Far-field Solution Research Articles

Far-field properties of non-paraxial elliptical Gaussian beams

By using the Rayleigh diffraction integral formula, a far-field asymptotic solution of the wave equation is found, which represents non-paraxial elliptical Gaussian beams. The basic properties of such beams are studied, and a comparison with the previous work is also made.

A mixed-type finite element approximation for radiation problems using fictitious domain method

In the finite element approximation of the exterior Helmholtz problem, we propose an approximation method to implement the DtN mapping formulated as a pseudo-differential operator on a computational artificial boundary. The method is then combined with the fictitious domain method. Our method directly gives an approximation matrix for the sesqui-linear form for the DtN mapping. The eigenvalues of the approximation matrix are simplified to a closed form and can be computed efficiently by using a continued fraction formula. Solution outside the computational domain and the far-field solution can also be computed efficiently by expressing them as operations of pseudo-differential operators. An inner artificial DtN boundary condition is also implemented by our method. We prove the convergence of the solution of our method and compare the performance with the standard finite element approximation based on the Fourier series expansion of the DtN operator. The efficiency of our method is demonstrated through numerical examples.

Accurate radiation boundary conditions for the two-dimensional wave equation on unbounded domains

A recursive sequence of radiation boundary conditions first given by Hagstrom and Hariharan [Appl. Numer. Math. 27 (1998) 403] for the time-dependent wave equation in a two-dimensional exterior region are re-derived based on direct application of the hierarchy of local boundary operators of Bayliss and Turkel [Commun. Pure Appl. Math. 33 (1980) 707] and a recursion relation for the expansion coefficients appearing in an asymptotic wave expansion. By introducing a decomposition into tangential Fourier modes on a circle we reformulate the sequence of local boundary conditions in integro-differential form involving systems of first-order temporal equations for auxiliary functions associated with each mode and the Fourier transform of the solution evaluated on the boundary. The auxiliary functions are recognized as residuals of the local boundary operators acting on the asymptotic wave expansion. Direct finite element implementations for the original local sequence of boundary conditions are compared to implementations of the Fourier transformed auxiliary functions. We show that both implementations easily fit into a standard finite element discretization provided that independent time integration algorithms are used for the interior and boundary equations with coupling through the boundary force vectors at each time step. For both of our direct and modal finite element implementations, the amount of work and storage is less than that required for the finite element calculation in the interior region within the boundary. One advantage of the tangential modal implementation is that far-field solutions may be computed separately for each Fourier mode without saving lengthy time-history data at interior points. Numerical studies confirm the progressive improvement in accuracy with increasing number of auxiliary functions included.

ACOUSTIC RADIATIONS FOR SUBMERGED ELASTIC STRUCTURES USING NATURAL MODE EXPANSIONS IN CONJUNCTION WITH RADIATION MODES APPROACH

This work formulates submerged elastic structures using in-vacuo vibrational mode expansions with which the acoustic impedance loading is derived based on radiation mode theory. The displacement of natural modes on the normal direction is expanded as linear combinations of a set of velocity radiation modes that the expansion coefficients characterize as the radiation characteristics of each vibration mode. This type of expansion allows one to represent the surface pressure by the corresponding set of pressure radiation modes. Thus, a symmetric impedance matrix associated with the natural vibration mode expansions is derived when a variational increment is applied to the virtual work done by the surface pressure against the normal displacement. The equation of the submerged structures is obtained according to Hamilton's principles. By incorporating the description of radiation modes, this equation of natural mode expansions is used to study the coupling among vibration modal amplitudes due to the modal cross-impedances and the convergence of near and farfield solutions. In addition, a slender submerged spheroidal shell vibrating axisymmetrically serves as a numerical example to demonstrate the effectiveness of the analysis procedure. This numerical example reveals that the acoustic impedances decrease with ascending mode numbers, causing the high order vibration modes to react independently. Moreover, the convergence of the surface pressure and normal velocity is examined on the basis of independent reaction of the vibration modes. Accurately predicting farfield solutions depends only on the convergence of the surface quantities whose components pertain to strong radiation modes. The numerical example indicates that the number of vibration modes used in the expansion for predicting farfield solutions is less than the modes required for the surface solution.

Ground penetrating radar antennas: theoretical and experimental directivity functions

The prediction of the directivity function of a GPR antenna still remains a partially unsolved problem because of the subject complexity. First, the far-field conditions are often not satisfied and second, the antenna design has little in common with the Hertzian dipole for which an analytical approach can be used. The authors' contribution is both theoretical and experimental. On one side, they solve the (electromagnetic) EM integral equations numerically to derive the wavefield components from near to far-field distances. On the other side, they experiment with two novel techniques for measuring the directivity functions in the near to far-field range on dry and saturated sand. Theoretical and experimental results show that neither the analytic approximation of far-field directivity for the numerical integration of near-field directivity can perfectly match the measured functions, although near-field solutions are generally more consistent. The mismatch should be attributed to the present-day GPR antenna design that includes absorbers and shields. Although the effects of these elements are not included in the present numerical near-field solutions, they believe the approach to be of practical value to predict an average directivity function. A smoothed version of the analytic far-field solution can also be used in the range of the near to far-field transition but near-field solutions are really recommended when TX-RX distances are shorter than five wavelengths.

Computation of far-field solutions based on exact nonreflecting boundary conditions for the time-dependent wave equation

In this work we show how to combine in the exact nonreflecting boundary conditions (NRBC) first derived by Grote and Keller, the calculation of the exterior (far-field) solution for time-dependent radiation and scattering in an unbounded domain. At each discrete time step, radial modes computed on a spherical artificial boundary which drive the exact NRBC for the near-field solution, are imposed as Cauchy data for the radial wave equation in the far-field. Similar to the far-field computation scheme used by Wright, the radial modes in the exterior region are computed using an explicit finite difference solver. However, instead of using an `infinite grid', we truncate the exterior radial grid at the far-field point of interest, and for each harmonic, impose the same exact NRBC used for the near-field truncation boundary, here expressed in modal form. Using this approach, two different methods for extrapolating the near-field solution to the far-field are possible. In the first, the near-field solution is computed using the exact NRBC, then, based on the solution for the radial modes evaluated on the artificial boundary, the exterior solution may be computed as a post-process. In the second, we show how to compute the far-field solution concurrently with the near-field solution and the NRBC. Numerical studies demonstrate that the method is highly accurate and efficient for direct time-domain computations of far-field solutions.

Receptivity of three-dimensional boundary-layers to localized wall roughness and suction

The receptivity of three-dimensional, incompressible, boundary-layer flows to localized roughness or suction is studied theoretically and numerically. Three-dimensional boundary-layer flows have a strong “nonparallel” component due to the curvature of the potential-flow streamlines. Our theoretical model extends the Fourier-transform methodology to nonparallel flows using a Taylor expansion of the laminar mean-flow at the location of the roughness. Both the near-field and the far-field solutions are contained in the model. Additionally, the use of MacLaurin expansions in the complex plane leads quickly to the receptivity factors of eigenmodes. The theoretical results are validated with solutions to the linearized Navier–Stokes equations efficiently obtained by replacing the actual wall forcing with a smoother, but equivalent, forcing. We find that the far-field response is proportional to both H̃(αe) and dH̃(αe)/dα, where H̃(α) is the Fourier transform of the forcing distribution, and αe is the eigenmode’s wave number. The receptivity coefficient for H̃(αe) decreases when nonparallelism is included in our models, while that for dH̃(αe)/dα increases.

Efficient Computation of Multi-Frequency Far-Field Solutions of the Helmholtz Equation Using Padé Approximation

For many problems in exterior structural acoustics, the solution is required to be computed over multiple frequencies. For some classes of these problems, however, it may be sufficient to evaluate the multiple frequency solutions over restricted regions of the spatial domain. Examples include optimization and inverse problems based on the minimization of a functional defined over a specified surface or sub-region. For such problems, which include both near-field and far-field computations, we recently proposed an efficient algorithm to compute the partial-field solutions at multiple frequencies simultaneously. In this paper, we consider the particular case of far-field computations and simplify the recently proposed algorithm by exploiting the symmetry of linear operators. The approach involves a reformulation of the Dirichlet-to-Neumann (DtN) map based finite-element matrix problem into a transfer-function form that can efficiently describe the far-field solution. A multi-frequency approximation of the transfer function is developed by constructing matrix-valued Pade approximation of the transfer function via a symmetric, banded Lanczos process. Numerical tests illustrate the accuracy of the approach for a wide range of frequencies and cost reductions of an order of magnitude when compared to commonly used factorization based methods.

EFFICIENT COMPUTATION OF MULTI-FREQUENCY FAR-FIELD SOLUTIONS OF THE HELMHOLTZ EQUATION USING PADÉ APPROXIMATION

For many problems in exterior structural acoustics, the solution is required to be computed over multiple frequencies. For some classes of these problems, however, it may be sufficient to evaluate the multiple frequency solutions over restricted regions of the spatial domain. Examples include optimization and inverse problems based on the minimization of a functional defined over a specified surface or sub-region. For such problems, which include both near-field and far-field computations, we recently proposed an efficient algorithm to compute the partial-field solutions at multiple frequencies simultaneously. In this paper, we consider the particular case of far-field computations and simplify the recently proposed algorithm by exploiting the symmetry of linear operators. The approach involves a reformulation of the Dirichlet-to-Neumann (DtN) map based finite-element matrix problem into a transfer-function form that can efficiently describe the far-field solution. A multi-frequency approximation of the transfer function is developed by constructing matrix-valued Padé approximation of the transfer function via a symmetric, banded Lanczos process. Numerical tests illustrate the accuracy of the approach for a wide range of frequencies and cost reductions of an order of magnitude when compared to commonly used factorization based methods.

The acoustics of diagnostic microbubbles: dissipative effects and heat deposition

We discuss the effectively detectable scattered intensity of ultrasound from diagnostic microbubble suspensions, taking dissipative mechanisms in the liquid medium into account. In particular, we conclude that neither non-linear wave steepening of the incident (driving) wave nor of the outgoing (scattered) wave has a large effect on the scattered signal from typical bubbles. It is shown that, paradoxically, the far-field solution of the wave field is sufficient to compute the magnitude of expected temperature rises in the medium due to acoustic heat deposition, although appreciable heating is limited to intermediate-field distances from the bubble surface.

Far-field propagation of an off-axis Gaussian wave

Based on a far-field asymptotic solution of the Helmholtz equation a vector theory to describe the propagation of an off-axis Gaussian wave is developed, the accurate formulas represented in terms of elementary functions are derived, and the propagation properties such as wave spot size and divergence angle are discussed in detail. The applicable range of scalar theory is also presented. A relative error criterion of optical intensity is given by epsilon = sin(2) alpha.

Effect of an air-back cavity on the sound field reflected by a vibrating plate

The effect of an air-back cavity on the spherical wave reflected by the stage floor is analyzed theoretically. An air-cavity backed infinite elastic plate is driven to vibration by a point force. A closed form expression for the far-field solution is derived by coupling the Helmholtz integrals for sound field and the equation of the plate motion. Numerical examples demonstrate four main features of the sound field. Two of them are absent for the plate without the air-cavity: one is a panel absorption dip owing to the plate and air-cavity resonance system. The other is small peaks and dips due to the acoustic resonances of the air-cavity.

Far-field approach to the transfer of polarized radiation in anisotropic, spatially dispersive, weakly inhomogeneous and dissipative media

The transport of polarized radiation in anisotropic, spatially dispersive, weakly inhomogeneous and dissipative media with embedded sources is dealt with in terms of the (second-rank) radiation intensity tensor obtained on the basis of the geometrical optics far-field solution of the wave equation. Such an approach accounts for the effects of both wave energy absorption and emission from the medium, the former being related to the anti-Hermitian part of the dielectric tensor of the medium and the latter being expressed in terms of the autocorrelation of the source current density, which allows one to treat both polarized and unpolarized, as well as coherent and incoherent, sources. The polarization properties of the radiation, described on the basis of the polarization vectors of the two propagating electromagnetic eigenmodes, are investigated with respect to the effects of both the spatial inhomogeneity and birefringence of the medium, the latter, in particular, being connected with the relative change of phase of the two propagating modes over the relevant ray path. Each of the effects is discussed, and the connections with standard notions such as the specific intensity of the radiation and Stokes parameters are investigated, the state of thermodynamic equilibrium being considered as an example. The specific case of the polarization degeneracy in an isotropic medium is addressed in some detail. The case of a non-uniformly magnetized medium is also considered.

Reflection of a spherical sound wave by an infinite elastic plate driven to vibration by a point force

This paper theoretically analyses the reflected sound field by an infinite elastic plate which generates its own sound field due to a point force excitation. The plate is a theoretical model for the stage floor. A closed form expression for the far-field solution was derived by coupling the Helmholtz equation for sound fields surrounding the plate and the equation of motion for the plate. The solution indicates that the field above the plate excited by a point force consists of the reflected sound by the plate at rest without the point force and the radiated sound from the plate excited by the point force. The general features of the field reflected by the vibrating plate with the point force excitation are discussed with the theoretical solution and numerical examples. The numerical examples are also presented to demonstrate the effects of the plate parameters on the sound field.

Electromagnetic scattering by an aggregate of spheres: Theoretical and experimental study of the amplitude scattering matrix

An accurate analytical representation of the amplitude scattering matrix is of key importance in the development of a reliable electromagnetic scattering formulation, because it enables one to derive rigorous analytical expressions for all scattering quantities. We compare a rigorous theory of scattering by aggregates of spheres [Y.-I. Xu, Appl. Opt. 36, 9496 (1997)] with a large set of laboratory microwave analog scattering measurements for multiple spheres obtained by Wang between 1968 and 1983. Close agreement is found for all of the experimental data tested, confirming that the four amplitude scattering matrix elements can be accurately evaluated by the theory that is based on a far-field solution. It also leads to the validation of a simplified noninteracting-scattering (NIS) approximation derived from the theory.

Sound radiation from a baffled elastic plate strip of infinite length with various concentrated excitation forces

Recently, problems with elevated structure noise arise around elevated motorways, i.e. noise in the audio-frequency range supposed to be caused by the vibration of the structural elements of an elevated motorway. The measurement data shows that steel plate girders may be the main source of the elevated structure noise. Therefore, the radiated sound field generated by a baffled elastic plate strip of infinite length, which is the simplest model of the web of a steel plate girder, is theoretically analysed in this paper. A far-field approximation solution is derived in a closed form. Since the actual condition of the strip excitation has never been clarified so far, three kinds of possible concentrated forces, a line force, a point force, and a concentrated moment, are included in the analysis. The main features of the radiated sound field by the strip are discussed in the light of numerical examples. A parametric study is carried out to clarify the effects of the various parameters of the strip. Furthermore, solutions to the sound radiation from an elastic plate strip of finite length, under the same excitation conditions, are derived and compared with those from the strip of infinite length above. The characteristics of the finite strip are found to be roughly described by those of the infinite strip.

Numerical solution for spherical laser-driven shock waves

The history of the flow behind a laser-driven shock is investigated in the context of variable energy blast waves. Thereby the total laser energy absorbed by the blast is assumed to vary proportionally to some power of time. Due to the high temperatures and pressures occurring in the initial phase of the flow a real gas model has been employed. It accounts for vibration, dissociation, electronic excitation, ionization and intermolecular forces. Radiative and conductive heat transfer are considered as well. The numerical computations were carried out using the method of characteristics. A self-similar strong shock solution serves as initial condition. It turns out that the exponent which determines the time-dependent addition of energy at the shock front is limited for physical reasons. The computed far-field solutions expand the temporal scope of the self-similar solution domain, which has been the main subject of the classical literature, into the non-self-similar domain at late time. The differences between the solutions obtained for real gas and perfect gas are less significant than in the case of the classical point explosion.

Global Far-Field Computational Boundary Conditions for C- and 0-Grid Topologies

Global far-field computational boundary conditions for inviscid external flow problems have been developed for C- and O-grid topologies. This analysis represents a unified approach for two-dimensional external flow problems. These boundary conditions are derived from analytic solutions of an asymptotic form of the steady-state Euler equations and have improved accuracy compared to characteristic boundary conditions commonly used in practice. The Euler equations are asymptotically linearized about a constant pressure, rectilinear flow condition, which is the true boundary condition at infinity. Previous work had developed higher-order boundary conditions for C-grid topologies by assuming small perturbations in both pressure and flow direction at and beyond the computational boundary, by decoupling the inflow and outflow analyses, and by linearizing the thermodynamic relations. This work lifts these restrictions, although some higher-order compressibility effects are neglected. It is based on a global mapping of the boundary and solution of the resulting Dirichlet-Neumann problem. Because the Euler equations are used to develop the boundary conditions, the flow crossing the boundary can be rotational. The boundary conditions can be used with any numerical Euler solution method and allow computational boundaries to be located very close to the nonlinear region of interest. This leads to a significant reduction in the number of grid points required for numerical solution. Numerical results are presented that show that the boundary conditions and far-field analytic solutions provide a smooth transition across a computational boundary to the true far-field conditions at infinity. They also demonstrate the synergism that can be realized from coupling analytic and computational methods.

Electromagnetic scattering by an aggregate of spheres: far field

In electromagnetic multisphere-scattering calculations the reexpansion method for seeking a single-field representation of the total scattered field is found impracticable because of severe numerical problems. We present a simple single-field expansion of the total scattered far field based on an asymptotic form of vector translational addition theorems. With this asymptotic expansion of the far field, we derive analytical expressions for the scattering properties of an arbitrary aggregate of spheres. Resulting formulas are free from numerical problems in practical applications. Theoretical predictions from this far-field solution for various aggregates of spheres that we tested agree favorably with laboratory microwave scattering measurements. Some numerical results are presented and compared with experimental data.

Artificial Boundary Conditions for Computation of Oscillating External Flows

In this paper, we propose a new technique for the numerical treatment of external flow problems with oscillatory behavior of the solution in time. Specifically, we consider the case of unbounded compressible viscous plane flow past a finite body (airfoil). Oscillations of the flow in time may be caused, for example, by the time-periodic injection of fluid into the boundary layer, which in accordance with experimental data, may essentially increase the performance of the airfoil. To conduct the actual computations, we have to somehow restrict the original unbounded domain, that is, to introduce an artificial (external) boundary and to further consider only a finite computational domain. Consequently, we will need to formulate some artificial boundary conditions (ABCs) at the introduced external boundary. The ABCs we are aiming to obtain must meet the following fundamental requirement. One should be able to uniquely complement the solution calculated inside the finite computational domain to its infinite exterior so that the original problem is solved within the desired accuracy. Our construction of such ABCs for oscillating flows is based on an essential assumption: the Navier--Stokes equations can be linearized in the far field against the free-stream background. To actually compute the ABCs, we represent the far-field solution as a Fourier series in time and then apply the difference potentials method (DPM) of V. S. Ryaben'kii. This paper contains a general theoretical description of the algorithm for setting the DPM-based ABCs for time-periodic external flows. Based on our experience in implementing analogous ABCs for steady-state problems (a simpler case), we expect that these boundary conditions will become an effective tool for constructing robust numerical methods to calculate oscillatory flows.