Wave Equation Model Research Articles

Military jet noise source imaging using multisource statistically optimized near-field acoustical holography

The identification of acoustic sources is critical to targeted noise reduction efforts for jets on high-performance tactical aircraft. This paper describes the imaging of acoustic sources from a tactical jet using near-field acoustical holography techniques. The measurement consists of a series of scans over the hologram with a dense microphone array. Partial field decomposition methods are performed to generate coherent holograms. Numerical extrapolation of data beyond the measurement aperture mitigates artifacts near the aperture edges. A multisource equivalent wave model is used that includes the effects of the ground reflection on the measurement. Multisource statistically optimized near-field acoustical holography (M-SONAH) is used to reconstruct apparent source distributions between 20 and 1250 Hz at four engine powers. It is shown that M-SONAH produces accurate field reconstructions for both inward and outward propagation in the region spanned by the physical hologram measurement. Reconstructions across the set of engine powers and frequencies suggests that directivity depends mainly on estimated source location; sources farther downstream radiate at a higher angle relative to the inlet axis. At some frequencies and engine powers, reconstructed fields exhibit multiple radiation lobes originating from overlapped source regions, which is a phenomenon relatively recently reported for full-scale jets.

Nonlinear stabilization through wave PDE dynamics with a moving uncontrolled boundary

In deep oil drilling, the length of the domain over which the wave equation models the torsional dynamics of the drill string keeps changing with time, and it also depends on the drill bit speed. Moreover, the drill bit speed cannot be controlled directly. In this context, we consider predictor-based design for the cascade system of a nonlinear ODE and a wave PDE with a moving uncontrolled boundary. In comparison with prior results on wave PDE–ODE cascades, this work differs by giving rise to a prediction horizon that is not given explicitly but has to be found from an implicit relationship involving the delay function and the future solution of the system. Stability analysis of the closed-loop system is conducted by constructing infinite-dimensional backstepping transformations and a Lyapunov functional. An explicit feedback law for compensating the wave actuator dynamics is obtained. For the moving boundary that depends on both the ODE’s state and time, a region of attraction is estimated. For the moving boundary that depends on time, a global stabilization for the closed-loop system is achieved. Finally, an example is given to illustrate the effectiveness of the proposed design technique.

Analytical Solution of the Classical Dam-Break Problem for the Gravity Wave–Model Equations

AbstractSaint-Venant Equations (SVE) are often simplified for the sake of practicality, faster computational times, or physical representation. The Gravity Wave Model (GWM), or Local Inertial Equations, is a simplification of the SVE whereby the convective terms are neglected. The aim of this work is to present an analytical solution for the dam-break problem based on the GWM set of equations without source terms using the method of characteristics (MOC) and compare it with the analytical solution of SVE. The formulas for the depth and velocity are derived and explained along with the wave propagation characteristics. Conclusions are drawn about the propagation and analytical solution.

Broadband omnidirectional absorption for linear liquid surface wave

We investigate the propagation of linear liquid surface wave (LLSW) over an omnidirectional absorber (OA), which has uneven bottom with appropriate gradient depth distribution. The OA consists of a guiding shell with gradient bottom topography and a central absorptive cavity. The non-dispersive shallow-water wave equation model and the dispersive mild-slope equation model are employed to investigate the absorption properties of the OA. The results show that almost omnidirectional incident LLSW can be spirally deflected from the interface to the center in the guiding shell and finally absorbed by the central cavity. The introducing of guiding shell enlarges the absorption cross section of the bare absorptive cavity from the inner diameter to the outer diameter of the OA, resulting in high absorption efficiency. Furthermore, the proposal can be constructed by simple modulation of the liquid bottom, which may significantly facilitate the engineering applications.

Three-dimensional acoustic wave equation modeling based on the optimal finite-difference scheme

Generally, FD coefficients can be obtained by using Taylor series expansion (TE) or optimization methods to minimize the dispersion error. However, the TE-based FD method only achieves high modeling precision over a limited range of wavenumbers, and produces large numerical dispersion beyond this range. The optimal FD scheme based on least squares (LS) can guarantee high precision over a larger range of wavenumbers and obtain the best optimization solution at small computational cost. We extend the LS-based optimal FD scheme from two-dimensional (2D) forward modeling to three-dimensional (3D) and develop a 3D acoustic optimal FD method with high efficiency, wide range of high accuracy and adaptability to parallel computing. Dispersion analysis and forward modeling demonstrate that the developed FD method suppresses numerical dispersion. Finally, we use the developed FD method to source wavefield extrapolation and receiver wavefield extrapolation in 3D RTM. To decrease the computation time and storage requirements, the 3D RTM is implemented by combining the efficient boundary storage with checkpointing strategies on GPU. 3D RTM imaging results suggest that the 3D optimal FD method has higher precision than conventional methods.

Acoustic particle displacement resonator

A simple device consisting of a waveguide and two loudspeakers is proposed for generation of low-frequency standing acoustic field with high amplitude of acoustic velocity and particle displacement, which is primarily intended to be used for stabilization of electric discharges in acoustic field. A coupled model of loudspeakers and nonlinear wave equation including waveguide radius variability, thermoviscous attenuation in boundary layer and minor losses is developed. The results of the conducted experiments validate the model revealing that the minor losses and acoustically generated turbulence in the boundary layer represent an important means of acoustic energy dissipation in this and similar applications.

Coupled wave-equation and eddy-current model for modelling and measuring propagating stress-waves

AbstractThe coupling of the propagating stress wave with the eddy current model is presented. The applied stress produces magnetization in the sample that can be measured outside the sample by measuring the resulting magnetic flux density. The stress and flux density measurements are made on a mechanically excited steel bar. The problem is modelled with the finite element method for both the propagating wave and the eddy current. Three aspects are considered: eddy current model using magnetization from the measurements, coupled wave and eddy current models, and coupled different dimensions in the wave model. The measured stress can be reproduced from the measured flux density by modelling. The coupled models work both for stress and flux couplings as well as for the different dimensionality couplings.

The local well-posedness, existence and uniqueness of weak solutions for a model equation for shallow water waves of moderate amplitude

This paper deals with the Cauchy problem for a nonlinear equation modeling the evolution of the free surface for waves of moderate amplitude in the shallow water regime, which was proposed by Constantin and Lannes (2009) [20]. Applying the pseudoparabolic regularization technique, the local well-posedness of strong solutions in Sobolev space Hs(R) with s>3/2 is established via a limiting procedure. Moreover, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space Hs(R) with 1<s≤3/2 is obtained.

Multisource statistically optimized near-field acoustical holography.

This paper presents a reduced-order approach to near-field acoustical holography (NAH) that allows the user to account for sound fields generated by multiple spatially separated sources. In this method, an equivalent wave model (EWM) of a given field is formulated to include combinations of planar, cylindrical, spherical, or other elementary wave functions in contrast to an EWM restricted to a single separable coordinate system. This can alleviate the need for higher-order functions, reduce the number of measurements, and decrease error. The statistically optimized near-field acoustical holography (SONAH) algorithm is utilized to perform the NAH projection after the formulation of the multisource EWM. The combined process is called multisource statistically optimized near-field acoustical holography (M-SONAH). This method is used to reconstruct simulated sound fields generated by combinations of a vibrating piston in a sphere and linear arrays of monopole sources. It is shown that M-SONAH can reconstruct near-field pressures in multisource environments with lower errors and fewer measurements than a strictly plane or cylindrical-wave formulation using the same simulated measurement.

Solution of Wave Equations on Transmission Lines where Leakage to Ground on the Line is Negligible

This paper presents the solution of wave equations on transmission lines where leakage to ground on the line is very small. As a result of the leakages to ground on the transmission lines which are negligible, the conductance and the inductance, which are responsible for leakages on the line, are set to zero in the model of the general wave equation of the transmission line. The Laplace transform method was now applied to transform the resulting partial differential equation into ordinary differential equation and the method of variation of parameters was used to get the particular solution to the problem.

Hybrid Kinematic-Dynamic Approach to Seismic Wave-Equation Modeling, Imaging, and Tomography

Estimation of the structure response to seismic motion is an important part of structural analysis related to mitigation of seismic risk caused by earthquakes. Many methods of computing structure response require knowledge of mechanical properties of the ground which could be derived from near-surface seismic studies. In this paper we address computationally efficient implementation of the wave-equation tomography. This method allows inverting first-arrival seismic waveforms for updating seismic velocity model which can be further used for estimating mechanical properties. We present computationally efficient hybrid kinematic-dynamic method for finite-difference (FD) modeling of the first-arrival seismic waveforms. At every time step the FD computations are performed only in a moving narrowband following the first-arrival wavefront. In terms of computations we get two advantages from this approach: computation speedup and memory savings when storing computed first-arrival waveforms (it is not necessary to make calculations or store the complete numerical grid). Proposed approach appears to be specifically useful for constructing the so-called sensitivity kernels widely used for tomographic velocity update from seismic data. We then apply the proposed approach for efficient implementation of the wave-equation tomography of the first-arrival seismic waveforms.

A multisource-type representation statistically optimized near-field acoustical holography method

A reduced-order approach to near-field acoustical holography (NAH) that accounts for sound fields generated by multiple spatially separated sources of different types is presented. In this method, an equivalent wave model (EWM) of a given field is formulated based on rudimentary knowledge of source types and locations. The statistically optimized near-field acoustical holography (SONAH) algorithm is utilized to perform the NAH projection after the formulation of the multisource EWM. The combined process is called multisource-type representation SONAH (MSTR SONAH). This method is used to reconstruct simulated sound fields generated by combinations of multiple source types. It is shown that MSTR SONAH can successfully reconstruct the near field pressures in multisource environments where other NAH methods result in large errors. The MSTR SONAH technique can be extended to general sound fields where the shapes and locations of sources and scattering bodies are known.

Super-Grid Modeling of the Elastic Wave Equation in Semi-Bounded Domains

AbstractWe develop a super-grid modeling technique for solving the elastic wave equation in semi-bounded two- and three-dimensional spatial domains. In this method, waves are slowed down and dissipated in sponge layers near the far-field boundaries. Mathematically, this is equivalent to a coordinate mapping that transforms a very large physical domain to a significantly smaller computational domain, where the elastic wave equation is solved numerically on a regular grid. To damp out waves that become poorly resolved because of the coordinate mapping, a high order artificial dissipation operator is added in layers near the boundaries of the computational domain. We prove by energy estimates that the super-grid modeling leads to a stable numerical method with decreasing energy, which is valid for heterogeneous material properties and a free surface boundary condition on one side of the domain. Our spatial discretization is based on a fourth order accurate finite difference method, which satisfies the principle of summation by parts. We show that the discrete energy estimate holds also when a centered finite difference stencil is combined with homogeneous Dirichlet conditions at several ghost points outside of the far-field boundaries. Therefore, the coefficients in the finite difference stencils need only be boundary modified near the free surface. This allows for improved computational efficiency and significant simplifications of the implementation of the proposed method in multi-dimensional domains. Numerical experiments in three space dimensions show that the modeling error from truncating the domain can be made very small by choosing a sufficiently wide super-grid damping layer. The numerical accuracy is first evaluated against analytical solutions of Lamb’s problem, where fourth order accuracy is observed with a sixth order artificial dissipation. We then use successive grid refinements to study the numerical accuracy in the more complicated motion due to a point moment tensor source in a regularized layered material.

On the quenching behaviour of a semilinear wave equation modelling MEMS technology

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Discrete and Continuous Dynamical Systems - Series A following peer review. The definitive publisher-authenticated version 2015, 35(3), pp. 1009-1037 is available online at: http://dx.doi.org/10.3934/dcds.2015.35.1009

A wave equation model for retrograde media with a bubble structure

A nonlinear wave equation is obtained for liquid-vapor mixtures with regard to arbitrary change of equilibrium sound speed in a mixture with growth and condensation of vapor bubbles, which has a limit transition to a wave equation for pure liquid at full collapse of vapor cavities. A particular case of a collapsing wave equation (CWE) of retrograde media is considered, e.g., liquid-vapor media with low vaporization heat and low dissipation, observed near the critical liquid-vapor point, where the main effect is variable sound speed. Based on the CWE numerical solutions it is shown that in such liquid-vapor media, one can observe previously unknown nonlinear wave structures with depression waves.

An efficient wavenumber–space–time domain finite-difference modeling of acoustic wave equation for synthesizing CMP gathers

The wavenumber–space–time domain wave equation is derived from the 2D time-domain wave equation in the horizontally stratified media by taking the Fourier transform of the equation with respect to one of the spatial variables. The derived wave equation is solved by the finite-difference method and analyzed in terms of stability and dispersion. Although there are lots of wavenumbers to be solved according to the sampling theorem, the modeling can be accelerated by limiting the range of wavenumber used for the inverse Fourier transform. The proposed method generates common mid-point (CMP) gathers directly without sorting process by applying the shifting theorem. We show that our algorithm is efficient and generates the same results as ones from the 2D modeling through two numerical examples. The proposed modeling algorithm can be used for the inversion of CMP gathers or amplitude-versus-offset inversion.

An optimal 5-point scheme for frequency-domain scalar wave equation

The classic 5-point scheme has been commonly applied in frequency-domain finite-difference (FD) modeling and full waveform inversion of 2D scalar wave equation. This scheme takes less computational costs but provides pretty lower accuracy than other more-point schemes. To improve the accuracy without increasing computational costs of this scheme, I develop a new 5-point scheme. Taylor-series expansion (TE) and conjugate gradient (CG) methods are adopted to calculate FD coefficients respectively. Dispersion analysis shows that the new scheme provides significantly greater accuracy than the classic one, and the CG-based new scheme generally has higher precision than the TE-based new one. Numerical experiments demonstrate the advantage of the proposed scheme. • A new 5-point scheme is proposed for 2D frequency-domain scalar wave equation. • Optimal finite-difference coefficients are obtained by conjugate gradient method. • The new scheme provides much greater accuracy than the classic one.

Smooth and singular multisoliton solutions of a modified Camassa–Holm equation with cubic nonlinearity and linear dispersion

We develop a direct method for solving a modified Camassa–Holm equation with cubic nonlinearity and linear dispersion under the rapidly decreasing boundary condition. We obtain a compact parametric representation for the multisoliton solutions and investigate their properties. We show that the introduction of a linear dispersive term exhibits various new features in the structure of solutions. In particular, we find the smooth solitons whose characteristics are different from those of the Camassa–Holm equation, as well as the novel types of singular solitons. A remarkable feature of the soliton solutions is that the underlying structure of the associated tau-functions is the same as that of a model equation for shallow-water waves introduced by Ablowitz et al (1974 Stud. Appl. Math. 53 249–315). Finally, we demonstrate that the short-wave limit of the soliton solutions recovers the soliton solutions of the short pulse equation which describes the propagation of ultra-short optical pulses in nonlinear media.

Optimal staggered-grid finite-difference schemes based on least-squares for wave equation modelling

Optimal staggered-grid finite-difference schemes based on least-squares for wave equation modelling

Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance

Exact explicit rogue-wave solutions of intricate structures are presented for the long-wave-short-wave resonance equation. These vector parametric solutions feature coupled dark- and bright-field counterparts of the Peregrine soliton. Numerical simulations show the robustness of dark and bright rogue waves in spite of the onset of modulational instability. Dark fields originate from the complex interplay between anomalous dispersion and the nonlinearity driven by the coupled long wave. This unusual mechanism, not available in scalar nonlinear wave equation models, can provide a route to the experimental realization of dark rogue waves in, for instance, negative index media or with capillary-gravity waves.