Acoustic Field Equations Research Articles

Broadband Scattering Characteristic Quantization Technique for Single Fish Based on a Split-Beam Echosounder

The utilization of broadband quantization data increases the possibility of practical applications for fish target recognition based on the acoustic scattering theory. However, the quantification of broadband data is more complex than that of narrowband systems, requiring consideration of the broadband characteristics of split-beam transducers, seawater absorption, and circuits. This paper elucidates the scatterer acoustic field equation, transducer power equation, and signal processing flow in split-beam broadband quantization technology for engineering applications. A broadband calculation model based on transducer parameters is proposed to enhance the generalization ability of broadband quantization technology to different types of sonar. The classical echo integration method is combined with Fourier transform to meet the requirements of target strength (TS) estimation under broadband signals. This paper includes a series of experiments to prove the rationality and effectiveness of the method. The results demonstrate that the provided calculation model can more accurately reflect the backscattering characteristics of the scatterer, but certain errors remain. This article analyzes the sources of errors and validates the effectiveness of the new TS calculation method.

Transient structural acoustics: The impact excitation of fluid loaded spherical shells via a generalized modal radiation impulse response approach.

A generalized modal radiation impulse response approach based on in vacuo eigenfunction expansions was recently developed to evaluate the space-time surface velocity vector field, instantaneous power, and energy transfer into a fluid resulting from the space-time force distribution of fluid loaded shell and panel structures. The impact excitation of a fluid loaded spherical shell is addressed in this paper to illustrate the generalized modal approach for an analytically tractable problem. In vacuo eigenfunctions of spherical shells are presented and a modal based pressure impulse response approach is developed to evaluate the associated transient pressure field. Coupled shell and acoustic field equations for the modal velocities in the time and frequency domains are developed to evaluate the modal velocities via the use of modal radiation impulse responses and impedances. The numerical results are presented to illustrate the characteristics of modal pressure impulse responses and modal radiation impulse responses, which couple the modal velocities, as well as the modal coupling effects of the fluid on the modal velocities, the energy transfer into a fluid, and the far field pressures for the breathing mode, rigid body mode, and higher order shell modes.

Influence of Relaxation Frequency on Acoustic Wave in Unconsolidated Sands and Acoustic Logging Simulation

Apart from consolidated rocks, the effect of relaxation on acoustic propagation in unconsolidated sands cannot be neglected. In this paper, we study the influence of relaxation frequency on the propagation of acoustic waves. We compute the frequency-dependent velocities and attenuation of P1-wave, P2-wave, and S-wave at different bulk or shear relaxation frequency for plane wave. In addition, we derive the integral solutions of acoustic field equations in cylindrical coordinate system to simulate acoustic logging. The reflected acoustic waveforms in a borehole are calculated at different bulk or shear relaxation frequency. Calculation results show that the increase of bulk relaxation frequency will cause the velocity of P1-wave to decrease slightly, and the velocity of P2-wave to decrease substantially. The change of bulk relaxation frequency has no effect on the velocity of S-wave. The increase of bulk relaxation frequency will cause the attenuation of P1-wave or P2-wave to decrease or increase in different wave frequency range. The change of bulk relaxation frequency has no effect on the attenuation of S-wave. The increase of shear relaxation frequency will cause the velocity of P1-wave to increase slightly, and the velocity of P2-wave or S-wave to decrease substantially. The increase of the shear relaxation frequency will cause the attenuation of P1-wave, P2-wave or S-wave to decrease. For acoustic field in a borehole surrounded by unconsolidated sands, the effect of bulk or shear relaxation frequency on the velocity of reflected waves in a borehole is negligible at the dimension of the distance from a logging source. The increase of bulk or shear relaxation frequency will cause the amplitude of the reflected waveforms from the borehole wall to increase.

Influence of Relaxation Frequency on Acoustic Wave in Unconsolidated Sands and Acoustic Logging Simulation

Apart from consolidated rocks, the effect of relaxation on acoustic propagation in unconsolidated sands cannot be neglected. In this paper, we study the influence of relaxation frequency on the propagation of acoustic waves. We compute the frequency-dependent velocities and attenuation of P1-wave, P2-wave, and S-wave at different bulk or shear relaxation frequency for plane wave. In addition, we derive the integral solutions of acoustic field equations in cylindrical coordinate system to simulate acoustic logging. The reflected acoustic waveforms in a borehole are calculated at different bulk or shear relaxation frequency. Calculation results show that the increase of bulk relaxation frequency will cause the velocity of P1-wave to decrease slightly, and the velocity of P2-wave to decrease substantially. The change of bulk relaxation frequency has no effect on the velocity of S-wave. The increase of bulk relaxation frequency will cause the attenuation of P1-wave or P2-wave to decrease or increase in differ...

Numerical solution of seismic exploration problems in the Arctic region by applying the grid-characteristic method

The goal of this paper is the numerical solution of direct problems concerning hydrocarbon seismic exploration on the Arctic shelf. The task is addressed by solving a complete system of linear elasticity equations and a system of acoustic field equations. Both systems are solved by applying the grid-characteristic method, which takes into account all wave processes in a detailed and physically correct manner and produces a solution near the boundaries and interfaces of the integration domain, including the interface between the acoustic and linear elastic media involved. The seismograms and wave patterns obtained by numerically solving these systems are compared. The effect of ice structures on the resulting wave patterns is examined.

Full wave finite-difference time-domain study of lossless acoustic bipolar cylindrical cloak with compressed geometry and complementary media

Full-wave finite-difference time-domain analysis of a lossless acoustic cloak with complementary media is studied for bipolar cylindrical geometry. The material parameters for the acoustic cloak are obtained by comparing two-dimensional Maxwell's equations in the transverse magnetic polarization and acoustic field equations for general curvilinear coordinates. The permittivity and permeability tensors for the corresponding electromagnetic fields are then obtained by using an effective medium approach in general relativity. The complementary media cancels the phase vectors across the boundaries of the illuminated acoustic waves. It is shown that for the lossless cloak the broadband and illumination direction independent cloaking can be achieved with the proposed structure.

A novel unsplit perfectly matched layer for the second-order acoustic wave equation

When solving acoustic field equations by using numerical approximation technique, absorbing boundary conditions (ABCs) are widely used to truncate the simulation to a finite space. The perfectly matched layer (PML) technique has exhibited excellent absorbing efficiency as an ABC for the acoustic wave equation formulated as a first-order system. However, as the PML was originally designed for the first-order equation system, it cannot be applied to the second-order equation system directly. In this article, we aim to extend the unsplit PML to the second-order equation system. We developed an efficient unsplit implementation of PML for the second-order acoustic wave equation based on an auxiliary-differential-equation (ADE) scheme. The proposed method can benefit to the use of PML in simulations based on second-order equations. Compared with the existing PMLs, it has simpler implementation and requires less extra storage. Numerical results from finite-difference time-domain models are provided to illustrate the validity of the approach.

Modeling of loudspeaker using hp-adaptive methods

Numerical modeling of harmonic acoustic field produced by a common commercial loudspeaker is carried out. The goal is to obtain the directional characteristic of the device. So far, these characteristics could only be obtained by measurements, which was very complicated and also expensive. Another benefit of the model consists in obtaining a reference mathematical model for other acoustic devices, such as diffusors or resonators. The continuous mathematical model is derived from the physical laws and is considered for the ideal fluids with properties of a perfect continuum. The model of the loudspeaker in the axial symmetry (2D) is described by the Helmholtz differential equation for harmonic acoustic field and for the correct solution the appropriate boundary conditions and material properties are also defined. The task is solved numerically by a fully adaptive higher-order finite element method developed by the hp-FEM group. The convergence of results is discussed for the different adaptive techniques and the advantages of the use of \(hp\)-adaptivity are demonstrated. Selected results of the computations are then compared with experiments performed in an anechoic chamber at the University of West Bohemia and based on the standard CSN EN 60268-5.

High Intensity Focused Ultrasound Optimal Device Design for Targeted Prostate Cancer Treatment: A Numerical Study

A parametric study was performed to design a device capable of treating small targeted regions within the prostate using high intensity focused ultrasound, while sparing the surrounding organs and minimizing the number of elements. The optimal focal length (L), operating frequency (f), element size (a) and central opening radius for lodging an imaging probe (r) of a device that would safely treat tissue within the prostate were obtained. Images from the Visible Human Project were used to determine simulated organ sizes and treatment locations. Elliptical tumors were placed throughout the simulated prostate and their lateral and axial limits were selected as test locations. Using graphics processors, the acoustic field and Bio-Heat Transfer Equation were solved to calculate the heating produced during a simulated treatment. L, f, a and r were varied from 45 to 75 mm, 2.25 to 3 MHz, 1.5 to 8 times the wavelength and 9 to 12.5 mm, respectively. The resulting optimal device was a 761-element concentric-ring transducer with L = 68 mm, f = 2.75 MHz, a = 2.05λ and r = 9 mm. Simulated thermal lesions showed that it was possible to treat target tumors consistent with reported locations and sizes for prostate cancer.

Thermoacoustic instability in a solid rocket motor: non-normality and nonlinear instabilities

An analytical framework is developed to understand and predict the thermoacoustic instability in solid rocket motors, taking into account the non-orthogonality of the eigenmodes of the unsteady coupled system. The coupled system comprises the dynamics of the acoustic field and the propellant burn rate. In general, thermoacoustic systems are non-normal leading to non-orthogonality of the eigenmodes. For such systems, the classical linear stability predicted from the eigenvalue analysis is valid in the asymptotic (large time) limit. However, the short-term dynamics can be completely different and a generalized stability theory is needed to predict the linear stability for all times. Non-normal systems show an initial transient growth for suitable initial perturbations even when the system is stable according to the classical linear stability theory. The terms contributing to the non-normality in the acoustic field and unsteady burn rate equations are identified. These terms, which were neglected in the earlier analyses, are incorporated in this analysis. Furthermore, the short-term dynamics are analysed using a system of differential equations that couples the acoustic field and the burn rate, rather than usingad hocresponse functions which were used in earlier analyses. In this paper, a solid rocket motor with homogeneous propellant grain has been analysed. Modelling the evolution of the unsteady burn rate using a differential equation increases the degrees of freedom of the thermoacoustic system. Hence, a new generalized disturbance energy is defined which measures the growth and decay of the oscillations. This disturbance energy includes both acoustic energy and unsteady energy in the propellant and is used to quantify the transient growth in the system. Nonlinearities in the system are incorporated by including second-order acoustics and a physics-based nonlinear unsteady burn rate model. Nonlinear instabilities are analysed with special attention given to ‘pulsed instability’. Pulsed instability is shown to occur with pressure coupling for burn rate response. Transient growth is shown to play an important role in pulsed instability.

Acoustic Radiation from Two Concentric Cylindrical Shells with Periodic Supports and a Viscoelastic Perforated Outer Coating

A solution is developed for the acoustic radiation from two concentric cylindrical shells which are connected by a set of periodic annular plates, and the outer of which is coated with a viscoelastic perforated layer. All the perforations in the coating have an identical cavity configuration with variable (thickness-dependent) cross-sections. The classical theory of Kennard type and the linear acoustic field equation are employed to simulate the two shells and their entrained fluid, respectively. It is assumed that only membrane motions exist in the annular plates, based on which a two dimensional membrane theory is employed to derive their reactive forces. The viscoelastic perforated coating is divided into many sub-layers each has an equivalent complex wave number by using a thick walled cylinder model. Transfer matrix method is then used to develop the relation between the quantities (velocity and pressure) at the inner and outer surfaces of the coating. The solution is found by using Fourier wave number transforms, and the stationary phase approximation is used to find an expression for the far field pressure. Comparison of numerical results for the two concentric infinite cylindrical shells without the outer coating to measured data from a cabin model is in good agreement. Far field acoustic pressure results are presented for two cylindrical shells with and without the viscoelastic perforated coating, which indicate that the viscoelastic perforated coating can effectively reduce the radiated sound pressure at middle and high frequency ranges. The influences of the coating parameters on far field acoustic pressure are also discussed.

The governing electrodynamic equations of electromagnetic acoustic transducers

This paper presents the governing electrodynamic equations of electromagnetic acoustic transducers (EMATs) and extends them to the derivation of the magnetic- and acoustic-field equations in terms of the magnetic vector potential (MVP) and the acoustic wave particle displacement vector (PDV), respectively. It also provides formulations for calculating forces and current densities in the case of two-dimensional (2D) models of EMAT configurations in Cartesian coordinates. Existing methods solve the governing electrodynamic equations for field quantities to analyze EMATs. However, they ignore skin and proximity effects and rely on simple 2D configurations for EMAT coils. Taking into account skin and proximity effects in complex 2D EMAT coil configurations requires the application of the finite element method (FEM). The FEM can be applied to solve the equations stated in terms of the MVP and PDV in the modeling of EMATs. These formulations and expressions facilitate the development and presentation of the FEM for the modeling and analysis of EMATs.

The reflected and transmitted acoustic field excited by a supersonic source in a two media configuration

The acoustic wave field due to a supersonic motion of a rigid object over a half-space is investigated. The analysis presented leads to closed form expressions for the reflected and transmitted conical waves as separate contributions. The linearized acoustic field equations are applied to obtain representations for the fields in a combined Laplace–Fourier transform domain. To these representations, which are mapped into the proper form, we apply the Cagniard–de Hoop technique in order to find closed form time–domain solutions for the reflected and transmitted acoustic fields. Attention is focused on supersonic effects, so the analysis concentrates on application of the Cagniard–de Hoop technique to obtain closed form space–time domain expressions from the contributions of poles, appearing in the transform domain representation for the reflected and transmitted acoustic wave field. It turns out that these pole contributions, next to wave solutions of the conical type also can give rise to headwaves, associated with the reflected conical wave. Numerical results for the reflected as well as the transmitted conical wave field are presented for an air–sea configuration with supersonic source velocities up to MACH 5.

Least-squares Legendre spectral element solutions to sound propagation problems.

This paper presents a novel algorithm and numerical results of sound wave propagation. The method is based on a least-squares Legendre spectral element approach for spatial discretization and the Crank-Nicolson [Proc. Cambridge Philos. Soc. 43, 50-67 (1947)] and Adams-Bashforth [D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (CBMS-NSF Monograph, Siam 1977)] schemes for temporal discretization to solve the linearized acoustic field equations for sound propagation. Two types of NASA Computational Aeroacoustics (CAA) Workshop benchmark problems [ICASE/LaRC Workshop on Benchmark Problems in Computational Aeroacoustics, edited by J. C. Hardin, J. R. Ristorcelli, and C. K. W. Tam, NASA Conference Publication 3300, 1995a] are considered: a narrow Gaussian sound wave propagating in a one-dimensional space without flows, and the reflection of a two-dimensional acoustic pulse off a rigid wall in the presence of a uniform flow of Mach 0.5 in a semi-infinite space. The first problem was used to examine the numerical dispersion and dissipation characteristics of the proposed algorithm. The second problem was to demonstrate the capability of the algorithm in treating sound propagation in a flow. Comparisons were made of the computed results with analytical results and results obtained by other methods. It is shown that all results computed by the present method are in good agreement with the analytical solutions and results of the first problem agree very well with those predicted by other schemes.

The transient conical wave field emitted by a supersonic source in a two media configuration

The acoustic conical wave field due to a supersonic motion of a rigid object over a half-space is investigated. The research presented here is of great importance for the modeling of the transmitted and reflected conical waves that are excited by supersonic airplanes that fly over sea or land. The media involved are acoustically modeled as homogeneous, isotropic fluids and the moving object is represented as an acoustic source of the dipole type. The linearized acoustic field equations are applied to obtain representations for the fields in a combined Laplace–Fourier transform domain. To these representations, which are mapped into the proper form, we apply the Cagniard–de Hoop technique in order to find closed-form time-domain solutions for the reflected and transmitted acoustic fields. In the Cagniard–de Hoop technique we use a new transformation in order to get separate expressions for the reflected and transmitted conical waves as well as the conical headwave that can occur under certain circumstances. These special wave phenomena come forward from the procedure with precursive fronts and clearly defined trailing edges. Numerical results for the reflected as well as the transmitted conical wave are presented in a configuration where conical headwaves can occur.

Spectral element analysis of sound propagation in a muffler

This paper presents a novel numerical method based on the least-squares spectral element formulation to analyze sound propagation in an expansion-chamber muffler with and without a mean flow. The method solves the linearized acoustic field equations derived from the full Navier–Stokes equations. Effects of the mean flow on the acoustic wave propagation in the muffler were taken into consideration. The method solves the first-order acoustic field equations in a finite number of elements which represent the chamber muffler. Within each element the method first approximates the solutions to these equations by a series of unknown coefficients with known basis functions, forms the residual of the approximation, and then minimizes the integral of the squares of the residual with respect to the unknown coefficients. The resultant system equations were discretized by the spectral element method for spatial derivatives and by the dual time stepping for the temporal derivative. Finally, the algebraic equations were solved by a Jacobin preconditioned conjugate gradient method. Numerical results were presented for pressure contours and transmission loss of sound waves propagating at various frequencies in the muffler. These results were compared with the analytical values available, and the comparison showed very good agreement.

A least-squares spectral element method for computational aeroacoustics

Because of its relatively small magnitude compared with the carrier flow field variations, an acoustic signal is not easily tracked without distortion and degradation to the phase and amplitude information contained therein. The spurious dispersion and dissipation effects introduced by most numerical methods for computational fluid dynamics are detrimental to an accurate calculation of the acoustic signal. To rectify such deficiency in current codes, a least-squares spectral element method was developed to solve the linearized acoustic field equations derived from the full Navier–Stokes equations. The method solves the first-order partial-differential equations and minimizes the integral of the squares of the residual over the domain of interest. Spatial derivatives are approximated by Legendre polynomials, and temporal discretization is performed by dual-time stepping. The resultant algebraic equations are solved by a Jacobi preconditioned conjugate gradient method. Numerical examples were presented for a Gaussian pulse in a uniform flow of Mach 0.5 above an infinite plate. The computed acoustic pressures on the wall at different time instances were compared against the exact analytical values, and the comparison showed very good agreement. It indicates that the proposed method is indeed suitable for computing the sound generation and propagation in an unsteady flow regime.

The use of the method of moments with finite difference operator in acoustics

Weighted residual methods are frequently used for solving a variety of problems in electromagnetics, solid mechanics, weather prediction, and fluid mechanics. Amidst these weighted residual methods, the electromagnetics research community frequently uses a technique known as the method of moments (MM). MM can be used to systematically form the whole set of expansion functions needed to construct the approximate solution of the problem based on one given basis function. This property of the method of moments can be exploited to solve acoustic field equations in irregular domains. The method of moments with finite difference operator approximation is explored in this paper. As illustrations of the applications of this method, both one- and two-dimensional acoustic problems are solved in this paper. These applications help to highlight the potential of the method of moments.

Sound scattering at a rough water–sediment interface in shallow water

The method of smoothing is used to study sound propagation and scattering in a realistic shallow-water environment with a randomly rough water–sediment interface. A mean field wave equation for the coherent acoustic field is obtained that has a complex-valued attenuation coefficient that is localized at the mean interface. The solution of the mean field equation, under highly idealized conditions, is derived and discussed. The branch point and pole singularities are found to be related to the head wave mode and the Biot–Tolstoy rough boundary wave mode, respectively. Also obtained is the coherent plane-wave reflection coefficient which is found to vanish when the acoustic frequency and the incident angle of the plane wave satisfy a certain resonant condition. For testing the mean field equation, a stochastic full-wave forward propagation PE model is developed for Monte Carlo simulations. The numerical results show that the coherent field obtained by an ensemble average over many realizations is in agreement with that obtained by solving the mean field equation just once. The Monte Carlo simulations also yield additional useful information about higher-order moments of the acoustic field. [Work supported by ONR.]

Iterative calculation of reflected and transmitted acoustic waves at a rough interface

A rigorous iterative technique is described for calculating the acoustic wave reflection and transmission at an irregular interface between two different media. The method is based upon a plane-wave expansion technique in which the acoustic field equations and the radiation condition are satisfied analytically, while the boundary conditions at the interface are satisfied numerically. The latter is accomplished by an iterative minimization of the integrated squared error in the boundary conditions by a conjugate gradient technique, leading to a converging and relatively simple scheme. The plane interface result can be used as starting value. Although in principle the method is rigorous, numerical examples show that in practice there is a lower bound on the error in the boundary conditions which can be achieved.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>