Horizontal Refraction Research Articles

On uniform asymptotic approximations of whispering gallery modes propagating along curved penetrable interfaces

While in the classical theory of whispering gallery waves it is assumed that they propagate along a cylindrical perfectly reflecting wall, in many real situations it is important to consider similar waves propagating along penetrable interfaces or curvilinear level sets of the refractive index. An example of waves of the latter kind are whispering gallery modes formed in a shallow-water waveguide due to the horizontal refraction. Such modes are counterpart of quasistationary states in quantum mechanics, as they do not exponentially decay at infinity, and their respective eigenvalues are complex. Recently it was shown that Maslov canonical operator (MCO) theory can be used to compute uniform asymptotic expansions of the solutions to various wave propagation problems. Such approximations admit convenient analytical representation in terms of special functions that can be used for investigating their properties and for other purposes. The MCO theory is applicable only in cases where the solution exponentially decays at infinity, e.g., for computing bound states of a given quantum system. Nevertheless, the technique outlined here allows to obtain analytical expressions for the radial profiles of whispering gallery modes propagating along a penetrable wall or along a curvilinear level set of the refractive index.

Modes selection based on holographic interferometry in presence of intense internal waves in shallow water

A holographic method is proposed for sound field modes selection and estimating their parameters (amplitude, horizontal wavenumber, group velocity, absorption coefficient) in shallow water by single scalar receiver in presence of intense internal waves (IIW). It is taken into account that IIW leads to horizontal refraction and modes coupling. The offered holographic method is based on the two-dimensional Fourier transform (2D-FT) of the real part of the moving-source sound field in time–frequency domain. It is shown in the paper that different sound field modes create different focal spots in the hologram domain in presence of IIW. The relationship of the mode phases and group velocities with the coordinates of focal-spot peaks is derived. The hologram domain allows to reconstruct the selected mode field and mode parameters in presence of IIW. The results of modes selection within framework of the numerical experiments are presented in the paper. The modal parameters are estimated. Analysis of influence of IIW on mode selection and mode parameters estimations is performed. It is shown that the effects of horizontal refraction and modes coupling do not lead to loss of the identity of mode parameters of the received signal. [This study was supported by Russian Science Foundation grant no. 22-79-10233. E.S. Kaznacheeva’s numerical experiment was supported grant no. МК-6144.2021.4]

Use of Linear Receiving Arrays for Observing Horizontal Refraction of Low-Frequency Sound in Shallow Water with a Strongly Inhomogeneous Waterlike Bottom

Use of Linear Receiving Arrays for Observing Horizontal Refraction of Low-Frequency Sound in Shallow Water with a Strongly Inhomogeneous Waterlike Bottom

Geotechnical Assessment of Dar Al Salam Complex in Hilla City, Babylon Using the Seismic Refraction

In this study, the horizontal seismic refraction survey method was used to calculate the elasticity coefficients of the soil of the Dar Al-Salaam complex site in the city of Hilla, Babil Governorate. A seismic survey was conducted along four lines to find the shear and longitudinal velocities. Two lines of length (115) meters in the direction north-south and others in the direction east-west, with a front, medium and reverse shock in each direction. The data were processed and interpreted by using the Reflex 2D Quick software. The results for the first layer were for the longitudinal velocity (Vp) (189-259 m/sec and for the second layer (342-411) m/sec and the thickness of the layer was (3.7) m. While the shear velocity of the first layer (VS) was (119-109) and the second layer (197-233.5) m/sec. The dynamic elasticity coefficients were calculated from these velocities, such as Poisson's ratio(δ) and Young’s modulus (E ), Bulk modulus (K), shear modulus (μ), The amount of the Poisson's ratio for the first layer ranged between (0.32-0.44) and its average in the first layer is 0.37 while the Poisson's ratio in the second layer ranged between (0.35-0.276) and its average in the second layer is 0.305, while the highest value of the Shear Modulus(u) was recorded in the second layer at 103.7 MPa and the average of Young’s Modulus of the first layer was recorded at 25.5 MPa, which indicated the second layer was characterized by its high hardness, strength, while the highest value of the Bulk modulus (K) was recorded in the second layer at 288.7 MPa and the average of Bulk modulus (K) of the first layer was recorded at 74.8 MPa ,while the highest value of the Young’s Modulus was recorded in the second layer at 209 MPa and the average of Young’s Modulus of the first layer was recorded at 57 MPa, which showed the improvement of engineering soil properties in the second layer in all stations of the study area.

Modeling underwater sound propagation in an arctic shelf region with an inhomogeneous bottom.

The seabed data from the Kara Sea (a part of the Arctic Shelf) are used to build a low-frequency (up to 250 Hz) acoustic waveguide model and study sound propagation in this region. A 30-m deep, well-mixed, and homogenous water layer over a flat seafloor is considered. The seabed's acoustic model is based on the spatial distribution of a sound speed recorded during a three-dimensional seismic survey in the Kara Sea, as well as density data from core sample analysis. One of the region's most distinctive features is the presence of large areas (up to several tens of square kilometers) where the bottom sound speed is close to that in water. In such a setting, the normal mode approach is applied to the sound propagation problem. The overall acoustic field is made up of propagating and leaky modes, which include quasi-modes. Numerical simulations show a high spatial variability of attenuation in the waveguide, 1.5 to 20 dB/km at 137 Hz. Even if the water depth is constant and the seafloor is uniform and smooth, mode coupling and horizontal refraction can occur owing to the bottom inhomogeneities. For higher modes, the predicted angle of refraction is up to 10°.

Acoustic spectrometry of bubbles in an estuarine front: Sound speed dispersion, void fraction, and bubble density.

Estuaries constitute a unique waveguide for acoustic propagation. The spatiotemporally varying three-dimensional front between the seawater and the outflowing freshwater during both flood and ebb constitutes an interfacial sound speed gradient capable of supporting significant vertical and horizontal acoustic refraction. The collision of these two water masses often produces breaking waves, injecting air bubbles into the water column; the negative vertical velocities of the denser saltwater often subduct bubbles to the bottom of these shallow waveguides, filling the water column with a bubbly mixture possessing a significantly lower effective sound speed. A field experiment was carried out in the mouth of Mobile Bay, Alabama in June 2021 to characterize estuarine bubble clouds in terms of their depth-dependent plume structure, frequency-dependent sound speed and attenuation, bubble size distribution, bubble number density, and void fraction. Results demonstrate that sound speed in the bubbly liquid consistently falls below the intrinsic sound speed of bubble-free water; specifically, the bubbly liquid 1.3 m below the surface in a front in this environment possesses effective sound speeds, void fractions, and bubble number densities of approximately 750 m/s, 0.001%, and 2 × 106 bubbles/m3, respectively.

Towards developing a generalized semi-coherent three-dimensional energy flux model

The energy flux method can be interpreted as an integration over propagation angle of a semi-classical (WKB) modal continuum, which yields an averaged incoherent intensity field with depth-dependent intensity bands that decay in range. Range dependence is typically incorporated via the adiabatic modes approximation and use of the ray invariant to map propagation angles. Though the energy flux method averages out nearly all of the field structure, it has the primary advantages of avoiding finding eigenvalues and eigenrays and its computational effort does not directly scale with frequency or range. The method has also been extended in the past decade to include a convergence factor derived from the interference of neighboring modes, producing a semi-coherent solution that captures some convergence structure analogous to high-frequency caustics. Since the development of the semi-coherent energy flux method, it has so far only been applied to Nx2D environmental models, but should theoretically be applicable in a 3D environment without the assumption of azimuthal symmetry or the exclusion of horizontal refraction. This paper will discuss the theoretical derivation and numerical implementation of a generalized semi-coherent three-dimensional energy flux model. [This study is funded by the NDSEG Fellowship program.]

Horizontal refraction and diffraction of underwater sound around an island.

The three-dimensional (3D) propagation effects of horizontal refraction and diffraction were measured on a tetrahedral hydrophone array deployed near the coast of Block Island, RI. Linear frequency modulated chirp signals, centered at 1 kHz with a 400 Hz bandwidth, were transmitted from a ship moving out of the acoustic shadow zone blocked by the island from the perspective of the hydrophone array. The observed shadow zone boundary was consistent with the prediction made by a 3D sound propagation model incorporating high-resolution bathymetry and realistic sound speed obtained from a data-assimilated regional ocean model. The 3D modal ray calculation provided additional insight into the frequency dependence of the signal spreading. This analysis found that the modes at higher frequencies can propagate closer to the coast of the island with shallower modal cutoff depths, where the sound energy penetrates the sloping seafloor at supercritical incidence. The evidence of horizontal caustics of the sound was shown in the parabolic equation and modal ray models by comparing to the arrival pattern observed in the data. The arrival angle measurements on the tetrahedral array show the complex propagation patterns, including the diffracted energy in the island shadow and acoustic energy refracted away from the island.

Three-dimensional coupled-mode model and characteristics of low-frequency sound propagation in ocean waveguide with seamount topography

Large-scale topography, such as a seamount, substantially impacts low-frequency sound propagation in an ocean waveguide, limiting the application of low-frequency acoustic detecting techniques. A three-dimensional (3D) coupled-mode model is developed to calculate the acoustic field in an ocean waveguide with seamount topography and analyze the 3D effect. In this model, a correction is introduced in the bottom boundary, theoretically making the acoustic field satisfy the energy conservation. Furthermore, a large azimuth angle calculation range is obtained by using the operator theory and higher-order Padé approximation. Additionally, the model has advantages related to the coupling mode and parabolic equation theory. The couplings corresponding to the effects of range-dependent environment are fully considered, and the numerical implementation is kept feasible. After verifying the accuracy and reliability of the model, low-frequency sound propagation characteristics in the seamount environment are analyzed. The results indicate lateral variability in bathymetry can lead to out-of-plane effects such as the horizontal refraction phenomenon, while the coupling effect tends to restore the abnormal sound field and produces acoustic field diffraction behind the seamount. This model effectively considers the effects of the horizontal refraction and coupling, which are proportional to the scale of the seamount.

A Study on the Estimation of Source Bearing in an ASA Wedge: Diminishing the Estimation Error Caused by Horizontal Refraction

The performance of warping transformation in diminishing the error in underwater source-bearing estimation, caused by horizontal refraction effects (also named 3D effects), is studied. First, the capability of warping transformation for separating the normal modes, as well as their direct and horizontally refracted paths, in a standard Acoustical Society of America (ASA) wedge is demonstrated. Second, the error values for source bearing estimation in three different manners, i.e., using the full signal, using its component corresponding to the first mode, and using the component of the latter corresponding only to the direct path are compared for the same wedge case. The results show that the estimation error can be significantly reduced by beamforming, with only the first mode, or using the direct path of the first mode in cases where there exists a horizontally refracted path of the first mode.

Gravity Wave Focusing on the Antarctic Polar Vortex Using Gaussian Beam Approximation in Horizontally Nonuniform Flows

AbstractRay path theory is an asymptotic approximation to the wave equations. It represents efficiently gravity wave propagation in nonuniform background flows so that it is useful to develop schemes of gravity wave effects in general circulation models. One of the main limitations of ray path theory to be applied in realistic flows is in caustics where rays intersect and the ray solution has a singularity. Gaussian beam approximation is a higher-order asymptotic ray path approximation that considers neighboring rays to the central one, and thus, it is free of the singularities produced by caustics. A previous implementation of the Gaussian beam approximation assumes a horizontally uniform flow. In this work, we extend the Gaussian beam approximation to include horizontally nonuniform flows. Under these conditions, the wave packet can undergo horizontal wave refraction producing changes in the horizontal wavenumber, which affects the ray path as well as the ray tube cross-sectional area and so the wave amplitude via wave action conservation. As an evaluation of the Gaussian beam approximation in horizontally nonuniform flows, a series of proof-of-concept experiments is conducted comparing the approximation with the linear wave solution given by the WRF Model. A very good agreement in the wave field is found. An evaluation is conducted with conditions that mimic the Antarctic polar vortex and the orography of the southern flank of South America. The Gaussian beam approximation nicely reproduces the expected asymmetry of the wave field. A much stronger disturbance propagates toward higher latitudes (polar vortex) compared to lower latitudes.

Horizontal Refraction of Acoustic Waves in Shallow-Water Waveguides Due to an Inhomogeneous Bottom Structure

Three-Dimensional (3-D) sound propagation in a shallow-water waveguide with a constant depth and inhomogeneous bottom is studied through numerical simulations. As a model of inhomogeneity, a transitional region between an acoustically soft and hard bottom is considered. Depth-averaged transmission loss simulations using the “horizontal rays and vertical modes” approach and mode parabolic equations demonstrate the horizontal refraction of sound in this region, even if the water column is considered homogeneous. The observed wave effect is prominent at low frequencies, at which the water depth does not exceed a few acoustic wavelengths. The obtained results within the simplified model are verified by the simulations for a real seabed structure in the Kara Sea.

Underwater Sound Propagation Modeling in a Complex Shallow Water Environment

Three-dimensional (3D) effects can profoundly influence underwater sound propagation in shallow-water environments, hence, affecting the underwater soundscape. Various geological features and coastal oceanographic processes can cause horizontal reflection, refraction, and diffraction of underwater sound. In this work, the ability of a parabolic equation (PE) model to simulate sound propagation in the extremely complicated shallow water environment of Long Island Sound (United States east coast) is investigated. First, the 2D and 3D versions of the PE model are compared with state-of-the-art normal mode and beam tracing models for two idealized cases representing the local environment in the Sound: (i) a 2D 50-m flat bottom and (ii) a 3D shallow water wedge. After that, the PE model is utilized to model sound propagation in three realistic local scenarios in the Sound. Frequencies of 500 and 1500 Hz are considered in all the simulations. In general, transmission loss (TL) results provided by the PE, normal mode and beam tracing models tend to agree with each other. Differences found emerge with (1) increasing the bathymetry complexity, (2) expanding the propagation range, and (3) approaching the limits of model applicability. The TL results from 3D PE simulations indicate that sound propagating along sand bars can experience significant 3D effects. Indeed, for the complex shallow bathymetry found in some areas of Long Island Sound, it is challenging for the models to track the interference effects in the sound pattern. Results emphasize that when choosing an underwater sound propagation model for practical applications in a complex shallow-water environment, a compromise will be made between the numerical model accuracy, computational time, and validity.

Three-Dimensional Sound Propagation in the South China Sea with the Presence of Seamount

Seamounts have important effects on sound propagation in deep water. A sound propagation experiment was conducted in the South China Sea in 2016. The three-dimensional (3D) effects of a seamount on sound propagation are observed in different propagation tracks. Ray methods (BELLHOP N×2D and 3D models) are used to analyze and explain the phenomena. The results show that 3D effects have obvious impacts on a sound field within a horizontal refraction zone behind the seamount because some sound beams cannot reach the receiver for the horizontal refraction effects, which impacts the sound field within a certain angle range behind the seamount. The arrival structure results show that the eigenrays after horizontal reflection will arrive at the receiver earlier than those obtained from the two-dimensional (2D) model within the horizontal refraction zone behind the seamount. This means that the horizontal reflection effect of a seamount will cause the shortening of sound propagation paths. Finally, in the reflection zone in front of the seamount, the 2D and 3D TL results show that the shape of the reflection zone is similar to an “arch” type, and the horizontal refraction of sound waves has little effect on the TLs in the reflection zone of a seamount.

Numerical modeling of the sound field interference pattern in the presence of intense internal waves

The purpose of the paper is numerical modeling of the sound field in the presence of intense internal waves to study the interference pattern in inhomogeneous ocean waveguide. It is assumed that intense internal waves are the reason for the sound field modes coupling and horizontal refraction. The results of sound field modeling are processed to obtain acoustic intensity distributions in the frequency-time domain (interferogram) for unmoving sources. The two-dimensional Fourier transform (2D-FT) is applied to analyze the interferogram. The result of the 2D-FT can be called the Fourier-hologram (hologram). The hologram allows us to coherently accumulate the sound intensity of the interferogram in the narrow area as focal spots. It is demonstrated in the paper that the position of focal spots in the hologram depends on the distance between the source and the receiver. The relationship between the interferogram and hologram structures and the unperturbed and scattered fields parameters is obtained in the paper. The hologram spectral density consists of the two disjoint regions corresponding to the scattered and scattered fields. The filtering of these areas allows us to transmit the non-distorted information through an inhomogeneous ocean environment. The numerical simulation results of interferograms and holograms in the presence of the internal waves are considered. [This research was supported by the RFBR (19-29-06075, 19-38-90326).]

Atmospheric parabolic equation with Galerkin discretization and boundary fitted grid for modeling infrasound propagation over topography

Studies utilizing the parabolic equation (PE) to model infrasound propagation over topography have traditionally implemented them as 2D or Nx2D PE models. These methods, to one extent or another, trade-off accuracy for enhanced computational speed. Three-dimensional parabolic equation (3DPE) models are seldom seen in atmospheric acoustic propagation studies compared to their 2D and Nx2D counterparts, despite the fact that theta coupling and proper modeling of horizontal refraction and diffraction in a 3DPE model should enable greater accuracy. In this work a 3DPE in cylindrical coordinates is developed under the alternating direction implicit scheme, with a Galerkin discretization and boundary fitted grid, such that the PE is capable of handling arbitrary and irregular topography features. Finally, 2D, Nx2D, and 3D implementations of the aforementioned PE model are examined for cases of propagation over a simple hill in homogeneous, downward refracting, and upward refracting atmospheres. Insights are drawn with regards to the accuracy at a variety of positions and ranges, and the computational speeds of the various methods.

Characteristics of Three-Dimensional Sound Propagation in Western North Pacific Fronts

Oceanic fronts involved by ocean currents led to strong gradients of temperature, density and salinity, which have significant effects on underwater sound propagation. This paper focuses on the impact of the oceanic front on three-dimensional underwater sound propagation. A joint experiment of ocean acoustic and physical oceanography at the western North Pacific fronts is introduced. The measurement data for sound waves passed through the oceanic front is processed. The results are analysed and compared with the numerical simulation. It was found that transmission loss presented some difference when the source was located in the front centre and sound waves propagated towards water mass on opposite sides of the front centre. And when the sound field is excited by the underwater explosion at a depth of 200 m, the effects of the horizontal refraction cannot be neglected. On the other hand, the transmission loss for sound pressure fell sharply and rose rapidly at the side of cold water masses.

Phased Array Analysis Incorporating the Continuous Wavelet Transform

ABSTRACT Nonlinear block thresholding of the continuous wavelet transform (CWT) of 2D phased array signals offers high time-resolution solutions for analyzing seismograms of local and regional seismic events. An initial denoising step on an array ensemble reveals the regions of the scale-time plane that contain high signal-to-noise arrivals. Individual seismic phase arrivals in ensemble, denoised seismograms can be partitioned using scale-time gating in which CWT wave packets of an individual seismic phase on the scale-time plane for a reference array element are time-correlated with all other elements to find an optimum time shift for the phase across all elements. The seismic phase is then clipped out of the CWT of each array element using this optimal time shift for further analysis. The seismogram can be separated into component seismic waves for a detailed view of wave characteristics such as slowness and arrival azimuth using conventional frequency–wavenumber methods. However, the process can be taken further using the CWT of each phase to construct high time-resolution signal beams over the CWT scale. Local explosion data from the 2016 Incorporated Research Institutions for Seismology (IRIS) Wavefields Community Experiment in northern Oklahoma are used to demonstrate these techniques in separating surface-wave modes and body waves and to examine the scattering of regional phases. High-resolution CWT processing in the 1–3 Hz band for northern Oklahoma reveals horizontal Rayleigh-wave refraction and multipathing. Rayleigh-wave phase-velocity dispersion measurements are used to construct a 1D velocity model under the IRIS experiment. Scale-time gating helps expose near-source surface reflections from a local M 3.6 earthquake that are used to verify the 4 km source depth obtained from a published regional moment tensor solution.

Application of GIS on Research of Horizontal Refraction in Polygonometry on Network

The article examines the influence of refraction on the results of measuring the horizontal angle in special places. In the conditions of the hot and arid climate of Uzbekistan in the polygonometric network of river banks, according to the results of field experiments, the new version of the developed methodology of the GIS makes it possible to increase the accuracy of measuring horizontal angles and lengths of lines. Significant temperature changes along the water and the coast, on rocks, in the upper layers of sand-concrete pavements are also explained by the properties of the heat and radiation balance.

On Adiabatic Sound Propagation in a Shallow Sea with a Circular Underwater Canyon

An idealized problem of sound propagation in a shallow sea with a curved underwater canyon is considered. The canyon is a circular feature of the bottom relief which causes horizontal refraction of acoustic waves. The sound field computation method for a waveguide with an inhomogeneity of this type is considered, and the geometry of horizontal rays trapped by the canyon is studied. The computation method is based on the separation of variables in the equation for the mode amplitudes and representation of the solution in the form of a double series. This technique is rather general in nature, because it can be used to compute the field in any problem where the waveguide has rotational symmetry. Furthermore, it is also shown which terms of the series correspond to horizontal rays trapped by the canyon.