Kirchhoff Formulation Research Articles

Efficient prediction methods for the micro-pressure wave from a high-speed train entering a tunnel using the Kirchhoff formulation.

The compression wave generated by a high-speed train emerging from the exit portal of a tunnel gives rise to an impulsive noise called a micro-pressure wave. In this study, new methods for the prediction of sonic-boom noise are proposed. The first method combines acoustic monopole analysis and the method of characteristics with the Kirchhoff method. The compression wave from a train entering a tunnel is calculated by an approximate compact Green's function, and the resultant noise at the tunnel exit is predicted by a linear Kirchhoff formulation. The second method couples the Kirchhoff formulation with the Euler equation, which is solved numerically for the generation and propagation of the compression wave. Numerical prediction of the compression wave, the propagation in the tunnel, and the micro-pressure wave obtained by the present methods are compared with measured data. The numerical results exhibit a reasonable agreement with the experimental data. The proposed methods in this study are shown to be very useful design tools for the nose shape of trains and the geometry of tunnels, and they can be utilized to minimize the pressure fluctuation in the tunnel and the corresponding booming noise.

PREDICTION AND VALIDATION ON THE SONIC BOOM BY A HIGH-SPEED TRAIN ENTERING A TUNNEL

A compression wave generated when a high-speed train emerges from the exit portal of a tunnel, causes an impulsive noise called micro-pressure wave. There are various countermeasures to reduce the micro-pressure wave, the most effective one being the proper design of the nose shape of the train. In the present study, we adopted a new method for the prediction of sonic boom noise, considering the effect of the nose shape on the resultant noise. The Euler equation is first solved, after which the linear Kirchhoff formulation is used for the prediction of farfield acoustics from the flow-field data.Experimental investigation is also carried out on the pressure fluctuations in the tunnel and the micro-pressure wave with parameters such as train speed, blockage ratio, nose shape of train and air-shafts. The computational prediction and experimentally measured data, compared in this paper, show a good agreement with each other. The results shows that several countermeasures could be used for the efficient reduction of the micro-pressure wave.

Gaussian imaging transformation for the paraxial Debye formulation of the focal region in a low-Fresnel-number optical system

The Debye formulation of focused fields has been systematically used to evaluate, for example, the point-spread function of an optical imaging system. According to this approximation, the focal wave field exhibits some symmetries about the geometrical focus. However, certain discrepancies arise when the Fresnel number, as viewed from focus, is close to unity. In that case, we should use the Kirchhoff formulation to evaluate accurately the three-dimensional amplitude distribution of the field in the focal region. We make some important remarks regarding both diffraction theories. In the end we demonstrate that, in the paraxial regime, given a defocused transverse pattern in the Debye approximation, it is possible to find a similar pattern but magnified and situated at another plane within the Kirchhoff theory. Moreover, we may evaluate this correspondence as the action of a virtual thin lens located at the focal plane and whose focus is situated at the axial point of the aperture plane. As a result, we give a geometrical interpretation of the focal-shift effect and present a brief comment on the problem of the best-focus location.

Analytical Comparison of the Acoustic Analogy and Kirchhoff Formulation for Moving Surfaces

The Lighthill acoustic analogy, as embodied in the Ffowcs Williams-Hawkings (FW-H) equation, is compared with the Kirchhoff formulation for moving surfaces. A comparison of the two governing equations reveals that the primary advantage of the Kirchhoff formulation (namely, that nonlinear flow effects are included in the surface integration) is also available to the FW-H method if the integration surface used in the FW-H equation is not assumed to be impenetrable. The FW-H equation is analytically superior for aeroacoustics because it is based on the conservation laws of fluid mechanics rather than on the wave equation. Thus, the FW-H equation is valid even if the integration surface is in the nonlinear region. This advantage is demonstrated numerically. With the Kirchhoff approach, substantial errors can result if the integration surface is not positioned in the linear region, and these errors may be hard to identify. Finally, new metrics, based on the Sobolev norm, are introduced that may be used to compare input data for both quadrupole noise calculations and Kirchhoff noise predictions.

Modelization of Kirchhoff scattering by a sound ray algorithm

The problem of the modelization of sound scattering by rough surfaces arises in room acoustics computer simulations, when sound diffusion is taken into account, as well as in other acoustical fields of interest. It is shown here how a sound ray algorithm can be applied to simulate a scattering process obeying the Kirchhoff tangent plane approximation. The Kirchhoff formulation for a random rough surface is briefly recalled, with the shadowing function proposed by Bass and Fuks. The scattered intensity is directly proportional to the statistical distribution of the surface slopes, which here is kept as general as possible. Then, it is established that this formulation is strictly equivalent to the result of a suitable sound ray algorithm, when the wavelength approaches zero in the Kirchhoff formulation. This is demonstrated in theory, and then applied to several Gaussian rough surfaces. Moreover, hints are given to extend the mathematical analysis of the sound ray technique to the second-order reflections on the rough surface. This is viewed as a way to take into account multiple scattering, which is neglected by the Kirchhoff approximation. [Work supported by FNRS, Natl. Fund for Scientific Research.]

A note on the Kirchhoff formulation of diffraction by a plane screen

A note on the Kirchhoff formulation of diffraction by a plane screen

A note on the kirchhoff formulation of diffraction by a plane screen

The inconsistency of the Kirchhoff diffraction theory is that the wave field and its normal derivative cannot simultaneously be imposed on the plane of the diffracting aperture. Considering a different surface drawn into free space, it is possible to show that the Kirchhoff integral formula becomes consistent with the boundary value imposed.

Analytical comparison of the acoustic analogy and Kirchhoff formulation for moving surfaces

The Lighthill acoustic analogy, as embodied in the Ffowcs Williams-Hawkings (FW-H) equation, is compared with the Kirchhoff formulation for moving surfaces. A comparison of the two governing equations reveals that the main Kirchhoff advantage (namely nonlinear flow effects are included in the surface integration) is also available to the FW-H method if the integration surface used in the FW-H equation is not assumed impenetrable. The FW-H equation is analytically superior for aeroacoustics because it is based upon the conservation laws of fluid mechanics rather than the wave equation. This means that the FW-H equation is valid even if the integration surface is in the nonlinear region. This is demonstrated numerically in the paper. The Kirchhoff approach can lead to substantial errors if the integration surface is not positioned in the linear region. These errors may be hard to identify. Finally, new metrics based on the Sobolev norm are introduced which may be used to compare input data for both quadrupole noise calculations and Kirchhoff noise predictions.

A superior Kirchhoff method for aeroacoustic noise prediction: The Ffowcs Williams–Hawkings equation

The Lighthill acoustic analogy, as embodied in the Ffowcs Williams–Hawkings (FW–H) equation, is compared with the Kirchhoff formulation for moving surfaces. A comparison of the two governing equations reveals that the main Kirchhoff advantage (namely, nonlinear flow effects are included in the surface integration) is also available to the FW–H method if the integration surface used in the FW–H equation is not assumed impenetrable. The FW–H equation is analytically superior for aeroacoustics because it is based upon the conservation laws of fluid mechanics rather than the wave equation. Hence, the FW–H equation is valid even if the integration surface is in the nonlinear region. This is demonstrated numerically for helicopter rotor applications in the paper. The Kirchhoff approach can lead to substantial errors if the integration surface is not positioned in the linear region (i.e., if the input data are not a solution to the wave equation). These errors may be hard to identify in some cases.

Comparison of Computational Aeroacoustic Prediction Methods for Transonic Rotor Noise

This paper compares two methods for predicting transonic rotor noise for helicopters in hover and forward flight. Both methods rely on a computational fluid dynamics (CFD) solution as input to predict the acoustic near and far fields. For this work, the same full-potential rotor code has been used to compute the CFD solution for both acoustic methods. The first method employs the acoustic analogy as embodied in the Ffowcs Williams-Hawkings equation, including the quadrupole term. The second method uses a rotating Kirchhoff formulation. Computed results from both methods are compared with one another and with experimental data for both hover and advancing rotor cases. The results are quite good for all cases tested. The Kirchhoff method was somewhat sensitive to the location of Kirchhoff surface, if the surface was positioned too close to the rotor blade. The acoustic analogy method was not as sensitive to the extent of volume included in the quadrupole calculation. The computational requirements of both methods are comparable; in both cases these requirements are much less than the requirements for the CFD solution.

Experimental studies of bistatic scattering from two-dimensional conducting random rough surfaces

Despite the recent development of analytical and numerical techniques for problems of scattering from two-dimensional rough surfaces, very few experimental studies were available for verification. The authors present the results of millimeter-wave experiments on scattering from two-dimensional conducting random rough surfaces with Gaussian surface roughness statistics. Machine-fabricated rough surfaces with controlled roughness statistics were examined. Special attention was paid to surfaces with large rms slopes (ranging from 0.35 to 1.00) for which enhanced backscattering is expected to take place. Experimentally, such enhancement was indeed observed in both the copolarized and cross-polarized returns. In addition, it was noticed that at moderate angles of incidence, the scattering profile as a function of observation angle is fairly independent of the incident polarization and operating frequency. This independence justifies the use of the geometric optics approximation embodied in the Kirchhoff formulation for surfaces with large surface radius of curvature. When compared with the experimental data, this analytical technique demonstrates good agreement with the experimental data.

Prediction of the far-field jet noise using Kirchhoff's formulation

The Kirchhoff method consists of the calculation of the nonlinear near- and midfields, usually numerically, with the far-field solutions found from a linear Kirchhoff formulation evaluated on a surface D surrounding the nonlinear field. The surface S is assumed to include all the nonlinear flow effects and noise sources

Comparison of two FDTD near-field to near-fieldtransformations applied to pulsed antenna problems

Recently, a time-domain, near-field to near-field transformation, based on the Stratton-Chu representation, was presented for use with the FDTD method. In the Letter, an alternative near-field to near-field transformation based on the Kirchhoff representation is described. Savings of both computation time and computer memory are achieved using the Kirchhoff formulation. Comparisons of the savings attained are illustrated through an example, the application of the transformation to a TEM horn antenna for pulse radiation.

Interpretation of Sea Beam backscatter data collected at the Laurentian fan off Nova Scotia using acoustic backscatter theory

The purpose is to use acoustic scattering theory and Sea Beam measurements to estimate seafloor roughness parameters. The Sea Beam backscatter data are from a test area in the Laurentian fan, a relatively flat region. Sidescan sonarlike images were reconstructed from the multibeam data. These images in the test area show two distinctly different types of areas (A) and (B). The backscatter model uses the Helmholtz–Kirchhoff formulation of scattering theory and correlation function C(r)=exp[−‖r/l‖n], where r is the displacement, l is the correlation length, and n is the exponent. A single rough interface model fits the backscatter data in (A). The root-mean-square roughness was 6–8 cm and the correlation lengths were 140–270 cm. The exponent n ranged from 0.95 to 1.5. The type (B) areas required a two-layer model: the interfaces in the first type of areas (A) is covered by a sediment having a smoother surface. The rms roughness of the covering sediments were about 3 cm, the exponent n was nearly 2 and correlation lengths l were 90–120 cm. These acoustic models are consistent with the geological setting and processes.

Optimum target detection in an uncertain reverberation-limited environment

In a reverberation-limited environment, can imperfect knowledge of the spatial correlation properties of the underlying seafloor geomorphology impact active target detection performance? This work seeks to address this question by examining the dependence of the joint probability density function of the scattered acoustic field at low to moderate incidence angles on the seafloor roughness correlation function. The objective is the development of an optimum decision theoretic approach to the problem of high-frequency active detection of a target suspended in the water column in the vicinity of the seafloor. Acoustic modeling is a geologically meaningful seafloor scattering environment is performed by applying the 3-D Helmholtz–Kirchhoff formulation to the anisotropic, power-law description of seafloor relief due to Goff and Jordan. The generality of the Bayesian approach permits uncertainty with respect to geoacoustic environmental properties to be defined probabilistically and incorporated into the detection algorithm. The method attempts to exploit the spatial coherence of the scattered sound field in constructing the joint pdf of the scattered pressure field measured at an N element array of sensors. Upper bounds on detection performance in terms of ROC curves are obtained as a function of environmental uncertainty. [Work supported by ONR, Ocean Acoustics.]

Distribution of magma beneath the East Pacific Rise near the 9°03′N overlapping spreading center from forward modeling of common depth point data

We have reprocessed six cross‐axis and three along‐axis common depth point (CDP) profiles near the 9°03′N overlapping spreading center (OSC) to understand the relationship between axial magma chamber (AMC) width and seafloor morphology. Travel time modeling of the AMC reflector reveals an asymmetric distribution of melt across the 9°03′N OSC. The variation of modeled AMC width beneath either OSC limb is minimal, but the width increases nearly fourfold across the offset attaining an estimated maximum width of 4.15 km near the 9°17′N ridge axis discontinuity. Additionally, melt distribution underlying the eastern rise limb is not symmetric with respect to the rise axis/neovolcanic zone but is displaced toward the western rise flank. Depth migration, based on a continuum velocity model consistent with postcritical reflections from the base of layer 2A, places the skewed AMC reflector beneath a nearly constant thickness sheeted dike section which dips ∼ 10° away from the rise axis. To confirm AMC continuity beneath the western rise flank, we use the Maslov synthetic seismogram method to show that amplitude enhancement of the AMC reflector is consistent with a continuous melt body underlying a thickening extrusive layer. Analysis of along‐strike CDP profiles indicates an AMC which is neither overlapping nor discontinuous when projected onto the along‐strike plane. Identifying intracrustal events on along‐axis CDP lines, however, requires extreme caution; we have modeled out‐of‐plane scattering using a Kirchhoff formulation, and we show that a coherent event identified beneath the overlap basin results from diffraction off the AMC which lies nearly 3 km to the west of the profile. We attribute the asymmetric pattern of melt to a decoupling of melt supply from preexisting weaknesses in the brittle upper crust. In this model, melt ascends upward (buoyancy forces) until deflected by the impermeable sheeted dike complex; melt then migrates updip, beneath the base of the sheeted dikes, toward the neovplcanic zone where fissuring produces a temporary conduit for emplacement. Discrete jumps in modeled AMC width toward the overlap basin represent a further displacement/defocusing of melt supply (western AMC edge) relative to the neovolcanic zone (eastern AMC edge). The asymmetric pattern of melt therefore represents a gradual, en‐echelon accommodation of melt supply across the 9 km of ridge axis offset at 9°03′N. Thus for asymmetric configurations, AMC width may not correlate solely with magmatic robustness but may signify the amount of decoupling which exists between melt supply and extrusive emplacement within the neovolcanic zone. Here we present a new model for OSC development which invokes a significant component of cross‐axis melt migration. Moreover, abrupt changes in AMC width near ridge axis discontinuities (e.g., 9°17′N deviation in axial linearity) suggest that any along‐axis melt migration is confined to subsegments of the ridge and seem to preclude the segment length migration of melt proposed in some current models. The transition of melt supply beneath the overlap basin might favor a continuous low‐velocity zone underlying this feature; if true, basin development may be related to the subsidence of a mechanically weak crustal lid. The proposed model for OSC development therefore views ridge axis discontinuities as the surficial response of misalignment and/or defocusing of melt supply in the uppermost mantle.

Hybrid ray-mode parametrization of acoustic scattering from submerged thin elastic shells with interior loading

A phenomenologically based hybrid ray-mode synthesis strategy is applied here to reconstruct the backscattered field from a submerged thin cylindrical elastic shell with internal structural loadings, which is insonified by a normally incident time-harmonic plane wave. The strategy is based on the Kirchhoff formulation for the acoustic field which expresses the scattered pressure explicitly in terms of the empty-shell Green’s function augmented by on-surface forces and bending moments that account for the internal effects. Earlier studies have shown that the prominent observable features in the scattered far field may be attributed to waves on the shell and resonant modes associated with the internal system. Accordingly, a ray formulation is utilized for the empty-shell Green’s function, while an angular harmonic (resonant mode) solution is used for the coupling forces and bending moments. This hybrid scheme is shown to reconstruct previously derived harmonic series solutions accurately and efficiently in the intermediate and high-frequency domains.

Characteristics pathlength Fresnel number equations and ideal lens wavefront Kirchhoff formulation: analysis of focused spherical wave diffraction fields

Derivation and applications of the characteristics optical pathlength Fresnel number equation for spherical waves focused by an ‘ideal’ circular lens illustrate the wealth of information which is quickly available from its use. A Kirchhoff wavefront diffraction formulation of this problem is developed and used to aid in this analysis. New characteristics about the diffraction field are revealed.

A rapid method for measuring tube attenuation

It has long been recognized that the acoustic impedance is the chief characteristic of the performance of acoustical materials. The impedance tube provides an easy way of measuring this characteristic at normal incidence. It is also known that the effect of tube attenuation must be included in the impedance tube analysis for accurate measurements. Although theoretical prediction can be deduced from the well-known Helmholtz–Kirchhoff formulation, disagreements with experimental results (up to 15%) have been reported. Previous results relied on locating and measuring the pressures as a number of maxima and minima along the standing wave pattern inside the tube. The method was tedious. A more rapid method for measuring the tube attenuation is possible based on the recently proposed three microphone method for impedance tube measurements [W. T. Chu, J. Acoust. Soc. Am. 92, 2315(A) (1992)]. The procedure will be presented together with results from both computer simulations and actual experiments.

On the robustness of hierarchic finite elements for Reissner-Mindlin plates

The discretization of the Reissner-Mindlin model for plates requires finite elements of class only C 0. This is perhaps the main advantage respect to the Kirchhoff formulation, where C 1 finite elements need to be used for a conforming approximation. However, despite its simple approach, the numerical approximation of the Reissner-Mindlin plate is not straightforward. The inclusion of transverse shear strain effect in standard finite element models introduces undesirable numerical effects. Standard low order finite elements are not able to meet the Kirchhoff constraint enforced when the thickness becomes smaller and therefore are subject to the locking phenomenon. The approximate solution is very sensitive to the plate thickness and unsatisfactory results are obtained for small thickness. We have solved the Reissner-Mindlin plate problem in its plain formulation with the use of high order finite elements. We have developed a hierarchic family of finite elements and have performed several numerical tests to analyze the behavior with respect to the thickness. In this paper we are particularly interested in the investigation of the robustness properties of the family of finite elements. In this direction we analyse the performance of the elements when very small values of the thickness of the plate are considered. The numerical results indicate a large range of robustness for the higher order elements.