Kirchhoff-Helmholtz Integral Research Articles

Scattering from elastic objects in a waveguide using the finite element technique.

Computation of scattering from an object in a waveguide requires the solution of the wave equation by simultaneously satisfying boundary conditions on the surface of the object as well as the boundaries of the waveguide. An accurate solution requires that the computational domain be discretized at least at one-tenth of the acoustic wavelength. For a computational domain tens or hundreds of wavelength large, this results in very large systems of linear equations whose solution is a daunting numerical task. To cope with the numerical complexity, in this paper we solve this problem by using a combination of the finite element and the spectral integral techniques. The finite element technique is used to compute the scattered field and the spectral integral technique is used to compute the waveguide Green’s function. The scattered field and the waveguide Green’s functions are used in the Helmholtz–Kirchhoff integral to compute the field everywhere in the waveguide. The accuracy of the technique is tested by comparing its results with those from a full finite element solution in 2-D. The technique is then extended to 3-D.

Time‐domain numerical simulation of scattering by objects in the seafloor.

Numerical simulation of scattering in the time domain offers potential implementation advantages in comparison with frequency‐domain methods. By using explicit numerical time integration schemes, elastic stresses and velocities can typically be advanced using only spatially local information which makes parallelization by domain decomposition straightforward to implement in comparison with the solution of a large linear system of equations. On the other hand, incorporation of frequency‐dependent sound speed and attenuation appears more challenging in the time domain with the general need to compute convolutions under non‐FFT friendly conditions. This talk presents efficiency gains for the elastodynamic finite integration technique (EFIT) which is similar to the finite‐difference time‐domain method (FDTD). We present results on using explicit decomposition of scattered and incident wave fields and far‐field projection in the time domain with the discrete Helmholtz–Kirchhoff integral. We also present incorporation of frequency‐dependent material parameters by efficient recursive convolution techniques that have been used successfully to create time‐domain perfectly matched layers [J. A. Roden and S. D. Gedney, Microwave Opt. Technol. Lett. 27, 334–338, (2000)]. As an application example, we present results on computing bistatic target strength variance for a two‐dimensional object located near a statistically rough seafloor. [Work sponsored by the Office of Naval Research.]

Sound absorption characteristics of a honeycomb-backed microperforated panel absorber: Revised theory and experimental validation

A microperforated panel (MPP) absorber is known as one of the most promising alternatives of the next-generation sound absorbers. However, MPPs are usually made of thin limp materials and need to be reinforced by a supporting element. The proposed to use a honeycomb attached behind an MPP for this purpose, and have shown that the honeycomb in the back cavity of an MPP absorber is not only useful to stiffen the MPP but improves its sound absorption performance, particularly at low frequencies. In the authors previous studies an electro-acoustical circuit model was used to analyse its sound absorption characteristics, however, the model inevitably includes an approximation, and more exact theory needs to be established for better prediction of its characteristics. In this paper, the absorber is analysed with the wave theory based on Helmholtz-Kirchhoff integral equations. First, the formalism based on the Helmholtz-Kirchhoff integral is presented. Next, an experiment is made to validate the theory. Finally, a parametric study is made to discuss the effect of the parameters of the sound absorbing system on its sound absorption characteristics

Modeling the low‐ to mid‐frequency scattering from a proud cylinder: Boundary and near‐field effects.

A 2:1 aspect ratio solid aluminum cylinder is placed on the planar interface between two fluids (water/sand and water/air), with a point source radiating at frequencies for which the acoustic wavelengths range from 0.5 to 12 cylinder lengths. The distance between the source and the target is approximately 100 wavelengths, relative to the center frequency, and a vertical receive array is placed near the source. The problem is studied using a finite‐element target scattering model, which is capable of treating axially symmetric objects via the decomposition of the unknowns into a Fourier basis around the axis of symmetry. Since the axis of the cylinder is parallel to the planar interface, the overall problem is not axially symmetric. Nevertheless, an approximate solution is obtained, which takes into account the interface reflection of the incident field, as well as the first bounce of the scattered field (via the Helmholtz–Kirchhoff integral with layered medium Greens functions). Small variations in the so...

Benchmark problems for acoustic scattering from elastic objects in the free field and near the seafloor

Results from a workshop organized in 2006 to assess the state of the art in target scatter modeling are presented. The problem set includes free-field scenarios as well as scattering from targets that are proud, half-buried, or fully buried in the sediment. The targets are spheres and cylinders, of size O(1 m), which are insonified by incident plane waves in the low-frequency band 0.1-10 kHz. In all cases, the quantity of interest is the far-field target strength. The numerical techniques employed fall within three classes: (i) finite-element (FE) methods and (ii) boundary-element (BE) techniques, with different approaches to computing the far field via discretizations of the Helmholtz-Kirchhoff integral in each case, and (iii) semianalytical methods. Reference solutions are identified for all but one of the seven test problems considered. Overall, FE- and BE-based models emerge as those being capable of treating a wider class of problems in terms of target geometry, with the FE method having the additional advantage of being able to deal with complex internal structures without much additional effort. These capabilities are of value for the study of experimental scenarios, which can essentially be envisioned as variations of the problem set presented here.

Finite element modeling of acoustic scattering from rough interfaces.

The finite element method is applied to the problem of acoustic scattering from rough surfaces. This method has the advantage that the surface can have almost any form whereas analytic approximation methods generally require constrained surface roughness parameters. Scattering from a rough pressure release boundary and from a rough interface between two fluid media is considered in two dimensions. The problem domain is truncated with perfectly matched layers. These greatly decrease the number of degrees of freedom in the finite element problem while minimizing unwanted reflections of outbound energy back into the physical domain. The Helmholtz–Kirchhoff integral is used to extend the solution to points outside the modeled domain. Using finite elements as well as an exact integral equation method, solutions are obtained for multiple realizations of the rough scattering surface. The scattering strengths of energy reflected back into the water and of energy penetrating the sea floor in the case of bottom scattering are calculated over a range of scattering angles. The two exact numerical methods show good agreement. Results are compared with those of analytic approximation methods. [Work sponsored by ONR, Ocean Acoustics.]

Three-dimensional diffraction of a thin metallic cylinder illuminated in conical incidence: application to diameter estimation

We present a model to determine the far-field diffraction pattern of a metallic cylinder of infinite length when it is illuminated in oblique incidence. This model is based on the Helmholtz-Kirchhoff integral using the Beckmann conditions for reflection. It considers the three-dimensional nature of the diffracting object as well as the material of which the cylinder is made. This model shows that the diffraction orders are placed in a cone of light. The amplitude at the far field can be divided into three terms: the first term accounts for Babinet's principle, that is, the contribution of the cylinder projection; the second term accounts for the three dimensionality of the cylinder; and the third term accounts for the material of which the cylinder is made. This model is applied to the diameter estimation of the cylinder. Since the amplitude of the Babinet contribution is much larger than the light reflected by the surface, the cylinder diameter can be obtained in a simple way. With this approximation, the locations of the diffraction minima do not vary when the cylinder is inclined. On the other hand, when the reflected light is considered the location of the minima and, hence, the estimation of the diameter, varies. Also, a modification of the diffraction minima is produced by the material of which the cylinder is made. Experimental results are also obtained that corroborate the theoretical approach.

Experiments and numerical modeling of low to midfrequency scattering from elastic objects near the sea floor

The scattering of low to mid-frequency sound (1-10's of kHz) from submerged elastic structures of size O(1m) is a topic of interest to the underwater acoustics community. In the first part of the presentation, a brief description of the relevant components of the EVA experiment is given. The purpose of the sea trial was the acquisition of high-fidelity echoes from submerged spherical and cylindrical targets, made of composite materials with internal layered structure. The second part of the presentation is focused on the finite-element modeling technique developed at NURC for investigating the scattering from axially symmetric submerged elastic objects. Particular attention is dedicated to the computation of the far field at a distance from the target via the Helmholtz-Kirchhoff integral, using the near field sampled on the target surface, together with Green's functions capable of describing a two-layered water-sediment fluid medium. Those geometries, for which the overall axial symmetry is broken by the presence of the water-sediment boundary, can be treated approximately by taking into account the boundary-reflected incident field, as well as the first order interaction between the target-scattered echo and the sea floor. The numerical technique is validated by comparison with data collected during the EVA trial.

Exact boundary integral equations for scattering of scalar waves from perfectly reflecting infinite rough surfaces

The derivation of integral equations for scattering from an infinite rough surface is not a straightforward procedure. The usual formalism involves a single plane wave incident on the surface and the free-space Green’s function. Problems arise due to waves propagating horizontal to the surface and to the inability to evaluate in a simple way the Helmholtz–Kirchhoff or Green’s surface integral on the upper hemisphere. One approach is to change the usual Sommerfeld radiation condition to exclude horizontal waves. In this paper we take a different approach. Instead of the single incident plane wave as the Born term we choose the flat surface result of incident plus reflected field. The additional field scattered from the rough surface satisfies the Sommerfeld condition. The full Green’s function is chosen as a combination of the free-space Green’s function and its image. The main result is that the Helmholtz–Kirchhoff integral using this new Born term and the image Green’s function can be evaluated exactly (and simply) on the hemisphere. This leads directly to integral equations on the total field (flat surface plus scattered fields) for the Dirichlet and Neumann problems. Simple coordinate-space representations for the kernels of these equations are also presented. Since we use image functions we have standing waves in the z-direction, so that the result is not strictly categorized as a diffraction result. Nevertheless, the integral relations induced by their choice yield an exact and simple result for the solution of an otherwise difficult problem.

The use of low-frequency noise in passive tomography of the ocean

A possible design of the mode tomography of the ocean with the use of a scheme requiring no expensive low-frequency radiators is considered. The design is based on the widely discussed method of estimating the Green’s function from the cross-coherence function of noise field received in a great number of observation points. The relationship between the Green’s function and the noise coherence function is derived from the Helmholtz-Kirchhoff integral. The use of the vertical multielement arrays composed of vector receivers is suggested to decrease the duration of noise signal accumulation required for a reliable determination of the Green’s function. The solution of the tomographic problem is based on the determination of the mode structure of acoustic field from the eigenvectors and eigenvalues of the cross-coherence matrix of the received noise field.

Semi-analytical computation of the acoustic field of a segment of a cylindrically concave transducer in lossless and attenuating media

Conventional ultrasound transducers used for medical diagnosis generally consist of linearly aligned rectangular apertures with elements that are focused in one plane. While traditional beamforming is easily accomplished with such transducers, the development of quantitative, physics-based imaging methods, such as tomography, requires an accurate, and computationally efficient, model of the field radiated by the transducer. The field can be expressed in terms of the Helmholtz-Kirchhoff integral; however, its direct numerical evaluation is a computationally intensive task. Here, a fast semianalytical method based on Stepanishen's spatial impulse response formulation [J. Acoust. Soc. Am. 49, 1627-1638 (1971)] is developed to compute the acoustic field of a rectangular element of cylindrically concave transducers in a homogeneous medium. The pressure field, for, lossless and attenuating media, is expressed as a superposition of Bessel functions, which can be evaluated rapidly. In particular, the coefficients of the Bessel series are frequency independent and need only be evaluated once for a given transducer. A speed up of two orders of magnitude is obtained compared to an optimized direct numerical integration. The numerical results are compared with Field II and the Fresnel approximation.

On the fundamentals of the virtual source method

The virtual source method (VSM) has been proposed as a practical approach to reduce distortions of seismic images caused by shallow, heterogeneous overburden. VSM is demanding at the acquisition stage because it requires placing downhole geophones below the most complex part of the heterogeneous overburden. Where such acquisition is possible, however, it pays off later at the processing stage because it does not require knowledge of the velocity model above the downhole receivers. This paper demonstrates that VSM can be viewed as an application of the Kirchhoff-Helmholtz integral (KHI) with an experimentally measured Green’s function. Direct measurement of the Green’s function ensures the effectiveness of the method in highly heterogeneous subsurface conditions.

Analysis of sound radiation from the coupling effect of vibrating noise and moving-vehicle noise on bridges

An acoustic finite element model of a bridge is developed to evaluate the noise generated by the traffic-induced vibration of the bridge. The dynamic response of a multi-girder bridge, modeled by a three-dimensional (3-D) frame element model, is analyzed with a 3-axle (8 degrees of freedom (DOF)) truck model and a 5-axle (13 DOF) tractor-trailer. The flat plate element is used to analyze the acoustic pressure due to the fluid–structure interactions between the vibrating surface and contiguous acoustic fluid medium. The radiation fields of noise with a specified distribution of vibrating velocity and pressure on the structural surface are also computed using the Kirchhoff–Helmholtz integral. Among the diverse parameters affecting the dynamic response of a bridge, vehicle velocity, vehicle weight, and spatial distribution of the road surface roughness are found to be the main factors that increase the level of vibration noise. In an attempt to illustrate the influence of the structural vibration noise of a bridge to total noise level around the bridge, the random function is used to generate the vehicle noise source including the engine noise and the rolling noise between the road and tire. The results show that the low-frequency noise produced by the vibrating bridge members amplifies the high-frequency vehicle noise by 4–7 dB. In addition, the amplification rate of noise increases with traveling speed and vehicle weight. Key words: acoustic pressure on surface, sound radiation, noise level, Kirchhoff–Helmholtz integrals, dynamic response, vehicle noise model, sound pressure level.

Diffraction of cylinders with longitudinal surface structures.

We present a model to determine the light scattered by a metallic cylinder with longitudinal structures when the cylinder is illuminated by a Gaussian light beam in oblique incidence. The model is based on an approximate solution to the Helmholtz-Kirchhoff integral by means of the stationary-phase method. We have studied the variations of the diffraction pattern in terms of the size of the defect and other geometrical parameters. The width of the beam and the misalignment between the beam and the cylinder have also been considered, as well as the optical properties of the surface.

Kirchhoff-approximate inversion of teleseismic wavefields

Summary We derive Kirchhoff-approximate inversion formulae for elastic wavefields that recover the location of discontinuity surfaces and associated material property contrasts in laterally varying, stratified media. The derivation is cast in the context of teleseismic wavefield scattering in a 2-D medium with allowance made for oblique incidence, although a fully general derivation for 3-D media follows in straightforward fashion. We exploit a little-used variant of the isotropic, elastic Kirchhoff-Helmholtz integral in which individual terms are directly identified with scattered P- and S-wave contributions prior to approximation using ray-theoretic forms. This approach yields relatively simple formulae for Kirchhoff-approximate forward modelling that bear a closer resemblance to their acoustic counterpart than standard elastic formulations cast in terms of traction and displacement. Using micro-local analysis, inversion formulae are readily derived using the generalized Radon transform. Our approach represents an extension of the Kirchhoff-approximate inversion scheme outlined by Beylkin & Burridge (1990) to S-waves and conversions. We demonstrate application of the method to field data recorded during the IRIS-PASSCAL CASC93 experiment.

The Kirchhoff-Helmholtz Integral Pair

The Kirchhoff–Helmholtz integral models the reflected acoustic wavefield by an integration along the reflector over the incident field multiplied by the specular plane-wave reflection coefficient. Based on the structural relationships between the reflector and the reflection-traveltime surface, we design an asymptotic inverse Kirchhoff–Helmholtz integral. Analogously to the forward integral, the proposed inverse consists of an integration along the reflection-traveltime surface over the recorded reflected field. We show that the new inverse integral asymptotically recovers the input to the standard Kirchhoff–Helmholtz integral, that is, the reflector position and the reflection coefficients along it. A simple numerical example demonstrates the inverse relationship between the proposed and the standard Kirchhoff–Helmholtz integrals. In this way, a new technique for kinematic (positioning) and dynamic (amplitude) wavefield inversion becomes available. This is realized by means of an integral operation that is most naturally related to its counterpart Kirchhoff–Helmholtz integral.

THE KIRCHHOFF–HELMHOLTZ INTEGRAL PAIR

The Kirchhoff–Helmholtz integral models the reflected acoustic wavefield by an integration along the reflector over the incident field multiplied by the specular plane-wave reflection coefficient. Based on the structural relationships between the reflector and the reflection-traveltime surface, we design an asymptotic inverse Kirchhoff–Helmholtz integral. Analogously to the forward integral, the proposed inverse consists of an integration along the reflection-traveltime surface over the recorded reflected field. We show that the new inverse integral asymptotically recovers the input to the standard Kirchhoff–Helmholtz integral, that is, the reflector position and the reflection coefficients along it. A simple numerical example demonstrates the inverse relationship between the proposed and the standard Kirchhoff–Helmholtz integrals. In this way, a new technique for kinematic (positioning) and dynamic (amplitude) wavefield inversion becomes available. This is realized by means of an integral operation that is most naturally related to its counterpart Kirchhoff–Helmholtz integral.

The Kirchhoff–Helmholtz integral for anisotropic elastic media

The Kirchhoff–Helmholtz integral is a powerful tool to model the scattered wavefield from a smooth interface in acoustic or isotropic elastic media due to a given incident wavefield and observation points sufficiently far away from the interface. This integral makes use of the Kirchhoff approximation of the unknown scattered wavefield and its normal derivative at the interface in terms of the corresponding quantities of the known incident field. An attractive property of the Kirchhoff–Helmholtz integral is that its asymptotic evaluation recovers the zero-order ray theory approximation of the reflected wavefield at all observation points where that theory is valid. Here, we extend the Kirchhoff–Helmholtz modeling integral to general anisotropic elastic media. It uses the natural extension of the Kirchhoff approximation of the scattered wavefield and its normal derivative for those media. The anisotropic Kirchhoff–Helmholtz integral also asymptotically provides the zero-order ray theory approximation of the reflected response from the interface. In connection with the asymptotic evaluation of the Kirchhoff–Helmholtz integral, we also derive an extension to anisotropic media of a useful decomposition formula of the geometrical spreading of a primary reflection ray.

Asymmetric transmissive behavior of liquid-crystal diffraction gratings

We have used computer simulations to predict that the beam-steering efficiency of a common liquid-crystal diffraction grating will depend on which side is presented to the incident beam. The finite-difference time-domain method and the Helmholtz-Kirchhoff diffraction integral were utilized to simulate the performance of an idealized configuration of the grating.

Diffraction by cylinders illuminated in oblique, off-axis incidence

Abstract In this paper we present a model to determine the light scattered by a metallic cylinder when it is illuminated with a light beam in oblique incidence. This model is based on an approximate solution to the Helmholtz-Kirchhoff integral by means of the Stationary Phase Method. The polarization of the beam, its width, and the misalignment between the beam and the cylinder are considered, as well as the reflection coefficient of the surface.