A mapping approach for handling sloping interfaces

A mapping approach for handling sloping interfaces in parabolic equation solutions is developed and tested. At each range, the medium is rigidly translated vertically so that a sloping interface becomes horizontal. To simplify the approach, the slope is assumed to be small and the extra terms that arise in the wave equation under the mapping are neglected. The effects of these terms can be approximately accounted for by applying a leading-order correction to the phase. The mapping introduces variations in topography, which are relatively easy to handle for the case of a pressure-release boundary condition. The accuracy of the approach is demonstrated for problems involving fluid sediments. The approach should also be accurate for problems involving elastic sediments and should be useful for solving three-dimensional problems involving variable topography.     

Parabolic equation solution of seismo-acoustics problems involving variations in bathymetry and sediment thickness

Recent improvements in the parabolic equation method are combined to extend this approach to a larger class of seismo-acoustics problems. The variable rotated parabolic equation [J. Acoust. Soc. Am. 120, 3534-3538 (2006)] handles a sloping fluid-solid interface at the ocean bottom. The single-scattering solution [J. Acoust. Soc. Am. 121, 808-813 (2007)] handles range dependence within elastic sediment layers. When these methods are implemented together, the parabolic equation method can be applied to problems involving variations in bathymetry and the thickness of sediment layers. The accuracy of the approach is demonstrated by comparing with finite-element solutions. The approach is applied to a complex scenario in a realistic environment.     

Experimental testing of the variable rotated elastic parabolic equation

A series of laboratory experiments was conducted to obtain high-quality data for acoustic propagation in shallow water waveguides with sloping elastic bottoms. Accurate modeling of transmission loss in these waveguides can be performed with the variable rotated parabolic equation method. Results from an earlier experiment with a flat or sloped slab of polyvinyl chloride (PVC) demonstrated the necessity of accounting for elasticity in the bottom and the ability of the model to produce benchmark-quality agreement with experimental data [J. M. Collis et al., J. Acoust. Soc. Am. 122, 1987-1993 (2007)]. This paper presents results of a second experiment, using two PVC slabs joined at an angle to create a waveguide with variable bottom slope. Acoustic transmissions over the 100-300 kHz band were received on synthetic horizontal arrays for two source positions. The PVC slabs were oriented to produce three different simulated waveguides: flat bottom followed by downslope, upslope followed by flat bottom, and upslope followed by downslope. Parabolic equation solutions for treating variable slopes are benchmarked against the data.     

An energy-conserving spectral solution

An energy-conserving spectral solution is derived and tested. A range-dependent medium is approximated by a sequence of range-independent regions. In each region, the acoustic field is represented in terms of the horizontal wave-number spectrum. A condition corresponding to energy conservation is derived for the vertical interfaces between regions. The accuracy of the approach is demonstrated for problems involving sloping ocean bottoms. The energy-conserving spectral solution is less efficient than the energy-conserving parabolic equation solution, but might be suitable for generalization to problems involving elastic bottoms.     

Single-scattering parabolic equation solutions for elastic media propagation, including Rayleigh waves

The parabolic equation method with a single-scattering correction allows for accurate modeling of range-dependent environments in elastic layered media. For problems with large contrasts, accuracy and efficiency are gained by subdividing vertical interfaces into a series of two or more single-scattering problems. This approach generates several computational parameters, such as the number of interface slices, an iteration convergence parameter τ, and the number of iterations n for convergence. Using a narrow-angle approximation, the choices of n=1 and τ=2 give accurate solutions. Analogous results from the narrow-angle approximation extend to environments with larger variations when slices are used as needed at vertical interfaces. The approach is applied to a generic ocean waveguide that includes the generation of a Rayleigh interface wave. Results are presented in both frequency and time domains.     

Statistical description of ray chaos in an underwater acoustic waveguide

An approximate analytical approach is developed to describe the chaotic behavior of ray trajectories in a deep-water acoustic waveguide up to three to five thousands of kilometers in length. The ray dynamics is investigated using the Hamiltonian formalism expressed in terms of the canonical action-angle variables. A realistic waveguide model is used, with refractive-index fluctuations due to the random field of internal waves. The Fokker-Planck equation is obtained for the action variable, and it is shown that the range dependence of this variable can be approximated by the Wiener random process, which represents the simplest model of diffusion. Formulas are derived for calculating the probability density of the coordinate and other ray characteristics. An approximate expression is found for the smoothed field intensity of a point source. For illustrating and testing the formulas obtained, their predictions are compared with the results of numerical solutions of ray equations and the results of field calculations by the parabolic equation method.     

A two-way parabolic equation that accounts for multiple scattering

A two-way parabolic equation that accounts for multiple scattering is derived and tested. A range-dependent medium is divided into a sequence of range-independent regions. The field is decomposed into outgoing and incoming fields in each region. The conditions between vertical interfaces are implemented using rational approximations for the square root of an operator. Rational approximations are also used to relate fields between neighboring interfaces. An iteration scheme is used to solve for the outgoing and incoming fields at the vertical interfaces. The approach is useful for solving problems involving scattering from waveguide features and compact objects.     

Modeling Rayleigh and Stoneley waves and other interface and boundary effects with the parabolic equation

An improved approach for handling boundaries, interfaces, and continuous depth dependence with the elastic parabolic equation is derived and benchmarked. The approach is applied to model the propagation of Rayleigh and Stoneley waves. Depending on the choice of dependent variables, the operator in the elastic wave equation may not factor or the treatment of interfaces may be difficult. These problems are resolved by using a formulation in terms of the vertical displacement and the range derivative of the horizontal displacement. These quantities are continuous across horizontal interfaces, which permits the use of Galerkin's method to discretize in depth. This implementation extends the capability of the elastic parabolic equation to handle arbitrary depth dependence and should lead to improvements for range-dependent problems.     

A single-scattering correction for large contrasts in elastic layers

The single-scattering solution is implemented in a formulation that makes it possible to accurately handle solid-solid interfaces with the parabolic equation method. Problems involving large contrasts across sloping stratigraphy can be handled by subdividing a vertical interface into a series of two or more scattering problems. The approach can handle complex layering and is applicable to a large class of seismic problems. The solution of the scattering problem is based on an iteration formula, which has improved convergence in the new formulation, and the transverse operator of the parabolic wave equation, which is implemented efficiently in terms of banded matrices. Accurate solutions can often be obtained by using only one iteration.     

On the acoustic modes in a duct containing a parabolic shear flow

The exact solution of the acoustic wave equation in an unidirectional shear flow with a parabolic velocity profile is obtained, representing sound propagation in a plane, parallel walled duct, with two boundary layers over rigid or impedance walls. It is shown that there are four cases, depending on the critical level(s) where the Doppler shifted frequency vanishes: (i) for propagation upstream the critical levels are outside the duct (case II); (ii) for propagation downstream there may be two (case IV), one (case I) or no (case III) critical level inside the duct. The acoustic wave equation is transformed in each of the four cases to particular forms of the extended hypergeometric equation, which has power series solutions, some involving logarithmic singularities. In the cases where critical levels occur, at real or ‘imaginary’ distance, matching of two or three pairs of solutions, valid over regions each overlapping the next, is needed. The particular case of the parabolic velocity profile is used to address general properties of sound in unidirectional shear flows. For example, it is shown that for ducted shear flows, there exist a pair of even and odd eigenfunctions, in the absence of critical levels. It is also proved, in more than one instance, that there is no single set of eigenvalues and eigenfunctions valid across one or two shear layers. This leads to the general conjecture, considering the acoustics of shear flows in ducts, that critical levels separate regions with distinct sets of eigenvalues and eigenfunctions.     

Response of infinite length rods and beams with periodically varying area

This paper develops a solution method for the longitudinal motion of a rod or the flexural motion of a beam of infinite length whose area varies periodically. The conventional rod or beam equation of motion is used with the area and moment of inertia expressed using analytical functions of the longitudinal (horizontal) spatial variable. The displacement field is written as a series expansion using a periodic form for the horizontal wavenumber. The area and moment of inertia expressions are each expanded into a Fourier series. These are inserted into the differential equations of motion and the resulting algebraic equations are orthogonalized to produce a matrix equation whose solution provides the unknown wave propagation coefficients, thus yielding the displacement of the system. An example problem of both a rod and beam are analyzed for three different geometrical shapes. The solutions to both problems are compared to results from finite element analysis for validation. Dispersion curves of the systems are shown graphically. Convergence of the series solutions is illustrated and discussed.     

The matched asymptotic expansion (MAE) technique applied to acoustic radiation from vibrating surfaces

The MAE technique has been used in the field of fluid mechanics for many years. Only recently this technique has been applied to acoustic problems, where it has been found to be an excellent and powerful tool in analyzing either scattering and diffraction or radiation from moving rigid objects (propellers). The purpose of this paper is to very briefly review the MAE technique as applied to low frequency acoustics in general, and then apply the resulting approach to a series of progressively more difficult problems which are of interest to many underwater acousticians. The analysis is first applied to two problems with single degrees of freedom for structural vibrations: (1) a sphere, both velocity and force driven, and (2) a circular piston in infinite rigid baffle. These are classical problems and the solutions as obtained by the MAE technique are then compared to the exact classical solutions. The MAE solutions are then generalized to a more difficult problem, with two degrees of freedom for the surface vibration, where two concentric pistons in an infinite rigid baffle are vibrating and coupled via the fluid. For each of the problems analyzed, the structural wavelength a is assumed to be small compared to the fluid wavelength (i.e., ka ? 1). The inner region close to the vibrating structure, in which the fluid motion is effectively incompressible in nature, is governed by the Laplace equation while the outer solution is governed by the Helmholtz equation. The inner and outer solutions are obtained independently and are then joined together by the MAE matching procedure. A composite solution is then obtained from a combination of the inner and the outer solutions. Agreements with the exact theory for the radiated pressure, surface resistance and reactance are shown to be excellent.     

Several Types of Similarity Solutions for the Hunter-Saxton Equation

We study separable and self-similar solutions to the HunterSaxton equation,a nonlinear wave equation which has been used to describe an instability in the director field of a nematic liquid crystal(among other applications).Essentially,we study solutions which arise from a nonlinear inhomogeneous ordinary differential equation which is obtained by an exact similarity transform for the HunterSaxton equation.For each type of solution,we are able to obtain some simple exact solutions in closed-form,and more complicated solutions through an analytical approach.We find that there is a whole family of self-similar solutions,each of which depends on an arbitrary parameter.This parameter essentially controls the manner of self-similarity and can be chosen so that the self-similar solutions agree with given initial data.The simpler solutions found constitute exact solutions to a nonlinear partial differential equation,and hence are also useful in a mathematical sense.Analytical solutions demonstrate the variety of behaviors possible within the wider family of similarity solutions.Both types of solutions cast light on self-similar phenomenon arising in the HunterSaxton equation.     

Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping

We consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time-asymptotically. The hyperbolic model may be viewed as isentropic Euler equations with friction term added to the momentum equation to model gas flow through a porous media. In this case our result justifies Darcy's law time-asymptotically. Our model may also be viewed as an elastic model with damping.Research supported in part by Energy Dept. grant DEFG 02-88-ER25053Research supported in part by NSF grant DMS 90-0226 and Army grant DAAL 03-91-G0017     

Multiscale finite-volume method for parabolic problems arising from compressible multiphase flow in porous media

The multiscale finite-volume (MSFV) method was originally developed for the solution of heterogeneous elliptic problems with reduced computational cost. Recently, some extensions of this method for parabolic problems have been proposed. These extensions proved effective for many cases, however, they are neither general nor completely satisfactory. For instance, they are not suitable for correctly capturing the transient behavior described by the parabolic pressure equation. In this paper, we present a general multiscale finite-volume method for parabolic problems arising, for example, from compressible multiphase flow in porous media. Opposed to previous methods, here, the basis and correction functions are solutions of full parabolic governing equations in localized domains. At the same time, to enhance the computational efficiency of the scheme, the basis functions are kept pressure independent and do not have to be recalculated as pressure evolves. This general approach requires no additional assumptions and its good efficiency and high accuracy is demonstrated for various challenging test cases. Finally, to improve the quality of the results and also to extend the scheme for highly anisotropic heterogeneous problems, it is combined with the iterative MSFV (i-MSFV) method for parabolic problems. As one iterates, the i-MSFV solutions of compressible multiphase problems (parabolic problems) converge to the corresponding fine-scale reference solutions in the same way as demonstrated recently for incompressible cases (elliptic problems). Therefore, the proposed MSFV method can also be regarded as an efficient linear solver for parabolic problems and studies of its efficiency are presented for many test cases.     

A single-scattering correction for the seismo-acoustic parabolic equation

An efficient single-scattering correction that does not require iterations is derived and tested for the seismo-acoustic parabolic equation. The approach is applicable to problems involving gradual range dependence in a waveguide with fluid and solid layers, including the key case of a sloping fluid-solid interface. The single-scattering correction is asymptotically equivalent to a special case of a single-scattering correction for problems that only have solid layers [Ku?sel et al., J. Acoust. Soc. Am. 121, 808-813 (2007)]. The single-scattering correction has a simple interpretation (conservation of interface conditions in an average sense) that facilitated its generalization to problems involving fluid layers. Promising results are obtained for problems in which the ocean bottom interface has a small slope.     

Analytical approach for solving the radiative transfer equation in two-dimensional layered media

This study presents an analytical approach for obtaining Green's function of the two-dimensional radiative transfer equation to the boundary-value problem of a layered medium. A conventional Fourier transform and a modified Fourier series which is defined in a rotated reference frame are applied to derive an analytical solution of the radiance in the transformed space. The Monte Carlo method was used for a successful validation of the derived solutions.     

A Variable Separation Approach to Solve the Integrable and Nonintegrable Models:Coherent Structures of the (2 + 1)-Dimensional KdV Eqnation

We study the localized coherent structures ofa generally nonintegrable (2 1 )-dimensional KdV equation via a variable separation approach. In a special integrable case, the entrance of some arbitrary functions leads to abundant coherent structures. However, in the general nonintegrable case, an additional condition has to be introduced for these arbitrary functions. Although the additional condition has been introduced into the solutions of the nonintegrable KdV equation, there still exist many interesting solitary wave structures. Especially, the nonintegrable KdV equation possesses the breather-like localized excitations, and the similar static ring soliton solutions as in the integrable case. Furthermor,in the integrable case, the interaction between two travelling ring solitons is elastic, while in the nonintegrable case we cannot find even the single travelling ring soliton solution.     

On the use of higher-order azimuthal schemes in 3-D PE modeling

In this paper, the issue of using higher-order finite difference schemes to handle the azimuthal derivative term in a three-dimensional parabolic equation based model is addressed.The three-dimensional penetrable wedge benchmark problem is chosen to illustrate the accuracy and efficiency of the proposed schemes. Both point source and modal initializations of the pressure field are considered. For each higher-order finite difference scheme used in azimuth, the convergence of the numerical solution with respect to the azimuth is investigated and the CPU times are given. Some comparisons with solutions obtained from another 3-D model [J. A. Fawcett, J. Acoust. Soc. Am. 93, 2627-2632 (1993)] are presented. The numerical simulations show that the use of a higher-order scheme in azimuth allows one to reduce the required number of points in the azimuthal direction while still obtaining accurate solutions. The higher-order schemes have approximately the same efficiency as a FFT-based approach (in fact, may outperform it slightly); however, the finite difference approach has the advantage that it may be more flexible than the FFT approach for various PE approximations.     

ADI spectral collocation methods for parabolic problems

We discuss the Crank–Nicolson and Laplace modified alternating direction implicit Legendre and Chebyshev spectral collocation methods for a linear, variable coefficient, parabolic initial-boundary value problem on a rectangular domain with the solution subject to non-zero Dirichlet boundary conditions. The discretization of the problems by the above methods yields matrices which possess banded structures. This along with the use of fast Fourier transforms makes the cost of one step of each of the Chebyshev spectral collocation methods proportional, except for a logarithmic term, to the number of the unknowns. We present the convergence analysis for the Legendre spectral collocation methods in the special case of the heat equation. Using numerical tests, we demonstrate the second order accuracy in time of the Chebyshev spectral collocation methods for general linear variable coefficient parabolic problems.