Mode functions for the wide-angle approximation to the parabolic equation

A new theoretical-numerical method has been developed which enables one to accurately and efficiently compute the entire two-dimensional (depth and range) low-frequency underwater acoustic field in realistic ocean environments. The theoretical development is based on the parabolic equation approximation [V.A. Fock, Electromagnetic Diffracfion Problems (Pergamon, 1965)] which automatically includes diffraction and all other full-wave effects, as well as depth- and range-dependent sound speeds and volume losses, and variable bathymetry. This parabolic wave equation is solved numerically using the split-step Fourier algorithm [R. H. Hardin and F. D. Tappert, SIAM Rev. 15, 423 (1973)] which is accurate, efficient, unconditionally stable, and is readily implemented. Operational computer programs based on this method have been used to reliably predict long-range underwater propagation effects in areas of practical interest. Representative results are displayed in the form of detailed computer-drawn decibel contours which illustrates SOFAR, BSR, surface duct, bottom bounce, and other modes of propagation in the presence of horizontal thermoclines, sea-mounts, and sea-ridges, migrating sound channel axes, multiple channels, and so forth.

Aiming at the low computational efficiency of the fixed-step solution of the standard parabolic equation (SPE) in the study of complex environmental radio wave propagation, a variable step size technology for the standard parabolic equation is proposed. Based on the relationship between the factors such as step size and frequency with the calculation error in the SSFT solution, the basic criteria for the SPE variable step size solution are proposed; secondly, the complex environment of SPE application is graded, and the corresponding step size is selected for different complex environment levels under the condition of satisfying the error requirement. Finally, the method is used to simulate the radio wave characteristics of typical complex environment. The calculation results verify the reliability and efficiency of the proposed method. It shows that the proposed method can greatly improve the computational efficiency of parabolic equations under the condition of ensuring the accuracy of calculation.

Based on the conventional split-step Fourier transform(SSFT),the recursive two-way parabolic equation(TWPE) modeling is presented for the problems of former algorithms' deficiencies and computational error.The traditional algorithms forecast only forward-propagating waves and neglect the backward-propagating waves,which causes greater error in the presence of irregular terrain boundary environment.The new algorithm enhances the computation accuracy and remedies the insufficiencies by the recursive two-way split-step method.Compared to traditional wide-angle parabolic equation(WAPE) and geometrical theory of diffraction(GTD),the correctness and applicability of the model are verified by simulation results in the complex environment including single knife-edge,double knife-edges and real terrain.

The split-step Fourier algorithm is widely used in underwater acoustics to solve parabolic equations (PEs) resulting from split-operator approximations to the one-way wave equation. Recently, Porter and Jensen [J. Acoust. Soc. Am. 94, 1510–1516 (1993)] reported on a seemingly benign propagation problem in which two wide-angle split-operator PEs performed poorly compared to the standard PE. For a source and a receiver located within a leaky surface duct, each of the wide-angle approximations gave rise to anomalously high values of transmission loss at ranges where the leakage energy is refracted back into the duct. Moreover, the two wide-angle PEs were more sensitive to the choice of reference wave number than the standard PE. This behavior is contrary to previous experience with such split-operator approximations in underwater acoustics applications. In this paper, we present a higher-order PE formulation that retains most of the efficiency of the split-step algorithm while improving the accuracy of the propagated solutions. In its simplest form, the new formulation modifies the standard split-step Fourier method by additionally solving a tridiagonal system of equations at each range step. The accuracy of our algorithm is illustrated with several numerical examples.

A modified wide-angle parabolic equation (MWAPE) [St. Mary et al., J. Comput. Phys. 71, 304–315 (1987)] is studied for a uniform plane-parallel waveguide with pressure-release surface and rigid bottom. In this situation, the MWAPE is separable, and the resulting fourth-order boundary value problem is obtained and solved, which leads to an exact solution of the MWAPE. It is shown that the depth-dependent eigenfunctions in this instance are identical to those resulting from both the standard parabolic equation and Helmholtz equation in the same channel. Modal wavenumbers for the MWAPE are shown to be equivalent to the first three terms of the Taylor expansion for the Helmholtz wavenumbers, which is one term better than the standard PE wavenumbers. This exact solution can serve as an accuracy benchmark for numerical implementations of the MWAPE.

In Chapter 2, we have seen how the parabolic equation could be interpreted in terms of Fourier transforms for the case of propagation in vacuum. The situation gets more complex when refractive index variations are present, since they cannot be dealt with directly with Fourier transform techniques. Section 3.2 introduces the powerful split-step/Fourier technique pioneered by Hardin and Tappert, which replaces the original problem with propagation through a sequence of phase screens. At this stage the discussion is limited to the narrow-angle case, starting from the standard parabolic equation.

A double-layer model is proposed to reduce the calculation amounts of the Linear Ship Map (LSM) model. The proposed model can be used for rapid and accurate calculation of the electromagnetic propagation characteristics in the complicated atmospheric environment over the sea. In the proposed model, the calculation regions are divided into the upper-layer and the lower-layer. The upper-layer is calculated by the Wide angle Parabolic Equation (WPE) model and the lower-layer is calculated by the LSM model. Through reducing the calculation height and optimizing the step length, the proposed model can be exact and rapid. By simulation, the proposed model is compared with LSM model in the smooth and the rough sea surface conditions. The results show that the proposed model can decrease the calculation time by 1/10 in the rough sea surface condition.

A normal mode theory is developed for the Brownian dynamics of weakly bending rods with preset hydrodynamic interactions. The rod is replaced by a chain of contiguous spheres whose radius is chosen to yield the appropriate uniform translational and rotational diffusion coefficients. Despite the inclusion of preset hydrodynamic interactions in the dynamical operator, its normal modes are not coupled by the potential energy, so their amplitudes remain pairwise “orthogonal” under equilibrium averaging. The uniform translational and rotational diffusion coefficients obtained from Langevin theory are shown to be identical to those obtained from the Kirkwood algorithm, despite their rather different appearance. An expression is given for the mean squared angular displacement 〈Δxm(t)2〉 of the mth bond vector around the instantaneous x axis (perpendicular to the end‐to‐end vector z). Necessary algorithms are presented for the numerical evaluation of all quantities. The normal mode theory is compared with Brownian dynamics simulations for the same model by examining 3〈Δxm(t)2〉 for the central bond vector of rods comprising 10 and 30 subunits with various persistence lengths. The normal mode theory works very well for all times for L/P ≲ 0.6, where P = κ/kBT is the persistence length and κ is the bending rigidity. With increasing flexibility, the domain of validity of the normal mode theory is restricted to shorter times, where violations of the weak bending approximation are less severe. However, increasing the length of the rod from 10 to 30 subunits yields improved agreement with the simulations for the same and even longer times. This latter effect is tentatively attributed to the greater fluctuating tension in the longer chains, which acts to retard the rotational relaxation in the simulations, but is not taken into account in the present normal mode theory.

The wide-angle parabolic equation gives a rigorous solution to the problem of diffraction by several buildings of arbitrary shape. Applications to propagation in urban microcellular environments are presented.

Radar systems performance under anomalous propagation conditions can be predicted if the atmosphere is properly known. This article introduces a tomographic approach to estimate refractivity in the troposphere with an adjoint-based inversion method. A new adjoint model is developed for the 2-D wide-angle parabolic equation using a variational adjoint approach, to invert refractivity from phaseless data measured with an array of radio receivers in an open-sea environment. The obtained adjoint model is validated considering propagation through a range-independent medium over a flat perfectly-electric-conducting surface at horizontal polarization. The ill-posedness of the regarded inverse problem is shown with the optimization landscapes. The parametric study indicates the potential use of this method as a refractivity gradient retrieval system under certain circumstances.

Wind Turbine Noise (WTN) variability is often related to the influence of local micrometeorology as well as to space and time variations in the ground properties. Thus, the PIBE Project (2020-2024) aims to estimate these variabilities and their associated uncertainties. Its objective is to enable engineering companies or wind farm developers to estimate more accurately the risk of noise. In the WP2 of the PIBE project, 3 approaches are explored: (i) a NUMerical approach through the development of a metamodel, (ii) an EXPerimental approach through a major in-situ measurement campaign lasting more than 1 year, and (iii) a STATistical approach consisting in the evaluation of WTN variability. The NUM approach consists in sound emission calculation of each wind turbine using the Amiet's theory, and sound propagation calculation using a wide-angle Parabolic Equation (WAPE) modelling. The EXP part of the PIBE project has led to the development of a large database, accessible via a dedicated Shiny application. The last approach (STAT) has been carried out using both an open-access online tool called WindTUNE and an open-source tool based on Code_TYMPANTM, namely OCP. The 3 approaches (NUM + EXP + STAT) are presented and the corresponding results are discussed.

An automatic detection method is employed to identify and track eddies in the Gulf of Mexico. The physical parameters of the eddies, such as lifespan, radius, and distribution position are first examined and used to determine the spatio–temporal evolution of a strong warm eddy separated from the Mexico current. Then, the influence of this strong warm eddy on sound propagation during its lifespan are comprehensively analyzed with the parabolic equation and explained by using the normal mode and ray theories. Additionally, the influence of mesoscale eddies on the redistribution of total depth-integrated energy among the normal modes in the deep water is also discussed. The variation of arrival angle is investigated to explain the spreading acoustic energy caused by eddies. Overall, the results show that warm eddies can change the propagation paths and cause the convergence zone to broaden and approach the sound source. Moreover, the warm eddy can disperse sound energy and cause the total depth-integrated energy to incline to a lower normal mode. Throughout the whole of these three periods (eddy generating, eddy maturing, and eddy terminating), the fluctuation in the transmission loss is up to 30 dB (depending on the relative location of eddy center to the source).

A wide‐angle parabolic wave equation solution, using shift‐map and finite‐difference techniques, is presented. The corresponding split‐step Fourier solution is well known. The solution using finite‐difference technique, where the standard parabolic wave equation is modified into the so‐called Claerbout equation allowing propagation angles up to 45° from the paraxial direction, is also well known. Here, we present an extension to that solution in which the shift‐map technique is incorporated into the finite‐difference scheme allowing a varying terrain to be considered. The result is a solution that corresponds to the well known split‐step solution, which is believed to perform well for terrain slopes up to 10°–15°, and discontinuous slope changes on the order of 15°–20°. This solution is of first order with respect to the terrain slope. The resulting difference scheme is for solving a tridiagonal system. When using the finite‐difference technique, it is also possible to find a second order solution with respect to the terrain slope. The resulting difference scheme in this case is for solving a pentadiagonal system. This new solution performs well for slopes up to about 15°–20° and discontinuous slope changes up to about 30°–40°, which is an improvement.

In the presence of currents, vertically stratified fluids become acoustically anisotropic in the horizontal plane and the group and phase velocities of the acoustic normal mode are generally not parallel. In this paper, the resulting 3-D effects and their implications for acoustic monitoring of ocean currents are discussed in terms of the normal modes. The sensitivity of various acoustic quantities to the vertical and in-plane and out-of-plane horizontal components of the flow velocity is estimated. Despite anisotropy, there is no azimuth coupling at long-range sound propagation in a stratified moving medium. When a waveguide varies in the horizontal plane, the 3-D effects due to current-induced anisotropy and horizontal refraction are superimposed. The relative importance of these two phenomena in shallow- and deep-water scenarios is analyzed by considering horizontal (modal) rays in the presence of currents. For the purposes of numerical simulations, a parabolic approximation is considered for sound in a moving fluid. A new parabolic equation is proposed for efficient numerical modeling of the 3-D acoustic effects in the ocean with currents. [Work supported by NSERC.]

Non-Fourier propagation of heat pulses in finite medium