MATHEMATICAL MODELING OF WAVE PHENOMENA: 3rd Conference on Mathematical Modeling of Wave Phenomena, 20th Nordic Conference on Radio Science and Communications

Fast Hybrid Algorithms for High Frequency Scattering
View Description Hide DescriptionThis paper deals with numerical methods for high frequency wave scattering. It introduces a new hybrid technique that couples a directional fast multipole method for a subsection of a scattering surface to an asymptotic formulation over the rest of the scattering domain. The directional fast multipole method is new and highly efficient for the solution of the boundary integral formulation of a general scattering problem but it requires at least a few unknowns per wavelength on the boundary. The asymptotic method that was introduced by Bruno and collaborators requires much fewer unknowns. On the other hand the scattered field must have a simple structure. Hybridization of these two methods retains their best properties for the solution of the full problem. Numerical examples are given for the solution of the Helmholtz equation in two space dimensions.

The Selection of Basis Functions Systems for Determination of Cutoff Frequency of Waveguides and Resonators of Complex Shape with the Help of R‐functions Method
View Description Hide DescriptionThe work focuses on the problem of determination of cutoff frequency of waveguides and resonators of a complex shape. The problem is sold by method of R‐functions. This approach has a lot of advantages, it possesses geometric flexibility, broad capabilities of numerical realization as for the production of the variation problem and for the selection of basis functions system as well. As basis functions the polynomials (trigonometrical, power, Tchebyshev of I and II types, Legendre, Gegenbauer) or local functions (atomic functions, splines) are used. The contrastive analysis of approximate boundary value problem solving is carried out in accordance to the basis functions system selected.

SOME APPLICATIONS OF WAVE‐FRONT SETS OF FOURIER LEBESGUE TYPES, I
View Description Hide DescriptionWe prove some micro‐local properties of solutions to hypoelliptic pseudo‐differential operators, using wave‐front sets of Fourier Lebesgue types.

Electro‐magnetic scattering in variously shaped waveguides with an impedance condition
View Description Hide DescriptionElectro‐magnetic scattering is studied in a waveguide with varying shape and cross‐section. Furthermore, an impedance or admittance condition is applied to two of the waveguide walls. Under the condition that variations in geometry or impedance take place in only one plane at the time, the problem can be solved as a two‐dimensional wave‐scattering problems.
By using newly developed numerical conformal mapping techniques, the problem is transformed into a wave‐scattering problem in a straight two‐dimensional channel. A numerically stable formulation is reached in terms of transmission and reflection operators.
Numerical results are given for a slowly varying waveguide with a bend and for one more complex geometry.

New Plane Wave Addition Theorems
View Description Hide DescriptionThe Multilevel Fast Multipole Algorithm (MLFMA) is a well known and very successful method for accelerating the matrix‐vector products required for the iterative solution of Helmholtz problems. The MLFMA is based on an addition theorem which suffers from the so‐called low‐frequency (LF) breakdown, due to numerical roundoff error. Here, a new addition theorem will be developed which does not suffer from an LF breakdown. Instead it suffers from a High‐Frequency (HF) breakdown. The new addition theorem is based on a novel set of distributions, the so called pseudospherical harmonics, closely related to the spherical harmonics. The so‐called translation operators can be calculated in closed form, which allows the easy implementation of an LF‐stable MLFMA.

Wide‐Angle Shift‐Map PE for a Piecewise Linear Terrain
View Description Hide DescriptionA wide‐angle parabolic wave equation solution, using shift‐map and finite‐difference techniques, is presented. The corresponding split‐step Fourier solution is well known. The solution using finite‐difference technique, where the standard parabolic wave equation is modified into the so‐called Claerbout equation allowing propagation angles up to 45° from the paraxial direction, is also well known. Here, we present an extension to that solution in which the shift‐map technique is incorporated into the finite‐difference scheme allowing a varying terrain to be considered. The result is a solution that corresponds to the well known split‐step solution, which is believed to perform well for terrain slopes up to 10°–15°, and discontinuous slope changes on the order of 15°–20°. This solution is of first order with respect to the terrain slope. The resulting difference scheme is for solving a tridiagonal system. When using the finite‐difference technique, it is also possible to find a second order solution with respect to the terrain slope. The resulting difference scheme in this case is for solving a pentadiagonal system. This new solution performs well for slopes up to about 15°–20° and discontinuous slope changes up to about 30°–40°, which is an improvement.

The Total Electric Field Compared to the Rectangular Component of the Electric Field in a Complex Environment
View Description Hide DescriptionStrong electromagnetic waves can cause upset in and even destruction of electronic equipment. Therefore, the vulnerability of electronic equipment has to be tested. To be able to quantify the test, an unequivocal test parameter has to be defined. In the case that a plane wave is stressed onto the equipment it is easy; the strength of the electric field is an unequivocal test parameter. In an electromagnetically complex environment with electric fields polarized along different directions it is not so obvious what the test parameter should be. A reverberation chamber is an example of such a complex environment. Some people in the community advocate that a rectangular component of the field is the best measure of the strength of the field, but others advocate for that the total field (the vector sum of three orthogonal field components) is the best. The most generic standard, the IEC 61000‐4‐21 prescribes the use of the rectangular component, but others prescribe the use of the total electric field. In this paper, we do not advocate the use of either of the two alternatives, but we compare them with the purpose of making it possible to perform a conversion between different standards. Theoretical results are illustrated and compared with measured results obtained in four different reverberation chambers.

Maxwellian Macroscopic Acoustics with Spatial Dispersion
View Description Hide DescriptionWhat should be the most general macroscopic linear theory of sound propagation in a connected fluidic domain not supporting the propagation of macroscopic transverse waves and defined by a viscothermal fluid in which rigid fixed inclusions of arbitrary shape are placed in periodic or stationary random manner? Based on considerations inspired by the long‐wavelength electromagnetic theory we discuss here what the general answer should be : a nonlocal “Maxwellian” acoustics with constitutive coefficients determined from microgeometry and fluid and solid physical constants. In forthcoming work the explicit construction of the postulated constitutive coefficients will be performed.

MAGNETIC FIELD MEASUREMENT SYSTEM FOR HPM RESEARCH
View Description Hide DescriptionOne method to characterize the radiated microwave field from a high‐power microwave (HPM) source is to measure the radiated high‐level electromagnetic field in several locations at a high sampling rate registering the frequency time dependence, thus being able to determine the radiated pattern and mode. A complete free‐field measurement system for measuring the magnetic field component in high‐level electromagnetic fields has been developed at FOI.
The system consists of a B‐dot sensor and a balun, both designed and constructed at FOI. The B‐dot sensor is designed as two cylindrical loop sensors with differential output. The balun is a microstrip design etched on a dual sided PTFE circuit board. Complete systems have been calibrated at SP Technical Research Institute of Sweden. A method to analyze the data from the free‐field systems has been developed.

Estimation of parameters of an inhomogeneous dielectric layer
View Description Hide DescriptionThis paper presents a general framework for sensitivity analysis for few‐parameter inverse problems using the Fisher information and the Cramér‐Rao bound. In particular, the one‐dimensional inverse problem of estimating the dispersive parameters of an inhomogeneous dielectric layer with linear spatial variation is studied. The analysis technique is particularly well‐suited for inverse problems using few parameters, and it is anticipated that the framework may be used as a basis for extensive numerical investigations and physical interpretations. The ill‐posedness of the inverse problem can be explicitely quantified by using the Fisher information analysis. As an example, the sensitivity analysis is used together with asymptotic theory to show that the inverse problem becomes extremely ill‐posed when the linear spatial variation vanishes.

Computation of the Fock scattering functions
View Description Hide DescriptionThe Fock functions are mainly used in diffraction theory. A problem relating to these functions arises when accurate numerical computation of integrals with certain exponentials is needed. An integration contour that essentially solves this problem for the Fock scattering functions is discussed.

ELECTROMAGNETIC SCATTERING THEORY FOR GRATINGS BASED ON THE WIENER‐HOPF‐FOCK METHOD
View Description Hide DescriptionThe diffraction problem of a plane electromagnetic wave incident on an ideally conducting grating is solved with the Wiener‐Hopf‐Fock method. In contrast to the standard Wiener‐Hopf method having a single integral equation, the boundary value problem is reduced to a system of integral equations. The short wave asymptotic solution of this system is obtained by means of the saddle point method and expressed in terms of the etalon integral. It contains a resonant denominator which determines the eigenfrequencies of the grating. A precision not below the inclusion of tertiary diffractions is provided by using the saddle point method and the etalon integral for the main contribution of the integrals in the solution. The given method has a clear advantage when the grating consists of a finite number of strips.

Line‐source excited pulsed acoustic wave reflection against the mass‐loaded boundary of a fluid
View Description Hide DescriptionA generalization of the modified Cagniard method (‘Cagniard‐DeHoop method’) is used to obtain closed‐form analytic time‐domain expressions for the line‐source excited pulsed acoustic wave pressure associated with the reflection against the mass‐loaded planar boundary of a semi‐infinite fluid. The expressions reveal the generation of anomalous waveforms near the boundary at large offsets from the source. Their occurrence can be interpreted as a surface effect. As the analysis shows, they are, however, not surface waves in the strict sense of the definition.

An analysis of the acoustic energy in a flow duct with a vortex sheet
View Description Hide DescriptionModelling the acoustic scattering and absorption at an area expansion in a flow duct requires the incorporation of the flow‐acoustic interaction. One way to quantify the interaction is to study the energy in the incident and the scattered field respectively. If the interaction is strong, energy may be transferred between the acoustic and the main flow field. In particular, shear layers, that may be the result of the flow separation, are unstable to low frequency perturbations such as acoustic waves. The vortex sheet model is an analytical linear acoustic model, developed to study scattering of acoustic waves in duct with sharp edges including the interaction with primarily the separated flows that arise at sharp edges and corners. In the model the flow field at an area expansion in a duct is described as a jet issuing into the larger part of the duct. In this paper, the flow‐acoustic interaction is described in terms of energy flow. The linear convective wave equation is solved for a two‐dimensional, rectangular flow duct geometry. The resulting modes are classified as “hydrodynamic” and “acoustic” when separating the acoustic energy from the part of the energy arising from the steady flow field. In the downstream duct, the set of modes for this complex flow field are not orthogonal. For small Strouhal numbers, the plane wave and the two hydrodynamic waves are all plane, although propagating with different wave speeds. As the Strouhal numbers increases, the hydrodynamic modes changes to get a shape where the amplitude is concentrated near the vortex sheet. In an intermediate Strouhal number region, the mode shape of the first higher order mode is very similar to the damped hydrodynamic mode. A physical interpretation of this is that we have a strong coupling between the flow field and the acoustic field when the modes are non‐orthogonal. Energy concepts for this duct configuration and mean flow profile are introduced. The energy is formulated such that the vortex sheet turns out as a sink for the acoustic field, but a source for the unstable hydrodynamic wave. This model is physical only close to the edge, due to an exponentially growing hydrodynamic mode. In a real flow, non‐linearities will limit the growth, but this is not included in the model.

Acoustic waves in variable sound speed profiles
View Description Hide DescriptionAn important topic in the area of airborne sound propagation is the prediction of sound propagation above an impedance ground with an atmospheric profile whose sound speed varies with height. Even if this problem is simple in concept, it leads to complications for general velocity profiles. This work illustrates the existence of a large class of realistic atmospheric profiles for which analytical solutions exist to be used as benchmark solutions for numerical methods. Spectral finite element results are discussed for sound propagation in a half‐space situated above a ground surface impedance.

Modeling a Clamped Boring Bar using Euler‐Bernoulli Beam Models with Various Boundary Conditions
View Description Hide DescriptionThis paper addresses modeling of a clamped boring bar using Euler‐Bernoulli beam theory. Euler‐Bernoulli beams with a number of different boundary conditions were used to model a clamped boring bar. Estimates of the boring bar’s natural frequencies and mode shapes were produced with each of the boring bar models. The estimates produced by the distributed‐parameter system models are compared with eigenfrequencies and mode shapes estimated based on experimental modal analysis of the actual boring bar clamped in a lathe.

Nonlinear Schrödinger Solitons with non‐zero velocities emerging from real symmetric initial conditions
View Description Hide DescriptionSolutions of the nonlinear Schrödinger equation for initial conditions in the form of two separated sech‐shaped in‐phase pulses are analyzed. It is found that this initial condition, with appropriate amplitude, may give rise to, not only stationary solitons, but also to symmetrically separating solitons, if the initial distance of separation is large enough. The condition for the generation of a separating soliton pair is derived from the Zakharov‐Shabat eigenvalue problem using a variational approach.

Basics of Nonlinear Time Reversal Acoustics
View Description Hide DescriptionAn investigation of time reversal of nonlinear and non‐dissipative or dissipative plane sound waves was made. Propagation and backpropagation is time reversal invariant only when the dissipation is zero and the wave has not been shocked. A wave that has shocked has irretrievably lost part of its content. Still, if the wave and its derivatives are considered continuous there remain in theory forever information about the original signal. Factors like numerical accuracy and noise naturally set limits in practical situations.

The kernel condition of a linearized pseudo‐relativistic Hartree equation, a numerical approach
View Description Hide DescriptionWe consider the nonlinear equation on describing the dynamics of pseudo‐relativistic boson stars in the mean‐field limit. Recently this equation, with an external potential has been used to describe the dynamics of boson stars under the influence of an external gravitational field. This analysis makes one explicit critical assumption. To the above differential equation we can associate an energy function. The assumption is on the size of the kernel of the Hessian of the energy functional when it is linearized around a soliton. In this paper we provide a numerical indicator that the assumption is satisfied. To achieve this goal, we need to numerically calculate the soliton for a range of normalized frequencies as well as and the spectrum of the linearization around a soliton of the Euler‐Lagrange equations describing the minimizer.