Nonlinear Progressive Wave Equation Research Articles

Models of nonlinear acoustics viewed as approximations of the Kuznetsov equation

We relate together different models of non linear acoustic in thermo-elastic media as the Kuznetsov equation, the Westervelt equation, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and the Nonlinear Progressive wave Equation (NPE) and estimate the time during which the solutions of these models keep closed in the \begin{document}$ L^2 $\end{document} norm. The KZK and NPE equations are considered as paraxial approximations of the Kuznetsov equation. The Westervelt equation is obtained as a nonlinear approximation of the Kuznetsov equation. Aiming to compare the solutions of the exact and approximated systems in found approximation domains the well-posedness results (for the Kuznetsov equation in a half-space with periodic in time initial and boundary data) are obtained.

Models of nonlinear acoustics viewed as an approximation of the Navier–Stokes and Euler compressible isentropic systems

The derivation of different models of non linear acoustic in thermo-ellastic media as the Kuznetsov equation, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and the Nonlinear Progressive wave Equation (NPE) from an isentropic Navier-Stokes/Euler system is systematized using the Hilbert type expansion in the corresponding perturbative and (for the KZK and NPE equations) paraxial \textit{ansatz}. The use of small, to compare to the constant state perturbations, correctors allows to obtain the approximation results for the solutions of these models and to estimate the time during which they keep closed in the L² norm. In the aim to compare the solutions of the exact and approximated systems in found approximation domains a global well-posedness result for the Navier-Stokes system in a half-space with time periodic initial and boundary data was obtained.

Models of nonlinear acoustics viewed as an approximation of the Navier-Stokes and Euler compressible isentropic systems

The derivation of different models of non linear acoustic in thermo-ellastic media as the Kuznetsov equation, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and the Nonlinear Progressive wave Equation (NPE) from an isentropic Navier-Stokes/Euler system is systematized using the Hilbert type expansion in the corresponding perturbative and (for the KZK and NPE equations) paraxial ansatz . The use of small, to compare to the constant state perturbations, correctors allows to obtain the approximation results for the solutions of these models and to estimate the time during which they keep closed in the L2 norm. The KZK and NPE equations are also considered as paraxial approximations of the Kuznetsov equation, which is a model obtained only by perturbations from the Navier-Stokes/Euler system. The Westervelt equation is obtained as a nonlinear approximation of the Kuznetsov equation. In the aim to compare the solutions of the exact and approximated systems in found approximation domains the well-posedness results (for the Navier-Stokes system and the Kuznetsov equation in a half-space with periodic in time initial and boundary data) were obtained.

Nonlinear acoustic pulse propagation in dispersive sediments using fractional loss operators.

The nonlinear progressive wave equation (NPE) is a time-domain formulation of the Euler fluid equations designed to model low-angle wave propagation using a wave-following computational domain. The wave-following frame of reference permits the simulation of long-range propagation and is useful in modeling blast wave effects in the ocean waveguide. Existing models do not take into account frequency-dependent sediment attenuation, a feature necessary for accurately describing sound propagation over, into, and out of the ocean sediment. Sediment attenuation is addressed in this work by applying lossy operators to the governing equation that are based on a fractional Laplacian. These operators accurately describe frequency-dependent attenuation and dispersion in typical ocean sediments. However, dispersion within the sediment is found to be a secondary process to absorption and effectively negligible for ranges of interest. The resulting fractional NPE is benchmarked against a Fourier-transformed parabolic equation solution for a linear case, and against the analytical Mendousse solution to Burgers' equation for the nonlinear case. The fractional NPE is then used to investigate the effects of attenuation on shock wave propagation.

Shock wave propagation along constant sloped ocean bottoms.

The nonlinear progressive wave equation (NPE) is a time-domain model used to calculate long-range shock propagation using a wave-following computational domain. Current models are capable of treating smoothly spatially varying medium properties, and fluid-fluid interfaces that align horizontally with a computational grid that can be handled by enforcing appropriate interface conditions. However, sloping interfaces that do not align with a horizontal grid present a computational challenge as application of interface conditions to vertical contacts is non-trivial. In this work, range-dependent environments, characterized by sloping bathymetry, are treated using a rotated coordinate system approach where the irregular interface is aligned with the coordinate axes. The coordinate rotation does not change the governing equation due to the narrow-angle assumption adopted in its derivation, but care is taken with applying initial, interface, and boundary conditions. Additionally, sound pressure level influences on nonlinear steepening for range-independent and range-dependent domains are used to quantify the pressures for which linear acoustic models suffice. A study is also performed to investigate the effects of thin sediment layers on the propagation of blast waves generated by explosives buried beneath mud line.

Nonlinear acoustic pulse propagation in dispersive sediments using fractional loss operators

The nonlinear progressive wave equation (NPE) is a time-domain formulation of Euler’s fluid equations designed to model low-angle wave propagation using a wave-following computational domain [ McDonald et al., J. Acoust. Soc. Am. 81]. The wave-following frame of reference permits the simulation of long-range propagation that is useful in modeling the effects of blast waves in the ocean waveguide. However, the current model does not take into account sediment attenuation, a feature necessary for accurately describing sound propagation into and out of the ocean sediment. These attenuating, dispersive sediments are naturally captured with linear, frequency-domain solutions through use of complex wavespeeds, but a comparable treatment is nontrivial in the time-domain. Recent developments in fractional loss operator methods allow for frequency-dependent loss mechanisms to be applied in the time-domain providing physically realistic results [Prieur et al., J. Acoust. Soc. Am. 130]. Using these approaches, the governing equations used to describe the NPE are modified to use fractional derivatives in order to develop a fractional NPE. The updated model is then benchmarked against a Fourier-transformed parabolic equation solution for the linear case using various sediment attenuation factors.

Nonlinear acoustic pulse propagation in range-dependent underwater environments

The nonlinear progressive wave equation (NPE) is a time-domain formulation of Euler’s fluid equations designed to model low angle wave propagation using a wave-following computational domain [B. E. McDonald et al., JASA 81]. The wave-following frame of reference permits the simulation of long-range propagation that is useful in modeling the effects of blast waves in the ocean waveguide. The standard formulation consists of four separate mathematical quantities that physically represent refraction, nonlinear steepening, radial spreading, and diffraction. The latter two of these effects are linear whereas the steepening and refraction are nonlinear. This formulation recasts pressure, density, and velocity into a single variable, a dimensionless pressure perturbation, which allows for greater efficiency in calculations. Nonlinear effects such as weak shock formation are accurately captured with the NPE. The numerical implementation is a combination of two numerical schemes: a finite-difference Crank-Nicholson algorithm for the linear terms of the NPE and a flux-corrected transport algorithm for the nonlinear terms. While robust, solutions are not available for sloping seafloors. In this work, range-dependent environments, characterized by sloping bathymetry, are investigated and benchmarked using a rotated coordinate system approach.

Nonlinear progressive wave equation for stratified atmospheres

The nonlinear progressive wave equation (NPE) [McDonald and Kuperman, J. Acoust. Soc. Am. 81, 1406-1417 (1987)] is expressed in a form to accommodate changes in the ambient atmospheric density, pressure, and sound speed as the time-stepping computational window moves along a path possibly traversing significant altitude differences (in pressure scale heights). The modification is accomplished by the addition of a stratification term related to that derived in the 1970s for linear range-stepping calculations and later adopted into Khokhlov-Zabolotskaya-Kuznetsov-type nonlinear models. The modified NPE is shown to preserve acoustic energy in a ray tube and yields analytic similarity solutions for vertically propagating N waves in isothermal and thermally stratified atmospheres.

Modification of the nonlinear wave progressive equation model for sonic booms in a stratified atmosphere.

The nonlinear progressive wave equation (NPE) model [McDonald and Kuperman, J. Acoust. Soc. Am. 81, 1406 (1987)] is modified to simulate sonic boom physics in a stratified atmosphere using a wave—following window whose dimensions are much less than atmospheric scale heights. (The NPE was originally formulated for time domain weak shock propagation in the ocean, where the ambient density is nearly constant.) This allows simulation of sonic boom propagation in a refracted ray tube in which the ambient density may vary considerably as the simulation window moves along. Energy conservation places a strong constraint on the form of the resulting nonlinear wave equation. Properties of the modified NPE and some illustrative solutions are presented. [Work supported by the Office of Naval Research.]

A comparison of wide-angle and narrow-angle progressive wave equations for modeling sonic boom propagation through turbulence.

The high-angle formulation of the nonlinear progressive wave equation model (NPE2) [McDonald, Wave Motion 31, 165–171 (2000)] has been adapted for atmospheric propagation and is used to perform numerical simulations of sonic boom propagation through atmospheric turbulence. Turbulence is incorporated into the model as perturbations of the local ambient sound speed, which have a random spatial distribution determined by a von Karman energy spectrum. Wave forms, peak overpressures, and wavefront profiles computed at fixed ranges from the onset of turbulence are compared to corresponding solutions produced by the original (narrow-angle) version of the NPE in order to quantitatively assess the importance of wide-angle accuracy for sonic boom propagation.

Weak shock propagation in the ocean and marine sediments.

The nonlinear progressive wave equation (NPE) [B. E. McDonald and W. A. Kuperman, JASA 81, 1406 (1987)] was initially developed to model weak shock waves in a refracting medium (specifically, ocean acoustic convergence zones). It has been refined to include high-angle and two-way propagation. Inclusion of two-way propagation improves agreement with empirical power laws for peak pressure from an explosive as a function of range and explosive charge. The NPE is currently being used to examine weak shock propagation into marine sediments for studies related to mine countermeasures. An intriguing behavior results from the stress-strain relation for Hertzian granular media, in which stress is proportional to the 3/2 power of strain rate. Numerical and analytic solutions with the Hertzian nonlinearity reveal its most nonlinear behavior (shock formation) near zero stress, whereas in a fluid, shock formation results from high stress. Numerical experiments are presented to determine whether shocks resulting from Hertzian nonlinearity can be observed with nominal values of frequency-linear attenuation common to granular media. [Work supported by the Office of Naval Research.]

Nonlinear propagation modeling of infrasound

The significance of nonlinear propagation effects on infrasound is studied using the Nonlinear Progressive Wave Equation (NPE) [B. E. McDonald and W. A. Kupperman, J. Acoust. Soc. Am., 81, 1406--1417 (1987)]. The NPE model accounts for nonlinear effects associated with a weak shock front, including shock-driven energy loss and self refraction. Numerical implementation is accomplished using a pseudospectral approach, which provides excellent computational efficiency while still maintaining acceptably small numerical errors. The nonlinear effects are isolated by generating NPE predictions with the nonlinear terms turned on and off. With the nonlinear terms off, the NPE reduces to a standard linear PE formulation. Waveform predictions through a realistic atmosphere are compared to ground truth observations to evaluate the NPE model performance and assess the influence of the nonlinear effects along the propagation path.

Stability of self-similar plane shocks with Hertzian nonlinearity

The nonlinear progressive wave equation [McDonald and Kuperman, J. Acoust. Soc. Am. 81, 1406-1417 (1987)] is expressed in a form to accommodate Hertzian nonlinearity of order n=3/2 typical of granular media. Stability of self-similar plane weak shocks against perturbations in three dimensions is demonstrated for arbitrary order nonlinearity with 1 <n <3. Investigation of stability in the presence of Hertzian nonlinearity is motivated by the nonlinearity coefficient being arbitrarily large in the unstressed state. Energy in the perturbation field is shown to fall off at least as fast as 1/t, while energy in the self-similar wave falls off slower than 1/t. For an initial wave profile increasing in the direction of propagation, it is shown that where the quadratic nonlinearity coefficient diverges, the slope of the wave goes to zero.

Propagation of shock waves from source to receiver

This paper describes a 3-stage technique to model the propagation to the far field of nonlinear shock waves produced by high-energy explosive sources. The stages are: (1) a 3-dimensional unsteady Euler method for the strong shock near the source, including a Flux-Corrected-Transport (FCT) method, (2) the Nonlinear Progressive-wave Equation (NPE) procedure for the degeneration of the shock into a linear acoustical wave, and (3) the Parabolic Equation (PE) method for propagation to the far field. Comparisons of calculated data with a linear analytical method are shown.

Long range nonlinear propagation in an ocean acoustic waveguide

The Nonlinear Progressive Wave Equation (NPE) [McDonald and Kuperman, J. Acoust. Soc. Am. 81, 1406–1417 (1987)] is an approximation of the Euler equations for nonlinear compressional waves in an inviscid fluid and, actually, is the nonlinear time domain counterpart of the frequency domain linear parabolic wave equation (PE) for small angle propagation. Simulations using a npe code were used to study both harmonic (high frequency) and parametric (low frequency) generation. This code was coupled with a linear adiabatic normal mode program which allows range-dependent cases treatment to study propagation in shallow or deep water with longer range propagation paths. Included in the modeling are both shock dissipation and linear attenuation in the sediment layer. The results of these studies are presented.

High-angle formulation for the nonlinear progressive-wave equation model

The nonlinear progressive-wave equation (NPE) model has been reformulated for wider propagation angle accuracy. The wide angle NPE results from range-integrating the second order time domain nonlinear wave equation rather than reducing it to a first order equation in time as in the original NPE. The numerical implementation of the second order NPE retains two time levels rather than one. The resulting model is refered to as NPE2. Benchmark tests reveal RMS errors in the model are reduced by nearly a factor of 10 for moderately high propagation angles as compared to errors in similar tests of the first order NPE.

Estimation of nonlinear effects in global scale ocean acoustic propagation.

The nonlinear progressive wave equation (NPE) [B. E. McDonald and W. A. Kuperman, J. Acoust. Soc. Am. 81, 1406–1417 (1987)] is used to derive quantitative estimates for the magnitudes and characteristic features of nonlinearities generated by finite amplitude acoustic signals during propagation across ocean paths of global scale. Nonlinear effects accumulate over long propagation paths, and are accentuated at convergence zone caustics. The goals are to estimate (1) levels of harmonic generation in ocean tomographic experiments and (2) waveform steepening in underwater explosions. Harmonic generation (1) is estimated from a linear baseline solution involving vertical eigenmode expansion. The result implies second harmonic magnification at odd numbered convergence zone ranges. Waveform steepening (2) is estimated by time‐domain numerical integration and related to modal excitation differences in the far field [J. Ambrosiano, D. R. Plante, B. E. McDonald, and W. A. Kuperman, J. Acoust. Soc. Am. 87, 1473–1481 (1990)]. [Work supported by ONR.]

Propagation of signals from strong explosions above and below the ocean surface.

In support of the Comprehensive Test Ban, research is underway on the long‐range propagation of signals from nuclear explosions in the deep underwater sound (SOFAR) channel. Our work has emphasized the variation of wave properties and source region energy coupling as a function of height or depth of burst. Initial calculations on CALE, a two‐dimensional hydrodynamics code developed at LLNL by Robert Tipton, were linked at a few hundred milliseconds to a version of NRL’s weak shock code, NPE, which solves the nonlinear progressive wave equation [B. E. McDonald and W. A. Kuperman, J. Acoust. Soc. Am. 81, 1406–1417 (1987)]. The wave propagation simulation was then followed down to 5000‐m depth and out to 10 000‐m range. In the future, calculations on a linear acoustics code will extend the propagation to greater distances. Until recently our research has considered only explosions in or above the deep ocean. New results on energy coupling and signal propagation in shallow water and the effects of other improvements will be presented. [Work performed under the auspices of the U. S. Department of Energy by the Lawrence Livermore National Laboratory under Contract W‐7405‐ENG‐48.]

Thermoviscous effects on transient and steady-state sound beams using nonlinear progressive wave equation models

Earlier NPE (nonlinear progressive wave equation) models were developed without considering dissipative effects of the media. In the present study, the thermoviscous dissipation term is derived and added to the NPE models. According to types of waves and coordinates under consideration, four formulations of dissipative versions of NPE are developed. Cylindrical and spherical versions of NPE which are suitable for describing propagation of sound beams are used to study the dissipative effects on a tone burst. The dissipative NPE version gives excellent agreement with experiment data by Moffett et al. [J. Acoust. Soc. Am. 47, 1473–1474 (1970); 49, 339–343 (1971)] and Lockwood et al. [J. Acoust. Soc. Am. 53, 1148–1153 (1973)]. Also, the dissipative NPE version can describe phenomena of self-demodulation. In addition, it is concluded that the number of wavelengths that are needed in an initial tone burst in order to obtain steady-state parts in the waveform is affected by the locations under consideration and the dissipation value.

A tutorial on the non-linear progressive wave equation (NPE). Part 2. Derivation of the three-dimensional Cartesian version without use of perturbation expansions

The NPE is obtained by a novel method which avoids the use of perturbation expansions (McDonald et al., Appl. Acoust., 43 (1994) 59–67). The single variable NPE is derived here from the continuity equation by assuming an irrotational flow and employing partially linearised versions of the momentum and state equations. This derivation sheds new light on the range of conditions of applicability of the NPE.