Kirchhoff-Helmholtz Integral Equation Research Articles

A novel numerical method for solving the Kirchhoff-Helmholtz integral equation based on the sound field representation using the reproducing kernel in the frequency domain

The paper introduces a unique numerical solution method for the Kirchhoff-Helmholtz integral equation. This method is based on the representation of the sound field using the reproducing kernel in the solution space of the homogeneous Helmholtz equation. When compared to the conventional boundary element method, the proposed method holds numerous advantages. This paper presents the analysis theory of the proposed method and demonstrates its validity by calculating bounded and unbounded sound fields. The accuracy of the calculation is assessed by computing the sound field in a rectangular room under various conditions. Furthermore, although some theoretical issues remain, the proposed method effectively avoids the non-uniqueness of solutions in unbounded domain problems.

A meshfree approach for vibro-acoustic analysis of combined composite laminated shells with variable thickness

The vibro-acoustic responses of combined composite laminated shell structures with variable thickness immersed in acoustic medium is investigated by using a meshfree method in this paper. The theoretical model of structural field is established by employing the energy principle in conjunction with the first-order shear deformation theory (FSDT) and the theoretical model of acoustic field are established by using Kirchhoff-Helmholtz integral equation. The displacement components of shell segments and the sound pressure variables of acoustic medium are described by using the meshfree Tchebychev point interpolation (TPIM) shape function and Double Fourier series in meridional and circumferential directions. The non-uniqueness problem of Kirchhoff-Helmholtz integral equation is overcome by adopting CHIEF method. The validation of the presented theoretical model is proved by comparing the results calculated by the proposed model with those results of existing literature and FEM/BEM. Based on the established theoretical model, vibro-acoustic responses of combined composite laminated shell structures with variable thickness immersed in acoustic medium are performed by analyzing the influences of thickness parameter, external load and boundary condition on the sound pressure level (SPL) and sound power level (SWL) of combined composite laminated shell structures. The above investigations can provide the theoretical thesis and technique guidance of preliminary design of combined composite laminated shell immersed in various acoustic medium.

An Acoustic Simulation Method of the Japanese Vowels /i/ and /u/ by Using the Boundary Element Method

This study aimed to establish and verify the validity of an acoustic simulation method during sustained phonation of the Japanese vowels /i/ and /u/. The study participants were six healthy adults. First, vocal tract models were constructed based on computed tomography (CT) data, such as the range from the frontal sinus to the glottis, during sustained phonation of /i/ and /u/. To imitate the trachea, after being virtually extended by 12 cm, cylindrical shapes were then added to the vocal tract models between the tracheal bifurcation and the lower part of the glottis. Next, the boundary element method and the Kirchhoff–Helmholtz integral equation were used for discretization and to represent the wave equation for sound propagation, respectively. As a result, the relative discrimination thresholds of the vowel formant frequencies for /i/ and /u/ against actual voice were 1.1–10.2% and 0.4–9.3% for the first formant and 3.9–7.5% and 5.0–12.5% for the second formant, respectively. In the vocal tract model with nasal coupling, a pole–zero pair was observed at around 500 Hz, and for both /i/ and /u/, a pole–zero pair was observed at around 1000 Hz regardless of the presence or absence of nasal coupling. Therefore, the boundary element method, which produces solutions by analysis of boundary problems rather than three-dimensional aspects, was thought to be effective for simulating the Japanese vowels /i/ and /u/ with high validity for the vocal tract models encompassing a wide range, from the frontal sinuses to the trachea, constructed from CT data obtained during sustained phonation.

Vibro-acoustic analysis of laminated composite cylindrical and conical shells using meshfree method

This paper intends to investigate the vibro-acoustic responses of cross-ply laminated composite shell including cylindrical and conical shells subjected to acoustic medium by employing meshfree method. The energy principle, first-order shear deformation theory (FSDT) and the Kirchhoff-Helmholtz integral equation are employed to deduce the theoretical formulations of the structural and acoustic models. The displacement and acoustic pressure of laminated composite shells and acoustic medium are described by employing the meshfree shape function and Fourier series. The interior CHIEF points is introduced for eliminating the non-unique solution with regard to the acoustic integral equations. The reliability and accuracy of presented method are verified for forecasting the vibro-acoustic characteristics of the laminated composite shells by comparing to the vibration and the acoustic response results of published articles. Furthermore, the parametric analysis is performed by investigating the influences of geometric dimensions, material properties, boundary conditions and external force types on the acoustic responses of shell, which is beneficial to the preliminary design of the shell structure immersed in acoustic medium.

Sound field reconstruction in urban environments: Application to enhance mapping of urban microspaces

In this work, reconstruction techniques for the spatial interpolation and extrapolation of sound fields in urban environments are presented. Gaussian processes are generally used for sound field reconstruction from limitedly observations of isotropic acoustic fields. However, this model is often not applicable for the anisotropic urban environments including urban street canyons and enclosed spaces, when the complexity of the sound field is high in the mid-frequency regime, unless diffusely reflecting boundaries are assumed. Two different techniques are compared for reconstructing the sound field: the least-squared method and the Kirchhoff-Helmholtz integral equation method. Of particular interest is the reconstruction of the sound field with a minimal number of irregularly and arbitrarily distributed microphone measurements. Therefore, the techniques will not require knowledge of the microphone positions. A successive series approximation approach is presented to enhance the microscale prediction of the Kirchhoff-Helmholtz integral equation method. The sound field reconstruction results from limited urban environment observations for both methods are presented and discussed.

Transmission loss of lined Helmholtz resonator with annular air gap: A Green's function based approach

The present article discusses a Green's function-based semi-analytical method to predict the transmission loss of a lined Helmholtz resonator with annular air gap. In the analysis, the walls of the chamber are assumed to be acoustically rigid except at the neck portion where it is treated as a piston source moving with uniform velocity. The Green's function is developed as the summation of eigenfunctions of the central duct. The cumulative effect of the lined portion and the annular air gap including the perforated screens is incorporated as the reflection coefficient in the eigenfunctions. By using the Kirchhoff-Helmholtz integral equation, the velocity potential generated by the piston inside the chamber is evaluated. A transfer matrix relating the acoustic pressure and volume velocity across the neck in the main duct is formulated. The effect of the neck length is included as an added inertance to the impedance in the transfer matrix. The results obtained from the proposed method are validated with the developed numerical models and the experimental data available in the literature. A parametric study has been conducted to investigate the effect of porosity of the perforated screens, thickness and flow resistivity of the absorptive material on the transmission loss of the chamber.

Broadband numerical simulation of scattering from rough pressure-release and fluid-fluid interfaces

The scattered field from the seafloor is often measured using short, broadband pulses, whereas many models for the mean scattered intensity are in the frequency domain. This As higher resolution seafloor mapping systems, i.e., synthetic aperture sonar, become more common, it is important to understand the effects of high resolution on measurements of both the averaged scattered intensity, and the distribution of the envelope of the scattered field. To address this problem, Monte Carlo simulations of the scattered field due to rough interfaces separating two fluids are perform. The integral equation governing the total pressure, the Helmholtz-Kirchhoff integral equation, is numerically solved using the boundary element method. Simulations are performed both in the limit of pressure release, as well as for more realistic sediment sound speed and density parameters. The von-Karman spectrum is used to generate random rough surfaces. Fourier synthesis is used to to generate time domain signals, which are then analyized in terms of the mean energy and amplitude distribution. The dependence of the scattering cross section and scintillation index on signal bandwidth (resolution) is examined. It is found that there is no observable dependence of the scattering cross section on bandwidth, but that there is significant dependence of the scintillation index.

Boundary Matching Filters for Spherical Microphone and Loudspeaker Arrays

Conversion of microphone array signals into loudspeaker array signals is an essential process in high-definition spatial audio. This paper presents the theory of boundary matching filters (BMFs) for spherical array signal conversion. BMFs adapt the physical boundary conditions used during recording to the ones required for reproduction by relying on a theoretical framework provided by the Kirchhoff–Helmholtz integral equation (KHIE). Computationally, array signal conversion is performed in a transform domain where sound fields are represented in terms of spherical harmonic functions. Related research on transform-domain signal conversion filters is interpreted in the context of the KHIE. The case of a rigid recording boundary and an open reproduction boundary is addressed. The proposed rigid-to-open BMFs provide a suitable basis for designing gain-limited filters to deal with the problem of excessive gains at certain frequency bands, observed when using high-resolution arrays. Spatial, spectral, and temporal effects in sound field reconstruction when finite numbers of transducers are used in anechoic conditions are investigated analytically and exemplified numerically. Results show that the proposed gain-limited rigid-to-open BMFs outperform the existing gain-limited filters based on Tikhonov regularization because they reduce the spatial discretization effects and yield impulse responses that are more localized around their main peaks.

Broadband multizone sound rendering by jointly optimizing the sound pressure and particle velocity.

In this paper, a recently proposed approach to multizone sound field synthesis, referred to as joint pressure and velocity matching (JPVM), is investigated analytically using a spherical harmonics representation of the sound field. The approach is motivated by the Kirchhoff-Helmholtz integral equation and aims at controlling the sound field inside the local listening zones by evoking the sound pressure and particle velocity on surrounding contours. Based on the findings of the modal analysis, an improved version of JPVM is proposed, which provides both better performance and lower complexity. In particular, it is shown analytically that the optimization of the tangential component of the particle velocity vector, as is done in the original JPVM approach, is very susceptible to errors and thus not pursued anymore. Furthermore, the analysis provides fundamental insights as to how the spherical harmonics used to describe three-dimensional sound fields translate into two-dimensional basis functions as observed on the contours surrounding the zones. By means of simulations, it is verified that discarding the tangential component of the particle velocity vector ultimately leads to an improved performance. Finally, the impact of sensor noise on the reproduction performance is assessed.

Broadband numerical simulations of scattering from rough surfaces

Numerical simulation of rough surface scattering via discretization of the Helmholtz-Kirchhoff integral equation (HKIE) is a powerful, yet computationally expensive method to study rough surface scattering. In this work, simulations are performed using Fourier synthesis of the numerical solution of the HKIE using the boundary element method (BEM). Each frequency component is solved using BEM employing a good approximation of a plane wave as an incident field. The surface length requirements on this incident field depend on angle, and all simulations are performed using plane waves whose angular width is less than one degree, and whose mean grazing angle is greater than 20 degrees. This requirement resulted in problems with over 22,000 degrees of freedom and approximately 3000 individual frequencies. This broadband technique is employed to study the dependence of the scattered intensity on surface parameters as well as the bandwidth of the transmitted pulse. Simulations are performed both on surfaces with truncated power law spectra, as well as rippled surfaces with power-law roughness superimposed.

Controlling 3D sound field generated by a 2D surface from its 1D boundary

As described by the Kirchhoff-Helmholtz integral equation, for a 3D space without internal sources, the inside sound field is completely determined by the sound pressure and its normal gradient on the boundary, so if only a portion of the boundary surface vibrates as the primary sound source while all the remaining portion of the surface is acoustically rigid, a sufficiently large number of secondary sound sources can be located evenly on the primary sound surface to control the 3D sound field completely. The question to be investigated in this research is whether it is possible to control the 3D sound field by only applying secondary sound sources along the 1D boundary of the 2D primary sound surface, for example, controlling the sound transmission from a door with secondary sound sources only on its frame. Simulation and experimental results will be presented to demonstrate the feasibility of the idea.

The Burton and Miller Method: Unlocking Another Mystery of Its Coupling Parameter

The phenomenon of irregular frequencies or spurious modes when solving the Kirchhoff–Helmholtz integral equation has been extensively studied over the last six or seven decades. A class of common methods to overcome this phenomenon uses the linear combination of the Kirchhoff–Helmholtz integral equation and its normal derivative. When solving the Neumann problem, this method is usually referred to as the Burton and Miller method. This method uses a coupling parameter which, theoretically, should be complex with nonvanishing imaginary part. In practice, it is usually chosen proportional or even equal to [Formula: see text]. A literature review of papers about the Burton and Miller method and its implementations revealed that, in some cases, it is better to use [Formula: see text] as coupling parameter. The better choice depends on the specific formulation, in particular, on the harmonic time dependence and on the fundamental solution or Green’s function, respectively. Surprisingly, an unexpectedly large number of studies is based on the wrong choice of the sign in the coupling parameter. Herein, it is described which sign of the coupling parameter should be used for different configurations. Furthermore, it will be shown that the wrong sign does not just make the solution process inefficient but can lead to completely wrong results in some cases.

A Kirchhoff approximation-based numerical method to compute multiple acoustic scattering of a moving source

Within the scope of a study of external noise propagation from moving ground vehicles, a numerical method is developed to compute the acoustic field emitted by a moving source in the presence of scattering objects such as roads, buildings or noise-shields. This method is developed with the purpose of being used in a vehicle design process and therefore it must have a low computational cost, which requires a certain number of approximations. The case of a fixed point source is studied first then the effect of a movement of the source is taken into account through the introduction of a retarded time. The acoustic source is assumed to be represented by one or many harmonic monopoles of possibly different frequency moving with a constant speed in a quiescent flow field. Scattering from nearby perfectly reflecting objects is computed through a Kirchhoff–Helmholtz integral equation applying the Kirchhoff approximation. A ray-surface intersection algorithm to compute shadow areas is proposed. The method is validated against analytical solutions and experimental results for a fixed source, and against a higher-order finite difference time-domain method for the multiple scattering of a moving source. Results are good and show that this method can potentially be used to predict urban noise.

Transient nearfield acoustical holography

Transient Nearfield Acoustical Holography Sean F. Wu Department of Mechanical Engineering, Wayne State University, Detroit, MI, 48202 This paper presents the general formulations for reconstructing the transient acoustic field generated by an arbitrary object with a uniformly distributed surface velocity in free space. These formulations are derived from the Kirchhoff-Helmholtz integral theory that correlates the transient acoustic pressure at any field point to those on the source surface. For a class of acoustic radiation problems involving an arbitrarily oscillating object with a uniformly distributed surface velocity, for example, a loudspeaker membrane, the normal surface velocity is frequency dependent but is spatially invariant. Accordingly, the surface acoustic pressure is expressible as the product of the surface velocity and the quantity that can be solved explicitly by using the Kirchhoff-Helmholtz integral equation. This surface acoustic pressure can be correlated to the field acoustic pressure using the Kirchhoff-Helmholtz integral formulation. Consequently, it is possible to use nearfield acoustic holography to reconstruct acoustic quantities in entire three-dimensional space based on a single set of acoustic pressure measurements taken in the near field of the target object. Examples of applying these formulations to reconstructing the transient acoustic pressure fields produced by various arbitrary objects are demonstrated.

Difference frequency scattered waves from nonlinear interactions of a solid sphere

In this work, the generation of difference frequency waves arising from the interaction of dual-incident beams on a solid sphere is considered. The high-frequency incident beams induce a radiation force onto the fluid-saturated sphere causing the scatterer to vibrate. An analysis on the contribution between the difference frequency sound and radiation force pressure is of particular interest. The scattered pressure due to the two primary waves are obtained as solutions to the Kirchhoff–Helmholtz integral equation for the fluid–solid boundary. Due to the contrasting material properties between the host fluid and solid sphere, high-order approximations are used to evaluate the integral equation.

An Energy Interpretation of the Kirchhoff-Helmholtz Boundary Integral Equation and its Application to Sound Field Synthesis

Most spatial audio reproduction systems have the constraint that all loudspeakers must be equidistant from the listener, a property which is difficult to achieve in real rooms. In traditional Ambisonics this arises because the spherical harmonic functions, which are used to encode the spatial sound-field, are orthonormal over a sphere and because loudspeaker proximity is not fully addressed. Recently, significant progress to lift this restriction has been made through the theory of sound field synthesis, which formalizes various spatial audio systems in a mathematical framework based on the single layer potential. This approach has shown many benefits but the theory, which treats audio rendering as a sound-soft scattering problem, can appear one step removed from the physical reality and also possesses frequencies where the solution is non-unique. In the time-domain Boundary Element Method approaches to address such non-uniqueness amount to statements which test the flow of acoustic energy rather than considering pressure alone. This paper applies that notion to spatial audio rendering by re-examining the Kirchhoff-Helmholtz integral equation as a wave-matching metric, and suggests a physical interpretation of its kernel in terms of common acoustic power flux density between waves. It is shown that the spherical basis functions (spherical harmonics multiplied by spherical Bessel or Hankel functions) are orthogonal over any arbitrary surface with respect to this metric. Finally other applications are discussed, including design of high-order microphone arrays and the coupling of virtual acoustic models to auralization hardware.

Reconstructing transient acoustic radiation from an arbitrary object with a uniform surface velocity distribution.

This paper presents the general formulations for reconstructing the transient acoustic field generated by an arbitrary object with a uniformly distributed surface velocity in free space. These formulations are derived from the Kirchhoff-Helmholtz integral theory that correlates the transient acoustic pressure at any field point to those on the source surface. For a class of acoustic radiation problems involving an arbitrarily oscillating object with a uniformly distributed surface velocity, for example, a loudspeaker membrane, the normal surface velocity is frequency dependent but is spatially invariant. Accordingly, the surface acoustic pressure is expressible as the product of the surface velocity and the quantity that can be solved explicitly by using the Kirchhoff-Helmholtz integral equation. This surface acoustic pressure can be correlated to the field acoustic pressure using the Kirchhoff-Helmholtz integral formulation. Consequently, it is possible to use nearfield acoustic holography to reconstruct acoustic quantities in entire three-dimensional space based on a single set of acoustic pressure measurements taken in the near field of the target object. Examples of applying these formulations to reconstructing the transient acoustic pressure fields produced by various arbitrary objects are demonstrated.

Theoretical Analysis of Open Spherical Microphone Arrays for Acoustic Intensity Measurements

Acoustic intensity is a vectorial measure of acoustic energy flow through a given region of interest. Three-dimensional measurement of acoustic intensity requires special microphone array configurations. This paper provides a theoretical analysis of open spherical microphone arrays for the 3-D measurement of acoustic intensity. The calculations of the pressure and the particle velocity components of the sound field inside a closed volume are expressed using the Kirchhoff-Helmholtz integral equation. The conditions which simplify the calculation are identified. This calculation is then constrained to a finite set of microphones positioned at prescribed points on an open sphere. Several open spherical array topologies are proposed. Their magnitude and directional errors and measurement bandwidths are investigated via numerical simulations. A comparison with conventional open-sphere 3-D intensity probes is presented.

An integral equation formulation for the diffraction from convex plates and polyhedra

A formulation of the problem of scattering from obstacles with edges is presented. The formulation is based on decomposing the field into geometrical acoustics, first-order, and multiple-order edge diffraction components. An existing secondary-source model for edge diffraction from finite edges is extended to handle multiple diffraction of all orders. It is shown that the multiple-order diffraction component can be found via the solution to an integral equation formulated on pairs of edge points. This gives what can be called an edge source signal. In a subsequent step, this edge source signal is propagated to yield a multiple-order diffracted field, taking all diffraction orders into account. Numerical experiments demonstrate accurate response for frequencies down to 0 for thin plates and a cube. No problems with irregular frequencies, as happen with the Kirchhoff-Helmholtz integral equation, are observed for this formulation. For the axisymmetric scattering from a circular disc, a highly effective symmetric formulation results, and results agree with reference solutions across the entire frequency range.

Analysis of pre-echo artifact in generating a focused source

A focused source is a virtual source that can provide an auditory illusion of sound radiating between the loudspeaker array and listener. When generating a focused source, a listener cannot avoid undesired pre-arriving direct waves from the control sources, which is known as pre-echo artifact. Investigation of the artifact can be seen in several researches; however, no mathematical definition of the pre-echo artifact is given so far and only the observation in limited case is known with computer simulation. The objective of this paper is to observe the cause and effect of pre-echo artifact analytically. The paper defines the pre-echo artifact mathematically, and the artifact at the arbitrary listening position is formulated with integral form based on Kirchhoff-Helmholtz integral equation. From the definition of the pre-echo, it is shown that the convergent wave of a focused source can be regarded as a special case of a pre-echo artifact. Furthermore, the derivation shows that the pre-echo artifact occurs in the case of using continuous array and is evolved due to the time-reversed nature of the solution.