Nonseparable Geometries Research Articles

The exact theory for scattering of waves by thick holes in a slab and other objects with non-separable geometries

The theory for scattering of electromagnetic waves is developed for scattering objects for which the natural modes of the field inside the object do not couple one-to-one with those outside the scatterer. Key feature of the calculation of the scattered fields is the introduction of a new set of modes. As an example, we calculate the reflected and transmitted fields generated by an electromagnetic plane wave that impinges upon a multilayer slab of which the layers are stacked perpendicular to the boundary planes. As this is the geometry of a thick plate with slits our theory encompasses the exact scattering theory of electromagnetic waves by a thick plate with slits.

Surface fields on nonseparable geometries.

The field on the surface of a structure having nonseparable geometry may be expanded in a set of eigenfunctions suitable for that particular geometry. The eigenfunctions are computed numerically using the boundary elements method to compute a matrix that relates the field on the surface of the structure to its normal derivative at vanishing frequency and diagonalizing the matrix. The field on the surface of a finite cylinder with hemisphical end-caps is expanded in such a set of eigenfunctions for both a simple cylinder and a cylinder containing frames. Results for the framed cylinder is compared to Fourier decomposition of the field on the cylindrical portion of the structure.

Regularization methods for near-field acoustical holography.

The reconstruction of the pressure and normal surface velocity provided by near-field acoustical holography (NAH) from pressure measurements made near a vibrating structure is a linear, ill-posed inverse problem due to the existence of strongly decaying, evanescentlike waves. Regularization provides a technique of overcoming the ill-posedness and generates a solution to the linear problem in an automated way. We present four robust methods for regularization; the standard Tikhonov procedure along with a novel improved version, Landweber iteration, and the conjugate gradient approach. Each of these approaches can be applied to all forms of interior or exterior NAH problems; planar, cylindrical, spherical, and conformal. We also study two parameter selection procedures, the Morozov discrepancy principle and the generalized cross validation, which are crucial to any regularization theory. In particular, we concentrate here on planar and cylindrical holography. These forms of NAH which rely on the discrete Fourier transform are important due to their popularity and to their tremendous computational speed. In order to use regularization theory for the separable geometry problems we reformulate the equations of planar, cylindrical, and spherical NAH into an eigenvalue problem. The resulting eigenvalues and eigenvectors couple easily to regularization theory, which can be incorporated into the NAH software with little sacrifice in computational speed. The resulting complete automation of the NAH algorithm for both separable and nonseparable geometries overcomes the last significant hurdle for NAH.

Issues of analytical continuation of the field interior to the target surface and a more general T-matrix for submerged fluid targets

The T-matrix or extended boundary condition method developed by P. Waterman for wave scattering from classical targets and extended by a host of researchers is a marvelous generalization of normal mode solutions allowed for separable geometries to extend to nonseparable geometries by means of mode coupling. The method is based on Huygens’ principle and partial wave expansions of physical quantities. In general the expansions don’t obey orthogonality and one must take care in the way expansions are made. One must never take expansions of a quantity and then represent expansions of their differential forms in terms of the partial wave differential forms using the same expansion coefficients lest one violate a mathematical principle (the Rayleigh hypothesis). One may circumvent any issues without resort to analytical continuation of interior solutions by including all solutions subject to interior and exterior Greens functions. This leads to matrix relations (constraining matrices) between the expansion terms which when used lead to new forms of the fluid T-matrix, for example. The new form differs from Waterman’s T-matrix for the case when the constraining matrices are not symmetric. Otherwise the original expression by Waterman is valid. These issues are discussed and examples are presented.

Helmholtz equation–least-squares method for reconstructing acoustic pressure fields

A method using spherical wave expansion theory to reconstruct an acoustic pressure field from a vibrating object is developed. The radiated acoustic pressures are obtained by means of an expansion of independent functions generated by the Gram–Schmidt orthonormalization with respect to the particular solutions to the Helmholtz equation on the vibrating surface under consideration. The coefficients associated with these independent functions are determined by requiring the assumed form of the solution to satisfy the pressure boundary condition at the measurement points. The errors incurred in this process are minimized by the least-squares method. Once these coefficients are specified, the acoustic pressure at any point, including the source surface, is completely determined. In this paper, this method is used to reconstruct the surface acoustic pressures based on the measured acoustic pressure signals in the field. It is shown that this method can be applied to both separable and nonseparable geometries, and the solutions thus obtained are unique. However, the convergence may be worse for an elongated object subject to high-frequency excitations. [Work supported by the Institute for Manufacturing Research at Wayne State University.]

Helmholtz equation–least-squares method for reconstructing the acoustic pressure field

A method using spherical wave expansion theory to reconstruct acoustic pressure field from a vibrating object is developed. The radiated acoustic pressures are obtained by means of an expansion of independent functions generated by the Gram–Schmidt orthonormalization with respect to the particular solutions to the Helmholtz equation on the vibrating surface under consideration. The coefficients associated with these independent functions are determined by requiring the assumed form of solution to satisfy the pressure boundary condition at the measurement points. The errors incurred in this process are minimized by the least-squares method. Once these coefficients are specified, the acoustic pressure at any point, including the source surface, is completely determined. In this paper, this method is used to reconstruct the surface acoustic pressures based on the measured acoustic pressure signals in the field. It is shown that this method can be applied to both separable and nonseparable geometries, and the solutions thus obtained are unique. However, the convergence may be worse for an elongated object subject to high-frequency excitations.

The Discretized Mie-Formalism for plane wave scattering on dielectric objects with non-separable geometries

The knowledge of the scattering behaviour of different irregularly shaped particles is of particular interest in various fields of physics, astronomy and chemistry. In this paper, we present a uniform description of electromagnetic wave scattering on dielectric axisymmetric objects and infinitely extended cylinders with non-circular cross section. The analysis is based on the Discretized Mie-Formalism which provides an appropriate method to solve the corresponding Helmholtz equation with non-separable boundary conditions. Numerical results for spheres, spheroids, Chebyshev particles and cylinders with circular and elliptical cross sections are given.

Asymptotic analysis of SH-wave modes in geologic resonators (sedimentary basins) of non-separable geometry

The problem investigated in this paper is that of estimating the normal modes of oscillation (resonant frequencies and wave functions) of scalar waves in a two-dimensional enclosure or resonator. Because the study is aimed to applications in geologic (seismic) situations, we consider SH-waves and wave resonators limited above by a flat reflector - the earth's free surface - and below by a curved one. The curved reflector is smooth, but otherwise of arbitrary shape. Commonly occuring sediment-filled valleys or sedimentary basins with such configuration act as seismic wave resonators to incoming earthquake waves. Determining the eigenmodes of basins of arbitrary geometry is a difficult problem in seismic wave propagation; even to simulate the dynamic response of basin models of simple non-separable geometry often requires the use of sophisticated numeric codes and high-performance computers. The accurate determination of the eigenfrequencies is important because in practice aby buildings or engineered structures with natural frequencies of oscillation close or equal to those of the resonant basin, may themselves resonate with the ground motion. This coincidental match of oscillation frequencies, known as “double resonance” is potentially very destructive, and in most cases leads to the total collapse of the building. It thus becomes important to clearly understand the effects of realistic basin geometries on seismic waves, and especially to accurately predict the resulting predominant frequencies of ground motion. We simplify the calculation of ground motions through asymptotic and numeric schemes that can be programmed on a personal computer. For this purpose we use analytic approximations based upon semiclassical quantization; the so-called EBK (Einstein-Brillouin-Keller) method for estimating the eigenmodes of non-linearly coupled mechanical oscillators. The EBK method solves for the eigenvalues (resonant frequencies) by imposing resonance conditions on the accumulated phase of the trapped waves (seismic ray trajectories) over topologically independent closed paths in phase space. To obtain the wave functions associated with the resonant frequencies, we propose a class of equations of motion whose asymptotic solutions are described by Hermite polynomials with an argument that is itself a function of the valley's geometry. The accuracy of the resonant frequencies and wave functions thus obtained is demonstrated by comparisons with finite element results. The approach can be easily applied to resonating alluvial valleys with an upper surface free of tractopm and a lower boundary that can be assumed rigid.

Ray methods for modal fields in general closed and open propagation environments

The author presents a systematic methodology for using modal solutions in nonseparable geometries. Ray parametrization is used as a technique for constructing approximate modal wave fields to treat guiding and ducting environments with varying transverse and longitudinal properties. Ideal modes are decoupled from, and propagate independently of, other modes in the set. When the channel does not conform with separability in one of the basic coordinate systems, ideal modes defined for separable conditions become coupled. If the departure from separability is weak, one may define adiabatic modes (AMs) that adapt smoothly, without coupling to other AMs, to the slow nonseparable longitudinal variations, retaining their distinct spectral identity throughout. Each AM fails in its own cutoff region, across which its wavefield changes from being propagating to becoming evanescent, and it must there be extended into an intrinsic mode (IM) which has greater spectral content.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Reconstruction of surface velocity on axisymmetric structures having nonseparable geometry

An algorithm is developed for reconstruction of the normal velocity on the surface of an arbitrary axisymmetric structure from measurements of pressure in the nearfield. This high‐resolutinn reconstruction procedure is better suited for broadband data than the singular value decomposition method previously implemented [W. A. Veronesi and J. D. Maynard, J. Acoust. Soc. Am. 85, 588‐598 (1989); Borgiotti et al., J. Acoust. Soc. Am. Suppl. 1 84, S173 (1988) ], enabling reduced computation time and providing a new expansion for the surface field with a set of functions that can be expressed as real functions, independent of frequency and independent of ones choice of a measurement surface. The boundary element method is applied to compute the orthonormal set of functions and a projection matrix relating nearfield pressure values to surface normal velocity.

Local, high-frequency analysis of the fields in a mode-stirred chamber

The fields inside a mode-stirred chamber are analyzed by a local plane-wave method. This approach has the advantage of being applicable to nonseparable geometries, for which the more standard modal method of analysis is not possible, and gives insight into the nature of the various contributions to the total field. Results are given for the quality factor and for the fields near the walls of the chamber. The expression for the quality factor agrees with previously published work. >

On the Exact Solution of Boundary Value Problems with Non-separable Geometries, Like a Rough Surface

A Dirichlet problem is considered for the Helmholtz equation and for a class of geometries for which the Helmholts operator does not separate (e.g. a rough surface). It is shown that, contrary to the widely held view, it is possible to obtain the solution of this problem in a closed form which resembles closely the solutions obtained for separable geometries—expansions generated by Sturm-Liouville theory. As with separable geometries, we show in particular that the expansion coefficients can be written explicitly as integrals containing a priori known functions—that matrix inversion is not required for the determination of the expansion coefficients. The problem includes as a special case the scattering of electromagnetic waves from a rough cylindrical surface which is the boundary of a perfect conductor. The method is very general and can be used for much more complicated boundary value problems, such as scattering by dielectric interface.

Numerical considerations in ray tracing and ray expansions of the acoustics wavefield

In separable geometries (range‐independent sound speed profile, flat bottom) the reduced wave equation may be solved using the WKB technique. The resulting integral expression is a plane wave or ray expansion the acoustic wavefield. In nonseparable geometries the Maslov technique provides an expression of the same form. Use of the stationary phase technique reduces these integral expressions to standard ray theory. We focus our attention on numerical considerations associated with: (1) computing the geometric ray information which appears in these integral wavefield expressions; and (2) evaluating the integrals. Many commonly occurring problems associated with (1) (ray tracing) may be avoided by appropriately parametrizing the model and rewriting the travel time in terms of the phase integral, τ (sometimes referred to as the delay or intercept time). Problems associated with (2) are particularly troublesome when caustics are present. These may be avoided entirely by transforming to the time domain. [Work supported by ONR.]

Helmholtz-equation eigenvalues and eigenmodes for arbitrary domains

An integral equation technique is employed to obtain eigenvalues and eigenmodes of the homogeneous Holmholtz equation for a two- or three-dimensional closed region of arbitrary shape with arbitrary first-order homogeneous boundary conditions. The method is described for general (i.e., nonseparable) geometries, with a discussion of the simplification introduced by having a separable geometry given in an Appendix. A numerical example is given for a nonseparable geometry, i.e., a two-dimensional right triangle of arbitrary enclosed angle with Neumann boundary conditions. Results for the special case of an isosceles right triangle agree very well with a known analytical solution.

Eigenvalues for Wave Motion in Irregularly Shaped Regions

Eigenvalues are obtained by means of a Helmholtz integral equation formulation for wave resonances in two- and three-dimensional enclosed regions. Examples of circular and spherical regions are given to compare this approach to the simpler but less general separation of variables method. An example of a triangularly shaped two-dimensional region is given to illustrate the advantages of this approach, i.e., that only surface values of the dependent variables appear, allowing a fine grid to be used with relatively few points, as compared to a standard finite difference approach for a nonseparable geometry.