Linear Acoustic Equation Research Articles

An open-source time-domain wave-based room acoustic software in Python based on the nodal discontinuous Galerkin method

In this work, we introduce an open-source implementation of a time-domain wave-based room acoustic modeling software package, named DG_RoomAcoustics. In this software, the linear acoustic equations are spatially discretized by the nodal discontinuous Galerkin method, and are integrated in time by either the explicit Runge-Kutta or the arbitrary high-order derivatives (ADER) integration schemes. Following the principles of object-oriented programming paradigm, the software is structured to ensure generic applicability and to facilitate future extensions with additional functionalities (e.g., different time integration schemes, boundary conditions). A comprehensive presentation of the physical and numerical aspects of the problem is provided. A detailed exposition of the code structure and components is presented, all of which are released under an open-source license, fostering community feedback and collaborative contributions for ongoing improvements. A brief overview of the current capabilities of the software is introduced. Future work regarding possible functionality extensions and performance optimizing are discussed.

SIMUS3: An open-source simulator for 3-D ultrasound imaging

Computational Ultrasound Imaging (CUI) has become increasingly popular in the medical ultrasound community, facilitated by free simulation software. These tools enable the design and exploration of transmit sequences, transducer arrays, and signal processing. We recently introduced SIMUS, a frequency-based ultrasound simulator within the open-source MUST toolbox, which offers numerical advantages and allows easy consideration of frequency-dependent factors. In response to the growing interest in simulating ultrasound imaging with 2-D matrix arrays, we present 3-D versions, PFIELD3 and SIMUS3. The linear acoustic equations driving these functions are described, with theoretical assumptions reviewed for user guidance. Comparative analyses with Field II, using a 32×32 element 3-MHz matrix array, highlight the performance of PFIELD3 and SIMUS3 under various transmission conditions. This work extends the capabilities of existing CUI tools and provides researchers with valuable resources for advanced ultrasound simulations.

The Lattice Boltzmann Method Using Parallel Computation: A Great Potential Solution for Various Complicated Acoustic Problems

This paper proposes the D2Q5 Lattice Boltzmann method (LBM) method, in two dimensions with five discrete lattice velocities, for simulating linear sound wave propagation in closed rooms. A second-order linear acoustic equation obtained from the LBM method was used as the model equation. Boundary conditions at the domain boundary use the bounce-back scheme. The LBM numerical calculation algorithm in this paper is relatively simpler and easy to implement. Parallelization with the GPU CUDA was implemented to speed up the execution time. The calculation results show that the use of parallel GPU CUDA programming can accelerate the proposed simulation 27.47 times faster than serial CPU programming. The simulation results are validated with analytical solutions for acoustic pulse reflected by the flat and oblique walls, the comparisons show very good concordance, and the D2Q5 LBM has second-order accuracy. In addition, the simulation results in the form of wavefront propagation images in complicated shaped rooms are also compared with experimental photographs, and the comparison also shows excellent concordance. The numerical results of the D2Q5 LBM are promising and also demonstrate the great capability of the D2Q5 LBM for investigating room acoustics in various complexities.

Estimating the Location of Acoustic Sources with Oceanic Internal Gravity Waves via Frequency-Difference Source Localization

Frequency-difference source localization (FDSL) is a relatively new method for locating sources emitting acoustic or electromagnetic waves from recorded data. Estimates of location are produced by matching a nonlinear function of the received data with the same nonlinear function of modeled data. This so-called matched-field approach depends on the accuracy of the modeled field. FDSL exhibits less sensitivity to model errors in some cases of interest. Fluctuations of the speed of sound in the ocean are affected by a wide variety of phenomena and internal gravity waves is one where their spectrum is usually well known. There is no analytical theory for predicting how these fluctuations of sound speed affect the accuracy of FDSL, leaving a numerical evaluation as the alternative for the focus of this paper. Experimental application of FDSL was previously made for an acoustic source of 100 Hz bandwidth surrounding 250 Hz for a 129 km section in the Philippine Sea with data recorded on a long vertical array [Geroski and Dowling, Long-range frequency-difference source localization in the Philippine Sea, J. Acoust. Soc. Am. 146 (2019) 4727–4739.]. The role of internal waves is quantified with a realistic FDSL simulation using models based on the parabolic and ray approximations of the linear acoustic wave equation. The modeled vertical variation of FDSL error is similar to observed while the modeled horizontal error is too small. The complicated software model is automatic but is computationally intensive, with our version needing about 1.5 mo to compute on a PC with 16 cpu cores.

Mixed Virtual Element approximation of linear acoustic wave equation

Abstract We design a Mixed Virtual Element Method for the approximated solution to the first-order form of the acoustic wave equation. In the absence of external loads, the semi-discrete method exactly conserves the system energy. To integrate in time the semi-discrete problem we consider a classical $\theta $-method scheme. We carry out the stability and convergence analysis in the energy norm for the semi-discrete problem showing an optimal rate of convergence with respect to the mesh size. We further study the property of energy conservation for the fully-discrete system. Finally, we present some verification tests as well as engineering applications of the method.

Mechanism of slow-wave generation and tuning in a perforation-modulated sonic black hole structure

A sonic black hole (SBH) in a retarding duct can be used to manipulate sound waves and achieve sound absorption. The SBH incorporates two indispensable physical processes: wave speed reduction and energy dissipation. In this study, a perforation-modulated SBH (PMSBH) retarding structure is proposed, in which the two physical phenomena can be balanced to enhance the black hole effects by modulating the perforation parameters. To fully investigate the mechanism of slow-wave generation and the effect of perforation parameters, an analytic model is established based on the Wentzel–Kramers–Brillouin (WKB) solutions to the linear acoustic wave equation. By examining the analytical solutions and comparing them with the numerical results obtained from transient finite element simulations, the effect of several important parameters on reducing the sound speed and enhancing sound absorption performance is revealed, offering a set of criteria and physical insights for modulating the parameters to design an optimal slow-sound absorber. Finally, an experiment is conducted to confirm the theoretical studies and demonstrate the performance of PMSBH. This study brings forward the concept of tunable design to improve the performance of SBH structures, which can benefit the design of sound wave manipulation and noise control devices.

A novel variational perturbation approach for formulating both linear and nonlinear acoustic model equations

A novel approach, based on variational principles, for obtaining linear acoustic model equations directly from an arbitrary Lagrangian is presented, followed by extension to the case of weakly nonlinear acoustics. The efficacy and ease of application of the methodology adopted is demonstrated for two different Lagrangians: one describing classical potential flow, the other relativistic flow including viscous dissipation. It is shown, following appropriate simplification, that corresponding classical, well-known wave equations available in the open literature, derived in the recognised standard manner, are recovered.

Efficient asymmetrical transmission through a metagrating for underwater acoustic waves

Acoustic asymmetrical transmission is a theoretical and engineering challenge because of the reciprocity of the linear acoustic wave equation. It can be achieved by systems breaking reciprocity or by reciprocal systems relying solely on spatial symmetry breaking. Metagratings are planar structures relying on Bragg's diffraction to reroute wave energy toward a desired direction and are eventually able to achieve asymmetrical transmission when build from an asymmetrical pattern of multiple basic elements. The challenge for water-like media is to combine the geometrical complexity of the structure with good acoustic impedance contrast and practical feasibility. In this work, we build a reciprocal metagrating from brass cylinders arranged according to a numerically optimized pattern and obtain highly efficient asymmetrical transmission for underwater acoustic waves. Around 200 kHz, the structure transmits nearly all incident energy toward a 45° angle when insonified from one side, but act as a near perfect reflector when insonified from the other. The effect relies entirely on the simple phenomena of linear wave diffraction and interference. The generality and efficiency of this device could be of interest for applications in underwater acoustics or medical ultrasounds.

Extended reacting boundary modeling of porous materials with thin coverings for time-domain room acoustic simulations

Modeling of acoustic boundary conditions has a significant impact on the accuracy of room acoustic simulations, which play an important role in the design phase of indoor built environments in order to improve the acoustical comfort. In this work, a numerical framework based on the discontinuous Galerkin (DG) method is presented for modeling extended reacting boundaries of porous absorbers covered by thin materials. The domain decomposition methodology is applied by treating the porous material as a subdomain. Equivalent fluid models are used to depict the acoustic properties of porous materials, whose effective density and compressibility as irrational functions are approximated by multipole rational functions in the frequency domain. By employing the auxiliary differential equation approach to calculate the time convolution, the augmented time-domain governing equations of porous materials can be expressed in the same unified hyperbolic form as the linear acoustic equations, which further enables a consistent upwind numerical flux formulation throughout the whole domain. The numerical coupling across the interface between propagation media is handled by solving the underlying Riemann problem. Compared to existing approaches with the DG method for extended reacting boundaries modeling for room acoustics, the derived upwind numerical flux formulation does not involve the computation of auxiliary variables. The presented framework yield a well-posed linear hyperbolic system with admissible boundary conditions as guided by the “uniform Kreiss condition” (Kreiss, 1970). Acoustic properties of the covering materials are illustrated by considering a limp permeable membrane model. A local time-stepping approach is utilized to improve computational efficiency. Numerical validations against analytical solutions in 1D are performed to verify the desired high-order convergence rate. A 3D case study on modeling spherical wave fronts demonstrates the broadband accuracy of the formulation.

Thermal ablation induced by low-intensity ultrasound for pulmonary vein isolation

Atrial fibrillation (AF) is currently the most common arrhythmia disease in clinic and its incidence increases with the age, which seriously endangers peoples health. Percutaneous catheter ablation is an important treatment strategy for AF, and pulmonary vein (PV) isolation has been widely accepted as the key approach of AF ablation since AFs are regarded the most important trigger of AF. The ablation scheme based on low-intensity ultrasound has the advantages of low biotoxicity and minimal invasion, while its applicability and parameter settings are still not well known. In this paper, we proposed a finite element model that could be used to simulate the ablation application of ultrasound intervention device for the purpose of PV isolation. The acoustic pressure and temperature fields generated during the process of low-intensity ultrasound pulmonary vein ablation were simulated based on linear acoustic wave equation and Pennes biological heat transfer equation, and then the corresponding thermal damage was quantitatively analyzed by taking into account of the equivalent heat dose equation. The results showed that the incident acoustic energy was mainly concentrated in the blood vessels of PV, and the temperature rise was mainly constrained in the inner part of the vessel walls, which suggested that high temperature rise could be realized nearby the central region of the low-intensity ultrasound transducer to achieve significant thermal damage in the inner part of the vessel walls. Moreover, the current study also discussed the impacts of different parameters on the ultrasound ablation effect, which would provide theoretical support and guidance for the optimization of ultrasound ablation strategy in clinics.

Amplitude reflections and interaction solutions of linear and nonlinear acoustic waves with hard and soft boundaries

In this study, the propagation of a fundamental plane mode in a bifurcated waveguide structure with soft–hard boundaries is analyzed by using the Helmholtz equation. The explicit solution is given to this bifurcated spaced waveguide problem by means of matching the potential across the boundary of continuity. Amplitudes of the reflected field in all those regions have been evaluated, and the energy balance has been derived. We have observed the reflection of the acoustic wave against the wavenumber and shown its variation with the duct width. Convergence of the problem has been shown graphically. In our analysis, we notice that the reflected amplitude decreases as the duct spacing increases; as a result, the acoustic energy will increase as the duct spacing increases. It is expected that our analysis could be helpful to give better understanding of wave reflection in an exhaust duct system. We then reduce the linear acoustic wave equation to the Kadomtsev–Petviashvili (KP) equation. Multiple-periodic wave interaction solutions of the KP nonlinear wave equation are investigated, and the energy transfer mechanism between the primary and higher harmonics is explained, which, to the best of our knowledge, is overlooked.

Ray Statement of the Acoustic Tomography Problem

The ray statement of the inverse problem of determining three unknown variable coefficients in the linear acoustic equation is studied. These coefficients are assumed to differ from given constants only inside some bounded domain. There are point pulse sources and acoustic receivers on the boundary of this domain. Acoustic signals are measured by a receiver near the moment of time at which the signal from a source arrives at the receiver. It is shown that this information makes it possible to uniquely determine all the three desired coefficients. Algorithmically, the original inverse problem splits into three subproblems solved successively. One of them is a well-known inverse kinematic problem (of determining the speed of sound), while the other two lead to the same integral geometry problem for a family of geodesic lines determined by the speed of sound.

The Inviscid Limit of Third-Order Linear and Nonlinear Acoustic Equations

We analyze the behavior of third-order-in-time linear and nonlinear sound waves in thermally relaxing fluids and gases as the sound diffusivity vanishes. The nonlinear acoustic propagation is modeled by the Jordan--Moore--Gibson--Thompson equation both in its Westervelt-type and in its Kuznetsov-type forms, that is, including general nonlinearities of quadratic type. As it turns out, sufficiently smooth solutions of these equations converge in the energy norm to the solutions of the corresponding inviscid models at a linear rate. Numerical experiments illustrate our theoretical findings.

Low Mach number limit for quasi-one-dimensional isentropic Euler flow

This paper is devoted to studying the low Mach number limit for the quasi-one-dimensional isentropic Euler equations in $BV\\cap~L^1$ space. Under assumptions that the total variation of the cross section of the nozzleis sufficiently small and the initial data is away from vacuum, by using the $L^1$-stability estimatesand the method of the standard Riemann semigroup, we rigorouslyprove that the solution can be expandedin the powers of small Mach number and that its coefficients satisfy a linear acoustic equation with the source term.

Stationarity Preservation Properties of the Active Flux Scheme on Cartesian Grids

Hyperbolic systems of conservation laws in multiple spatial dimensions display features absent in the one-dimensional case, such as involutions and non-trivial stationary states. These features need to be captured by numerical methods without excessive grid refinement. The active flux method is an extension of the finite volume scheme with additional point values distributed along the cell boundary. For the equations of linear acoustics, an exact evolution operator can be used for the update of these point values. It incorporates all multi-dimensional information. The active flux method is stationarity preserving, i.e., it discretizes all the stationary states of the PDE. This paper demonstrates the experimental evidence for the discrete stationary states of the active flux method and shows the evolution of setups towards a discrete stationary state.

Analytical and numerical solution of the pressure response to an unsteady heat release pulse in 1D

This paper analyzes the pressure response to an unsteady heat release source in an unconfined one-dimensional domain. An analytical model based on the acoustic wave equation with an unsteady heat source is derived, and then compared against results from highly-resolved numerical simulations. Two different heat release profiles, a Gaussian spatial distribution with either a step or a Gaussian temporal distribution, are used to model the heat source. The analytical solutions predict two regimes in the pressure response depending on the Helmholtz number, which is defined as the ratio of the acoustic time over the duration of the heat release pulse. A critical Helmholtz number is found to dictate the pressure response regime. For compact cases, in the subcritical regime, the amplitude of the pressure pulse remains constant in space. For noncompact cases, above the critical Helmholtz number, the pressure pulse reaches a maximum at the center of the heat source, and then decays in space converging to a lower far field amplitude. At the limits of very small and very large Helmholtz numbers, the heat release response tends to be a constant pressure process and a constant volume process, respectively. The parameters of the study are chosen to be representative of the extreme conditions in a rocket combustor. The analytical models for both heat source profiles closely match the simulations with a slight overprediction. The differences observed in the analytical solutions are due to neglecting mean flow property variations and the absence of loss mechanisms. The numerical simulations also reveal the presence of nonlinear effects such as weak shocks that cannot be captured by the linear acoustic wave equation.

Analysis and Performance Evaluation of Adjoint-guided Adaptive Mesh Refinement for Linear Hyperbolic PDEs Using Clawpack

Adaptive mesh refinement (AMR) is often used when solving time-dependent partial differential equations using numerical methods. It enables time-varying regions of much higher resolution, which can selectively refine areas to track discontinuities in the solution. The open source Clawpack software implements block-structured AMR to refine around propagating waves in the AMRClaw package. For problems where the solution must be computed over a large domain but is only of interest in a small area, this approach often refines waves that will not impact the target area. We seek a method that enables the identification and refinement of only the waves that will influence the target area. Here we show that solving the time-dependent adjoint equation and using a suitable inner product allows for a more precise refinement of the relevant waves. We present the adjoint methodology in general and give details on the implementation of this method in AMRClaw. Examples and a computational performance analysis for linear acoustics equations are presented. The adjoint method is compared to AMR methods already available in AMRClaw, and the advantages and disadvantages are discussed. The approach presented here is implemented in Clawpack, in Version 5.6.1, and code for all examples presented is archived on Github.

A staggered time integrator for the linear acoustic wave equation using the Jacobi-Anger expansion

In this study, we introduce a staggered time integrator with the Fourier pseudospectral method to solve the first-order linear wave equation, which is accelerated by using the Jacobi-Anger expansion. The proposed method can reduce the computational cost by approximately half compared to the scheme whose temporal order of accuracy is extended by the Lax-Wendroff method under the equivalent modeling conditions, such as time step length, grid interval and maximum wave propagation speed. This is because the Jacobi-Anger expansion can effectively approximate the sinusoidal function, which in our case is the sine function in the wavenumber domain. Based on the wavenumber domain analysis of the proposed method, a strategy to optimally design the simulation parameters to maintain a preset level of accuracy is also introduced. According to the strategy, as the time step length increases, not only the computational cost of the simulation is reduced, but also the accuracy of the numerical solution is improved. Numerical simulations are also performed by using both homogeneous and heterogeneous models to validate the introduced strategy, which supports the practicality of the proposed method.

Sound transmission control through a hybrid smart double sandwich plate structure

A three-dimensional analytical model is developed for control of sound transmission through an acoustically baffled simply supported hybrid smart double sandwich panel partition of rectangular planform. The hybrid structure includes spatially distributed piezoelectric (PZT) and electro-rheological fluid (ERF) actuator layers arranged in a non-collocated configuration in the incident or transmitted panels. The basic formulation utilizes the relevant classical thin plate theories, Kelvin-Voigt viscoelastic damping model, Maxwell’s equations of electrodynamics, and the linear acoustic wave equations coupled by the pertinent structure/fluid compatibility relations. The conventional multi-input multi-output sliding mode control (MIMOSMC) strategy is subsequently applied to improve the sound transmission loss (STL) through the composite partition by smart (semi-active) variation of the stiffness/damping characteristics of the ERF core layer in collaboration with the (active) uniform force PZT actuator layer according to the control commands. Extensive numerical simulations present both the uncontrolled and controlled STL performances of the composite sandwich structure for four different arrangements of the active and semi-active control elements (i.e. PZT/PZT, ERF/ERF, ERF/PZT, and PZT/ERF) as well as for the single partition (ERF, PZT) sandwich panels at selected incident angles and air cavity dimensions. Four different types of vibroacoustic resonances for the composite structure are identified and briefly discussed, namely, the mass-air-mass resonance, the sandwich panel resonances, the cavity standing-wave resonances, and the coincidence resonance. The remarkable overall advantages of all four MIMOSMC-based double panel control configurations in significant attenuation of sound transmission at very low frequencies, and in the vicinity of the coupled mass-air-mass resonances as well as at the panel resonance frequency dips, are demonstrated. In particular, the proposed hybrid smart double panel (ERF/PZT or PZT/ERF) configuration (which integrates the high performance, precision, and capability of the active PZT-based panel with simplicity, reliability, and stability robustness of the semi-active ERF-based panel) is observed to display a considerably smoother broadband sound proof ability in comparison with the conventional full active (PZT/PZT) system. More importantly, it is particularly capable of delivering up to 90% of the full active control performance, with inherently less energy and actuation demands. Limiting cases are considered and validity of the formulation is rigorously confirmed by comparison with the FEM results as well as with the available data.

Identification of Radiation Damping Using a Coupled FE/BE Formulation and a Nonlinear Eigenvalue Solver

AbstractWhile acoustic radiation damping is a rather small and insignificant aspect of many engineering applications, it is the primary energy‐dissipating mechanism for lightweight structures with large radiating surfaces. Attempts to reduce the vibrational response of those lightweight structures by additional mechanical damping can only be successful if the extent of mechanical damping is comparable or larger than the extent of radiation damping. In this regard, engineers are in need of reliable and flexible methods for the quantification of radiation damping, as well as for the modeling of its effect in an early stage of the design process. In this paper, the modal radiation loss factors are obtained from an eigenvalue analysis of the coupled structural acoustic system. The equations of linear elasticity and acoustics are addressed by finite and boundary element methods respectively. The nonlinear eigenvalue problem is solved by contour integration. The thereby obtained modes represent the structural modes subject to acoustic loading. The modal radiation damping values can be obtained from the complex eigenvalues. The approach is verified based on the example of a honeycomb sandwich panel in air.