the trailing edge to a zero-porosity leading edge can, in fact, generate lower levels of trailing-edge noise than a plate whose porosity remains constant at the trailing-edge value

Anastasia V Kisil - One of the best experts on this subject based on the ideXlab platform.

Xiang Zhang - One of the best experts on this subject based on the ideXlab platform.