Helmholtz Equation In Spherical Coordinates Research Articles

Using learned priors to regularize the Helmholtz equation least-squares method

The Helmholtz equation least-squares (HELS) method is a valuable tool for estimating equivalent sound sources of a radiating object. It solves an inverse problem by mapping measured pressures to a set of basis functions satisfying the Helmholtz equation in spherical coordinates. However, this problem is often ill-posed, necessitating additional regularization methods, in which often variations of Ridge or Lasso are used. These conventional methods do not explicitly consider the distribution underlying the source radiations (besides sparsity) and are often used in the context of obtaining only a point estimate, even in the presence of ambiguity in the data. In this work, we propose the use of empirical priors through a normalizing flow model to enhance the inversion results obtained with the HELS method. We first validate our approach using numerical data and subsequently demonstrate its superior performance in interpolating a measured violin directivity compared to Lasso and Ridge methods, even when optimal regularization parameters are selected.

Localized plasmon resonance

A new analytical solution of the Helmholtz equation in spherical coordinates for an infinite homogeneous space, as well as the solution of the boundary problem for a piecewise-homogeneous medium (space divided into homogeneous domains with various dielectric permittivities of the spherical symmetry borders) has been obtained. The application of the derived formulas to spherical nanoparticles of metal makes it possible to record a mathematical relation to the localized plasmon resonance, which connects the resonance parameters of such a system and the electromagnetic waves. The equation of plasmon resonance confirms the known experimental result of significant amplification of an electromagnetic field at the localized plasmon resonance. The condition of the optical resonance obtained in the work is applicable to layered spheres with an arbitrary number of layers of any size and material.

Acoustical tweezers using single spherically focused piston, X-cut, and Gaussian beams

Partial-wave series expansions (PWSEs) satisfying the Helmholtz equation in spherical coordinates are derived for circular spherically focused piston (i.e., apodized by a uniform velocity amplitude normal to its surface), X-cut (i.e., apodized by a velocity amplitude parallel to the axis of wave propagation), and Gaussian (i.e., apodized by a Gaussian distribution of the velocity amplitude) beams. The Rayleigh-Sommerfeld diffraction integral and the addition theorems for the Legendre and spherical wave functions are used to obtain the PWSEs assuming weakly focused beams (with focusing angle α ⩽ 20°) in the Fresnel-Kirchhoff (parabolic) approximation. In contrast with previous analytical models, the derived expressions allow computing the scattering and acoustic radiation force from a sphere of radius a without restriction to either the Rayleigh (a ≪ λ, where λ is the wavelength of the incident radiation) or the ray acoustics (a ≫λ) regimes. The analytical formulations are valid for wavelengths largely exceeding the radius of the focused acoustic radiator, when the viscosity of the surrounding fluid can be neglected, and when the sphere is translated along the axis of wave propagation. Computational results illustrate the analysis with particular emphasis on the sphere's elastic properties and the axial distance to the center of the concave surface, with close connection of the emergence of negative trapping forces. Potential applications are in single-beam acoustical tweezers, acoustic levitation, and particle manipulation.

The Asymptotic Eigenvalues of First-Order Spectral Differentiation Matrices

We complete and extend the asymptotic analysis of the spectrum of Jacobi Tau approximations that were first considered by Dubiner. The asymptotic formulas for Jacobi polynomials PN(α ,β ) ,α ,β > -1 are derived and confirmed by numerical approximations. More accurate results for the slowest decaying mode are obtained. We explain where the large negative eigenvalues come from. Furthermore, we show that a large negative eigenvalue of order N2 appears for -1 1 . The eigenvalues for Legendre polynomials are directly related to the roots of the spherical Bessel and Hankel functions that are involved in solving Helmholtz equation inspherical coordinates.

Shear wave induced resonance elastography of spherical masses with polarized torsional waves

Shear Wave Induced Resonance (SWIR) is a technique for dynamic ultrasound elastography of confined mechanical inclusions. It was developed for breast tumor imaging and tissue characterization. This method relies on the polarization of torsional shear waves modeled with the Helmholtz equation in spherical coordinates. To validate modeling, an in vitro set-up was used to measure and image the first three eigenfrequencies and eigenmodes of a soft sphere. A preliminary in vivo SWIR measurement on a breast fibroadenoma is also reported. Results revealed the potential of SWIR elastography to detect and mechanically characterize breast lesions for early cancer detection.

Orthogonal Polynomials and Related Special Functions Applied in Geosciences and Engineering Computations

In applications of mathematics involving either the Laplace or the Helmholtz equation in spherical coordinates the associated Legendre equation occurs. Its solutions are called associated Legendre functions. They have some relations to classical Legendre orthogonal polynomials and classical Jacobi orthogonal polynomials. The classical Legendre polynomials and the associated Legendre functions have been used by some authors at the approximation of length of curves and cylindrical surfaces and the approximation of Earth shape in geosciences and engineering computations. In the present paper we investigate some relations between these functions pointing out possible applications in geosciences.

Patch Near-field Acoustic Holography by Extrapolation Using Orthogonal Spherical Wave Source

In order to reduce the reconstructed errors, the near-field acoustic holography requires the measurement aperture larger than the sound source area. This limitation causes heavy measurement work and makes the NAH difficult to implement on large-scale structure. To solve this problem Patch near-field acoustic holography based on hologram pressure extrapolation is developed. A new patch near-field acoustic holography (PNAH) based on exploration using orthogonal spherical wave source is proposed. According to the solution of Helmholtz equation in spherical coordinates, the sound field can be approximately expressed as a linear sum of a series of orthogonal spherical wave function, so that the sound field generated by vibrator is approximated by superposing fields generated by a series of orthogonal spherical wave sources of different orders and the pressure exploration is realized by radiating process of these spherical wave sources. After near-field exploration the fast Fourier transform (FFT) method is implemented for reconstructing the normal velocity or the field on and near the radiator. The results of numerical simulations show that the method can work well in the small measurement aperture, reduce the requirement of measurement aperture area, and is convenient for application.

Near‐field acoustic holography in conical coordinates.

Near‐field acoustical holography (NAH) techniques can be optimized if the method capitalizes on the geometry of the noise source under investigation. Helmholtz‐equation least squares method (HELS) uses the solutions of Helmholtz equation in spherical coordinates as basis functions for the pressure field. HELS is an efficient NAH technique if the source and the measurement surfaces are spherical in nature. For nonspherical cases, such as radiation from a plate or bar, it takes a large number of functions to represent the field. In these cases, there is also a question about where to place the origin of the wave functions. In search of a HELS‐type method that could be applied to nonspherical sources, a study into the features of conical coordinates has been conducted. Because the Helmholtz equation is separable in conical coordinates, the solutions can be used, in a manner similar to HELS, as basis functions to represent the pressure field. For conical coordinates, the basis functions are spherical Hankel functions and Lame functions. This HELS‐type formulation in conical coordinates could be a natural choice for NAH on conical sources and may be appropriate for NAH on jet noise. [Work supported by Blue Ridge Research and Consulting and Air Force Research Laboratory.]

Recursions for the Computation of Multipole Translation and Rotation Coefficients for the 3-D Helmholtz Equation

We develop exact expressions for the coefficients of series representations of translations and rotations of local and multipole fundamental solutions of the Helmholtz equation in spherical coordinates. These expressions are based on the derivation of recurrence relations, some of which, to our knowledge, are presented here for the first time. The symmetry and other properties of the coefficients are also examined and, based on these, efficient procedures for calculating them are presented. Our expressions are direct and do not use the Clebsch--Gordan coefficients or the Wigner 3-j symbols, although we compare our results with methods that use these to prove their accuracy. For evaluating an Nt term truncation of the translated series (involving $O(N_t^2)$ multipoles), our expressions require $O(N_t^3)$ evaluations, compared to previous exact expressions that require $O(N_t^5)$ operations.

Surface Green's function of the Helmholtz equation in spherical coordinates — Abstract

The surface Green's function belonging to the non‐spherical exterior boundary value problem of Helmholtz's equation in spherical coordinates is derived. This is performed in two ways, first by applying the Separation of Variables method, and, second, by using the Method of Lines as a special Finite‐Difference technique. With this Green's function we are able to resolve some contradictions concerning conceptual aspects of the Separation of Variables method, the Finite‐Difference methods, and the Boundary Integral Equation methods which have been developed for rigorously solving non‐separable boundary value problems. The necessary mathematical background, the relation to Waterman's T matrix, and simplifications due to certain symmetry properties of the boundary surface will be discussed. In this paper we focus on the scalar problem. The extension to the vector case for electromagnetic wave scattering is in preparation and will be published later.

Surface Green's Function of the Helmholtz Equation in Spherical Coordinates

The surface Green's function belonging to the non-spheri- cal exterior boundary value problem of Helmholtz's equation in spher- ical coordinates is derived. This is performed in two ways, first by applying the Separation of Variables method, and, second, by using the Method of Lines as a special Finite-Difference technique. With this Green's function we are able to resolve some contradictions con- cerning conceptual aspects of the Separation of Variables method, the Finite-Difference methods, and the Boundary Integral Equation methods which have been developed for rigorously solving non-sepa- rable boundary value problems. The necessary mathematical back- ground, the relation to Waterman's T matrix, and simplifications due to certain symmetry properties of the boundary surface will be discussed. In this paper we focus on the scalar problem. The extension to the vector case for electromagnetic wave scattering is in preparation and will be published later.

Sensitivity studies for imaging a spherical object embedded in a spherically symmetric, two-layer turbid medium with photon-density waves

We present analytic expressions for the amplitude and phase of photon-density waves in strongly scattering, spherically symmetric, two-layer media containing a spherical object. This layered structure is a crude model of multilayered tissues whose absorption and scattering coefficients lie within a range reported in the literature for most tissue types. The embedded object simulates a pathology, such as a tumor. The normal-mode-series method is employed to solve the inhomogeneous Helmholtz equation in spherical coordinates, with suitable boundary conditions. By comparing the total field at points in the outer layer at a fixed distance from the origin when the object is present and when it is absent, we evaluate the potential sensitivity of an optical imaging system to inhomogeneities in absorption and scattering. For four types of background media with different absorption and scattering properties, we determine the modulation frequency that achieves an optimal compromise between signal-detection reliability and sensitivity to the presence of an object, the minimum detectable object radius, and the smallest detectable change in the absorption and scattering coefficients for a fixed object size. Our results indicate that (l) enhanced sensitivity to the object is achieved when the outer layer is more absorbing or scattering than the inner layer; (2) sensitivity to the object increases with the modulation frequency, except when the outer layer is the more absorbing; (3) amplitude measurements are proportionally more sensitive to a change in absorption, phase measurements are proportionally more sensitive to a change in scattering, and phase measurements exhibit a much greater capacity for distinguishing an absorption perturbation from a scattering perturbation.

What is the confocal parameter?

A novel derivation is presented of the Gaussian beam as a limit of the solution to the full wave equation. Usually, the functional form of the Gaussian beam is found by a two-step process. First, the Green's function of the paraxial wave equation is identified. Then, since the paraxial wave equation is invariant under translation, the z-axis variable is replaced by z+jb. It is shown that when starting with a solution of the full three-dimensional Helmholtz equation in spherical coordinates, performing the transformation z to z+jb corresponds physically to causing the phase fronts of the solution to become ellipsoids. The separation of the foci of the ellipsoids is 2b, where b is the confocal parameter of the beam. In the paraxial limit the ellipsoidal solution becomes a Gaussian beam. Adopting this approach to Gaussian beams allows a simple, geometrical interpretation of the optical resonator stability criterion.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>