Transient Acoustic Wave Propagation Research Articles

Theoretical and numerical study on transient acoustic wave propagation across ice layers in the Arctic Ocean.

The study of transient acoustic wave propagation across the Arctic Ocean ice layer provides theoretical guidance for the design of trans-ice acoustic communication systems. In this study, the Arctic Ocean was modeled as an ice-water composite structure, where the ice and water are regarded as an elastic solid and liquid, respectively. An analytical transient solution for acoustic wave propagation in this structure was derived using the eigenfunction expansion method. Further, the numerical procedures were presented and used to analyze the acoustic wave propagation characteristics across the ice layer. The results show that waveforms corresponding to the radial displacements are more severely distorted than the axial displacements. The amplitudes of the radial and axial displacements decreased rapidly with increasing propagation distance. The ice thickness had a greater impact on the radial displacement than axial displacement; the thicker the ice, the greater the distortion for both radial and axial displacements.

The edge-based smoothed FEM with [formula omitted]-Bathe implicit temporal discretization scheme for the analyses of underwater wave propagation problems

The underwater acoustic wave propagation problem is an important issue in ocean engineering field. The classical finite element approach with edge-based smoothed gradient smoothing technique is incorporated into the novel ρ∞-Bathe implicit direct time integration scheme in this paper for solving underwater transient acoustic wave propagation problems. This edge-based smoothed FEM (ES-FEM) and ρ∞-Bathe method are employed to perform the space and time domain discretization, respectively. Due to the appropriate softening effects from the ES-FEM, the space discretization errors can be markedly reduced. Meanwhile, the possible numerical error related to the temporal discretization also can be significantly suppressed by the novel ρ∞-Bathe method which possesses the controllable numerical damping effects. Through detailed analysis of the total dispersion error, we find that the numerical results from the proposed scheme can converge monotonically to the exact solutions when employing the decreasing temporal discretization intervals and the possible numerical anisotropy effects are also effectively suppressed. These excellent numerical performance clearly distinguish the present method from the conventional approaches and make it to be particularly suitable for complicated acoustic wave propagation analysis. Finally, various typical numerical experiments are performed to illustrate the capacities of the proposed scheme in tackling underwater transient acoustic wave propagation problems.

Transient Acoustic Wave Propagation Problems in Multilayered Pavement Using a Time Discontinuous Galerkin Finite Element Method

This paper presents a modified time discontinuous Galerkin finite element method (MDGFEM) for transient acoustic wave propagation problem of multilayered pavement. The pavement consists of cement concrete pavement, semi-rigid base, and natural soil. The multilayered pavement is modeled as poroelastic mediums and assumed to be water-saturated. The well-known generalized Biot’s theory is employed to describe the wave propagation problem. The present MDGFEM, based on the artificial damping scheme, employs the Hermite (P3) functions and the linear (P1) shape functions to interpolate the global nodal vector and its temporal gradient respectively in a time step. Numerical results of 1D and 2D problems show that the MDGFEM can filter out spurious numerical oscillations before and after waves, boundaries of the hole, and the interface between the layers more effectively for the propagation of acoustic waves in multilayered pavement. Compared with widely used time-continuous methods such as the Newmark method, the method proposed in this paper presents better capabilities in the fluid–structure interaction behavior of multilayer pavements and provides a more accurate solution, which contributes to the further development of non-destructive testing of multilayer pavement structures.

Transient acoustic wave propagation in bone-like porous materials using the theory of poroelasticity and fractional derivative: a sensitivity analysis

In this paper, the transient acoustic wave propagation in a bone-like porous material saturated with a viscous fluid was investigated using Biot’s theory. Due to the interaction between the viscous fluid and solid skeleton, the damping behavior is proportional to a fractional power of frequency, i.e., the dynamic tortuosity was written in terms of the fractional power of frequency. Furthermore, to describe the viscous interaction of fluid and solid in the time domain, the fractional derivative was used. The fast and slow waves, which are the solutions to Biot’s equations, were described by fractional calculus in the time domain. The reflection and transmission operators were expressed in the Laplace domain and inverted into the time domain using Durbin’s numerical inversion. Once the numerical implementation was validated, the effects of porosity and viscosity on the stress, and reflected and transmitted waves were investigated. The results showed that by increasing the porosity the stress in a bone-like material filled with either air or bone marrow increases. The transmitted pressure decreases by increasing the porosity. The reflected pressure decreases for low viscous fluid when the porosity increases while it increases when the viscosity of the fluid is high. In addition, the results showed the importance of taking into account the fractional derivatives in the transient wave propagation in such porous materials.

Perfectly matched layers for convex truncated domains with discontinuous Galerkin time domain simulations

This paper deals with the design of perfectly matched layers (PMLs) for transient acoustic wave propagation in generally-shaped convex truncated domains. After reviewing key elements to derive PML equations for such domains, we present two time-dependent formulations for the pressure–velocity system. These formulations are obtained by using a complex coordinate stretching of the time-harmonic version of the equations in a specific curvilinear coordinate system. The final PML equations are written in a general tensor form, which can easily be projected in Cartesian coordinates to facilitate implementation with classical discretization methods. Discontinuous Galerkin finite element schemes are proposed for both formulations. They are tested and compared using a three-dimensional benchmark with an ellipsoidal truncated domain. Our approach can be generalized to domains with corners.

Generalized equation for transient-wave propagation in continuous inhomogeneous rigid-frame porous materials at low frequencies

This paper provides a temporal model for the propagation of transient acoustic waves in continuous inhomogeneous isotropic porous material having a rigid frame at low frequency range. A temporal equivalent fluid model, in which the acoustic wave propagates only in the fluid saturating the material, is considered. In this model, the inertial effects are described by the inhomogeneous inertial factor [A. N. Norris, J. Wave Mat. Interact. 1, 365-380 (1986)]. The viscous and thermal losses of the medium are described by two inhomogeneous susceptibility kernels which depend on the viscous and thermal permeabilities. The medium is one-dimensional and its physical parameters (porosity, inertial factor, viscous, and thermal permeabilities) are depth dependent. A generalized wave propagation equation in continuous inhomogeneous material is established and discussed.

Transient acoustic wave propagation in air-saturated porous media at low frequencies

This paper provides a temporal model of the direct scattering problem for the propagation of transient low frequency waves in a homogeneous isotropic slab of porous material having a rigid frame. In this model, the inertial effects are described by the low frequency approximation of the tortuosity. The viscous and thermal losses of the medium are modeled by viscous and thermal permeabilities. The propagation equation is derived and solved analytically in the time domain. An original expression of Green’s function of the porous medium is obtained. The transmission scattering operator of the first transmitted wave is calculated and compared with experimental data.

Effect of spatial dispersion on transient acoustic wave propagation in 3D

Spatial dispersion is the variation of wave speed with wavelength. It sets in when the acoustic wavelength approaches the natural scale of length of the medium, which could, for example, be the lattice constant of a crystal, the repeat distance in a superlattice, or the grain size in a granular material. In centrosymmetric media, the first onset of dispersion is accommodated by the introduction of fourth order spatial derivatives into the wave equation. These lead to a correction to the phase velocity which is quadratic in the spatial frequency. This paper treats the effect of spatial dispersion on the point force elastodynamic Green’s functions of solids. The effects of dispersion are shown to be most pronounced in the vicinity of wave arrivals. These lose their singular form, and are transformed into wave trains known as quasi-arrivals. The step and ramp function wave arrivals are treated, and it is shown that their unfolded quasi-arrival forms can be expressed in terms of integrals involving the Airy function.

On the propagation of transient acoustic waves in isothermal bubbly liquids

The dynamic propagation of acoustic waves in a half-space filled with a viscous, bubbly liquid is studied under van Wijngaarden's linear theory. The exact solution to this problem, which corresponds to the compressible Stokes' 1st problem for the van Wijngaarden–Eringen equation, is obtained and analyzed using integral transform methods. Specifically, the following results are obtained: (i) van Wijngaarden's theory is found to be ill-suited to describe air bubbles in water; (ii) At start-up, the behavior of the bubbly liquid is similar to that of a class of non-Newtonian fluids under shear; (iii) Bounds on the pressure field are established; (iv) For large time, the solution exhibits Taylor shock-like (i.e., nonlinear) behavior.

A time-domain model of transient acoustic wave propagation in double-layered porous media

This paper concerns a time-domain model of transient wave propagation in double-layered porous materials. An analytical derivation of reflection and transmission scattering operators is given in the time domain. These scattering kernels are the medium’s responses to an incident acoustic pulse. The expressions obtained take into account the multiple reflections occurring at the interfaces of the double-layered material. The double-layered porous media consist of two slabs of homogeneous isotropic porous materials with a rigid frame. Each porous slab is described by a temporal equivalent fluid model, in which the acoustic wave propagates only in the fluid saturating the material. In this model, the inertial effects are described by the tortuosity; the viscous and thermal losses of the medium are described by two susceptibility kernels which depend on the viscous and thermal characteristic lengths. Experimental and numerical results are given for waves transmitted and reflected by double-layered porous media formed by air-saturated plastic foam samples.

Propagation of Transient Acoustic Waves in Layered Porous Media: Fractional Equations for the Scattering Operators

Acoustic waves scattering from a rigid air-saturated porous medium is studied in the time domain. The medium is one dimensional and its physical parameters are depth dependent, i.e., the medium is layered. The loss and dispersion properties of the medium are due to the fluid-structure interaction induced by wave propagation. They are modeled by generalized susceptibility functions which express the memory effects in the propagation process. The wave equation is then a fractional telegraphist’s equation. The two relevant quantities are the scattering operators—transmission and reflection operators—which give the scattered fields from the incident wave. They are obtained from Volterra equations which are fractional equations for the scattering operators.

Reducing the dispersion error in acoustic wave modeling with modified integration rules

Numerical analysis of transient acoustic wave propagation is often performed using finite element or finite difference methods along with central difference time stepping. Such analysis results in wave velocities that are different from exact ones. For conventional algorithms, this dispersion error is second order with respect to discretization parameters. In this paper, modified integration rules are combined with a modified central difference scheme to reduce this error to fourth order. Essentially, the locations of integration points used for evaluating the system matrices are selected such that the wave velocity error is minimized. It was found that the optimal integration points are square roots of 2/3 for the stiffness matrix, which eliminates the anisotropy in the second-order error. The optimal integration points for mass matrix, which depend on the wave velocity, the mesh size, and the time-step size, are also identified to completely eliminate the second-order error. The resulting dispersion error is thus reduced to fourth order. Numerical experiments illustrate that the proposed method has superior performance over existing methods, not only for structured square meshes, but also for unstructured meshes.

Propagation of Sound in Inhomogeneous Media: Exact, Transient Solutions in Curvilinear Geometries

Exact solutions for one-dimensional, transient acoustic wave propagation in curvilinear geometries with mean temperature variations are presented in this paper. These solutions are obtained by using a transformation of the spatial and acoustic variables in a manner suggested by the WKB approximation. The analysis is performed for spherical and cylindrical co-ordinate systems. Temperature profiles admitting exact traveling wave type solutions are derived. Although these solutions resemble the approximate, “high frequency,” WKB form of solution of the wave equation, they have the interesting property that they are exact, regardless of the scale of the acoustic disturbance relative to that of the inhomogeneity. Calculations showing the propagation of waves in cylindrical and spherical geometries are presented.

A FAMILY OF EXACT TRANSIENT SOLUTIONS FOR ACOUSTIC WAVE PROPAGATION IN INHOMOGENEOUS, NON-UNIFORM AREA DUCTS

This paper presents a family of exact solutions for quasi-one-dimensional, transient acoustic wave propagation in ducts with mean temperature and area variations in the absence of mean flow. These solutions are obtained using a transformation of the spatial and acoustic variables in a manner suggested by the WKB approximation. Exact travelling wave-type solutions are obtained for a class of temperature and area profiles. These solutions differ from the classical travelling wave solution, however, in that the acoustic pressure and velocity are not algebraically related by the local value of the acoustic impedance,ρ̄(x)c̄(x). Although these solutions resemble the approximate, “high frequency”, WKB form of solution of the wave equation, they have the interesting property that they are exact, regardless of the scale of the acoustic disturbance relative to that of the inhomogeneity.

Transient acoustic wave propagation in rigid porous media: a time-domain approach

Wave propagation of acoustic waves in porous media is considered. The medium is assumed to have a rigid frame, so that the propagation takes place in the air which fills the material. The Euler equation and the constitutive relation are generalized to take into account the dispersive nature of these media. It is shown that the connection between the fractional calculus and the behavior of materials with memory allows time-domain wave equations, the coefficients of which are no longer frequency dependent, to be worked out. These equations are suited for direct and inverse scattering problems, and lead to the complete determination of the porous medium parameters.

The pseudospectral time-domain (PSTD) algorithm for acoustic waves in absorptive media

A technique based on the combination of Fourier pseudospectral method and the perfectly matched layer (PML) is developed to simulate transient acoustic wave propagation in multidimensional, inhomogeneous, absorptive media. Instead of the finite difference approximation in the conventional finite-difference time-domain (FDTD) method, this technique uses trigonometric functions, through an FFT (fast Fourier transform) algorithm, to represent the spatial derivatives in partial differential equations. Traditionally the Fourier pseudospectral method is used only for spatially periodic problems because the use of FFT implies periodicity. In order to overcome this limitation, the perfectly matched layer is used to attenuate the waves from other periods, thus allowing the method to be applicable to unbounded media. This new algorithm, referred to as the pseudospectral time-domain (PSTD) algorithm, is developed to solve large-scale problems for acoustic waves. It has an infinite order of accuracy in the spatial derivatives, and thus requires much fewer unknowns than the conventional FDTD method. Numerical results confirms the efficacy of the PSTD method.

Shallow water acoustic studies using an air‐suspended water waveguide model

A novel experimental technique is introduced that allows multimode interference and coupling due to boundary topography and to the presence of scatterers in shallow water acoustic waveguides to be studied in the absence of radiation and elastic bottom effects. The air‐suspended water waveguide model consists of a water layer bounded above by air and below by a thin rubber membrane supported by pressurized air to counteract the weight of the water. Experimental data on the transmission of transient ultrasonic signals along the ‘‘perfect’’ waveguide are presented and compared with theory. Both experimental and numerical results are used to demonstrate that multiple nondispersed discrete pulses in a waveguide are equivalent to a summation of highly dispersed waveguide modes. Preliminary experimental results are also given demonstrating the propagation of transient acoustic waves in a free‐fluid layer waveguide with sloped or periodic boundaries.

Transient acoustic wave propagation in a fluid layer with stepwise range dependent perfectly reflecting boundaries

The coupled mode solution technique JR. B. Evans, J. Acoust. Soc. Am. 74, 188–195 (1983)] is adapted to the problem of transient acoustic wave propagation in a single fluid layer with perfectly reflecting boundaries. Properties of one of the boundaries are allowed to vary in a stepwise range dependent manner. The transient response due to an impulsive line source is synthesized from solutions to a global matrix formulation of the coupled mode problem for two‐dimensional waveguides. Several numerical examples are presented that model: (i) waveguide scattering behavior of both single and multiple scatterers placed near the boundary of the fluid layer and (ii) propagation in a waveguide with a variable impedance boundary comprised of alternate free and rigid strips. The numerical results are compared with experimental observations taken from measurements performed at ultrasonic frequencies using an air‐suspended waveguide [J. R. Chamuel and G. H. Brooke, J. Acoust. Soc. Am. Suppl. 1 77, S13 (1985)].

Transient Acoustic Wave Propagation in Stratified Fluids

Abstract : Transient acoustic wave propagation is analyzed for the case of plane-stratified fluids having density rho(y) and sound speed c(y) at depth y. For infinite fluids it is assumed that the (in general discontinuous) functions rho(y), c(y) are uniformly positive and bounded and satisfy abs.val (rho(y) - rho(at infinity)) or = C(+ or - y) to the - alpha power, abs. val. (c(y) - c(at infinity)) or = C(+ or - y) to the - alpha power for + or - y 0, where alpha 3/2. Semi-infinite and finite layers are also treated. The acoustic potential is a solution of the wave equation del-squared u/del t-squared - c-squared(y) rho(y) del dot (1/rho(y)grad(u)) = f(t,x,y) where x = (x1,x2) are horizontal coordinates and f(t,x,y) characterizes the wave sources. The principal results of the analysis show that u is the sum of a free component, which behaves like a diverging spherical wave for large t, and a guided component which is approximately localized in regions abs. val. (y - y sub j) h sub j where c(y) has minima and propagates outward in horizontal planes like a diverging cylindrical wave. (Author)