Normal Mode Solutions Research Articles

Normal Modes Computation by Generalized Reflection‐Transmission Coefficients Method in Planar Layered Half Space

AbstractA new approach is presented to calculate the dispersion curves and normal mode solutions of multilayered solid earth model with low velocity layers based on the generalized reflection‐transmission coefficient method. It is efficient to tackle these two problems by utilizing the single top‐layer secular function proposed by Chen in 1993. However, it is difficult to obtain the lower modes by merely using the single top‐layer secular function due to numerical singularity while the earth model contains low velocity layers. In this paper, we proposed using the secular function set, instead of the single secular function, to solve this problem. With secular function set, different modes can be calculated respectively through corresponding secular function. Numerical experiments show that the new approach is efficient, stable and reliable, thus it makes the generalized reflection‐transmission coefficients method applicable to computation of normal mode in various stratified earth models.

Acoustic waves in a medium bounded by curved surfaces

A method for obtaining quasi-analytic solutions to the three-dimensional (3D) Helmholtz equation for the case of an acoustic medium bounded by two identical curved surfaces is presented. The method can be extended to a semi-infinite medium with a curved boundary for the study of Rayleigh waves on a non-planar surface, albeit the solution procedure entails the numerical matrix method. The formulation of the method is based on the differential-geometry argument employing the curved coordinates ( u 1 , u 2 , u 3 ) where u 1 and u 2 are along the local tangent plane of one of the bounding surfaces z = z ( x ) and u 3 is perpendicular to the local tangent plane. This choice of coordinates allows the 3D Helmholtz equation, subject to boundary conditions specified on non-planar surfaces, to be solved with relative ease. Normal-mode solutions are shown for the case of a fluid layer with two pressure-release boundaries, where the bounding surfaces are given by the ramp, the Gaussian, and the sinusoidal functions, respectively.

Pressure corrections for potential flow analysis of capillary instability of viscous fluids

Funada & Joseph (Intl J. Multiphase Flow, vol. 28, 2002, p. 1459) analysed capillary instability assuming that the flow is irrotational but the fluids are viscous (viscous potential flow, VPF). They compared their results with the exact normal-mode solution of the linearized Navier–Stokes equations (fully viscous flow, FVF) and with the irrotational flow of inviscid fluids (inviscid potential flow, IPF). They showed that the growth rates computed by VPF are close to the exact solution when Reynolds number is larger than and are always more accurate than those computed using IPF. Recently, Joseph & Wang (J. Fluid Mech., vol. 505, 2004, p. 365) presented a method for computing a viscous correction of the irrotational pressure induced by the discrepancy between non-zero irrotational shear stress and the zero-shear-stress boundary condition at a free surface. The irrotational flow with a corrected pressure is called the viscous correction of VPF (VCVPF). Here we compute the pressure correction for capillary instability in cases in which one fluid is viscous and the other fluid is a gas of negligible density and viscosity. The growth rates computed using VCVPF are in remarkably good agreement with the exact solution FVF.

Acoustic radiation by a submerged source in shallow water—Influence of the boundaries

The acoustic radiation and scattering characteristics of an acoustic source in an unbounded deep water environment are different from those in a shallow water environment, where one has to take into account the influence of the boundaries on the source radiation impedance. In this paper the influence of the waveguide (surface and bottom of the ocean in a shallow water environment) on the radiation from an elastic noise source is investigated by means of a superposition method. The radiated sound is initially determined from the scattering of an incident wave from the elastic target. Using the superposition method [A. Sarkissian, J. Acoust. Soc. Am. (1994)], the scattered field, including multiple scattering effects, is computed and represented by a series of point sources positioned inside the scatterer. A number of calibration points located at the surface of the scatterer are selected to determine the source strength of each point source using a least square approximation. Having obtained the source strength of each point source, the combined propagation field in the waveguide from all point sources and the influence back on the source are determined using normal mode or ray tracing solutions common in underwater sound propagation. [Work supported by ONR.]

The counter‐propagating Rossby‐wave perspective on baroclinic instability. II: Application to the Charney model

AbstractThe constant‐density Charney model describes the simplest unstable basic state with a planetary‐vorticity gradient, which is uniform and positive, and baroclinicity that is manifest as a negative contribution to the potential‐vorticity (PV) gradient at the ground and positive vertical wind shear. Together, these ingredients satisfy the necessary conditions for baroclinic instability. In Part I it was shown how baroclinic growth on a general zonal basic state can be viewed as the interaction of pairs of ‘counter‐propagating Rossby waves’ (CRWs) that can be constructed from a growing normal mode and its decaying complex conjugate. In this paper the normal‐mode solutions for the Charney model are studied from the CRW perspective.Clear parallels can be drawn between the most unstable modes of the Charney model and the Eady model, in which the CRWs can be derived independently of the normal modes. However, the dispersion curves for the two models are very different; the Eady model has a short‐wave cut‐off, while the Charney model is unstable at short wavelengths. Beyond its maximum growth rate the Charney model has a neutral point at finite wavelength (r=1). Thereafter follows a succession of unstable branches, each with weaker growth than the last, separated by neutral points at integer r—the so‐called ‘Green branches’. A separate branch of westward‐propagating neutral modes also originates from each neutral point. By approximating the lower CRW as a Rossby edge wave and the upper CRW structure as a single PV peak with a spread proportional to the Rossby scale height, the main features of the ‘Charney branch’ (0<r<1) can be deduced without prior calculation of the normal modes. Furthermore, CRWs from this branch are seen to make a smooth transition into the boundary and interior PV structure of the neutral modes appearing at r=1. The behaviour of the other branches and neutral points is essentially the same when viewed from the CRW perspective, but with cancelling interior PV structures reducing the self and mutual interaction of the CRWs. The underlying dynamics determining the nature of all the solutions is the difference in the scale‐dependence of PV inversion for boundary and interior PV anomalies, the Rossby‐wave propagation mechanism and the CRW interaction. The behaviour of the Charney modes and the first neutral branch, which rely on tropospheric PV gradients, are arguably more applicable to the atmosphere than modes of the Eady model where the positive PV gradient exists only at the tropopause. Copyright © 2004 Royal Meteorological Society

The Effect of Viscosity on the Stability of Planar Vortices with Fine Structure

We examine the viscous response of sharp‐edged vortices surrounded by fine structure. Such compact vortices admit normal mode solutions, and when weak vorticity is superimposed around the edge of the vortex a quasi‐mode appears. A critical layer is formed at the location at which the angular velocity of the fluid matches the frequency of the normal mode. Fine‐scale vorticity within this layer, combined with the presence of viscosity, can affect the stability of the quasi‐mode in a number of ways. Here we discuss briefly one situation for which viscosity can have a destabilizing effect.

Coupling spectral elements and modes in a spherical Earth: an extension to the ‘sandwich’ case

We present an extension to the coupling scheme of the spectral element method (SEM) with a normal-mode solution in spherical geometry. This extension allows us to consider a thin spherical shell of spectral elements between two modal solutions above and below. The SEM is based on a high-order variational formulation in space and a second-order explicit scheme in time. It combines the geometrical flexibility of the classical finite-element method with the exponential convergence rate associated with spectral techniques. In the inner sphere and outer shell, the solution is sought in terms of a modal solution in the frequency domain after expansion on the spherical harmonics basis. The SEM has been shown to obtain excellent accuracy in solving the wave equation in complex media but is still numerically expensive for the whole Earth for high-frequency simulations. On the other hand, modal solutions are well known and numerically cheap in spherically symmetric models. By combining these two methods we take advantage of both, allowing high-frequency simulations in global Earth models with 3-D structure in a limited depth range. Within the spectral element method, the coupling is introduced via a dynamic interface operator, a Dirichlet-to-Neumann operator which can be explicitly constructed in the frequency and generalized spherical harmonics domain using modal solutions in the inner sphere and outer shell. The presence of the source and receivers in the top modal solution shell requires some special treatment. The accuracy of the method is checked against the mode summation method in simple spherically symmetric models and shows very good agreement for all type of waves, including diffracted waves travelling on the coupling boundary. A first simulation in a 3-D D″-layer model based on the tomographic model SAW24b16 is presented up to a corner frequency of 1/12 s. The comparison with data shows surprisingly good results for the 3-D model even when the observed waveform amplitudes differ significantly from those predicted in the spherically symmetric reference model (PREM).

Viscoelastic potential flow analysis of capillary instability

Analysis of the linear theory of capillary instability of threads of Maxwell fluids of diameter D is carried out for the unapproximated normal mode solution and for a solution based on viscoelastic potential flow. The analysis here extends the analysis of viscous potential flow [Int. J. Multiphase Flow 28 (2002) 1459] to viscoelastic fluids of Maxwell type. The analysis is framed in dimensionless variables with a velocity scale based on the natural collapse velocity V= γ/ μ (surface tension/liquid viscosity). The collapse is controlled by two dimensionless parameters, a Reynolds number J= VDρ/ μ= ργD/ μ 2=( Oh) 2 where Oh is the Ohnesorge number, and a Deborah number Λ 1= λ 1 V/ D where λ 1 is the relaxation time. The density ratio ρ a/ ρ and μ a/ μ are nearly zero and do not have a significant effect on growth rates. The dispersion relation for viscoelastic potential flow is cubic in the growth rate σ and it can be solved explicitly and computed without restrictions on the Deborah number. On the other hand, the iterative procedure used to solve the dispersion relation for fully viscoelastic flow fails to converge at very high Deborah number. The growth rates in both theories increase with Deborah number at each fixed Reynolds number, and all theories collapse to inviscid potential flow (IPF) for any fixed Deborah number as the Reynolds number tends to infinity.

An analytical study of linear and non-linear convection in Boussinesq–Stokes suspensions

The Rayleigh–Benard situation in Boussinesq–Stokes suspensions is investigated using both linear and non-linear stability analyses. The linear and non-linear analyses are based on a normal mode solution and minimal representation of double Fourier series, respectively. The effect of suspended particles on convection is delineated against the background of the results of the clean fluid. The realm of non-linear convection warrants the quantification of heat transfer and this has been achieved on the Rayleigh–Nusselt plane. Possibility of aperiodic convection is discussed.

Resonance radiation of submerged infinite cylindrical shell

The resonance sound radiation from submerged infinite elastic cylindrical shell, excited by internal harmonic line force, is investigated. The shell radiation power is presented in terms of resonant modal radiation derived from resonance radiation theory (RRT). The resonance radiation formulae are derived from classical Rayleigh normal mode solution, which are useful for understanding the mechanism of sound radiation from submerged shells. As an example, numerical calculation of a thin steel cylindrical shell is done by using these two methods. It seems that the results of RRT solutions are in good agreement with that of Rayleigh normal mode solutions.

Modal mapping in range and azimuth for shallow-water waveguides

It is well established that the normal mode solution for sound propagation in shallow water is a function of the local waveguide properties. Much work has been done on the inverse problem where the local modes are measured and used as input data for geoacoustic inversion methods. In the range-dependent case, short sliding-window transform methods based on the Hankel transform have been implemented to extract the corresponding range-varying wave number spectra. A major objective of this work is to extend the range-dependent work to fully three-dimensionally varying environments. Recently, experiments were conducted using cw sources operating in the frequency range 20–475 Hz. The acoustic field was measured on an array of freely drifting hydrophone buoys equipped with GPS navigation. Using the complex pressure measured on these synthetic aperture horizontal arrays, local mode information is extracted as a function of range and azimuth from the source. Emphasis is placed on modal variability for tracks both along and across bathymetric contours at several frequencies. [Work supported by ONR.]

Issues of analytical continuation of the field interior to the target surface and a more general T-matrix for submerged fluid targets

The T-matrix or extended boundary condition method developed by P. Waterman for wave scattering from classical targets and extended by a host of researchers is a marvelous generalization of normal mode solutions allowed for separable geometries to extend to nonseparable geometries by means of mode coupling. The method is based on Huygens’ principle and partial wave expansions of physical quantities. In general the expansions don’t obey orthogonality and one must take care in the way expansions are made. One must never take expansions of a quantity and then represent expansions of their differential forms in terms of the partial wave differential forms using the same expansion coefficients lest one violate a mathematical principle (the Rayleigh hypothesis). One may circumvent any issues without resort to analytical continuation of interior solutions by including all solutions subject to interior and exterior Greens functions. This leads to matrix relations (constraining matrices) between the expansion terms which when used lead to new forms of the fluid T-matrix, for example. The new form differs from Waterman’s T-matrix for the case when the constraining matrices are not symmetric. Otherwise the original expression by Waterman is valid. These issues are discussed and examples are presented.

Energy-Conserving and Reciprocal Solutions for Higher-Order Parabolic Equations

The energy conservation law and the flow reversal theorem are valid for underwater acoustic fields. In media at rest the theorem transforms into well-known reciprocity principle. The presented parabolic equation (PE) model strictly preserves these important physical properties in the numerical solution. The new PE is obtained from the one-way wave equation by Godin12 via Pade approximation of the square root operator and generalized to the case of moving media. The PE is range-dependent and explicitly includes range derivatives of the medium parameters. Implicit finite difference scheme solves the PE written in terms of energy flux. Such formalism inherently provides simple and exact energy-conserving boundary condition at vertical interfaces. The finite-difference operators, the discreet boundary conditions, and the self-starter are derived by discretization of the differential PE. Discreet energy conservation and flow reversal theorem are rigorously proved as mathematical properties of the finite-difference scheme and confirmed by numerical modeling. Numerical solution is shown to be reciprocal with accuracy of 10–12 decimal digits, which is the accuracy of round-off errors. Energy conservation and wide-angle capabilities of the model are illustrated by comparison with two-way normal mode solutions including the ASA benchmark wedge.

ENERGY-CONSERVING AND RECIPROCAL SOLUTIONS FOR HIGHER-ORDER PARABOLIC EQUATIONS

The energy conservation law and the flow reversal theorem are valid for underwater acoustic fields. In media at rest the theorem transforms into well-known reciprocity principle. The presented parabolic equation (PE) model strictly preserves these important physical properties in the numerical solution. The new PE is obtained from the one-way wave equation by Godin12 via Padé approximation of the square root operator and generalized to the case of moving media. The PE is range-dependent and explicitly includes range derivatives of the medium parameters. Implicit finite difference scheme solves the PE written in terms of energy flux. Such formalism inherently provides simple and exact energy-conserving boundary condition at vertical interfaces. The finite-difference operators, the discreet boundary conditions, and the self-starter are derived by discretization of the differential PE. Discreet energy conservation and flow reversal theorem are rigorously proved as mathematical properties of the finite-difference scheme and confirmed by numerical modeling. Numerical solution is shown to be reciprocal with accuracy of 10–12 decimal digits, which is the accuracy of round-off errors. Energy conservation and wide-angle capabilities of the model are illustrated by comparison with two-way normal mode solutions including the ASA benchmark wedge.

MOTION OF A MOVING ELASTIC BEAM CARRYING A MOVING MASS—ANALYSIS AND EXPERIMENTAL VERIFICATION

In this paper, vibrational motion of an elastic beam fixed on a moving cart and carrying a moving mass is investigated. The equations of motion of the beam–mass–cart system are derived and the coupled dynamic equations are solved by the unconstrained modal analysis. In modal analysis, the exact normal mode solutions corresponding to the eigenfrequencies for each position of the moving mass and the ratios of the weight of the beam–mass–cart system are used. Proper transformation of time solutions between the normal modes for a position and those for the next position of the moving mass is also considered. Numerical simulations are carried out to obtain open-loop responses of the system in tracking pre-designed paths of the moving mass. The simulation results show that the model predicts the dynamic behavior of the beam–mass–cart system well. Experiments are carried out to show the validity of the proposed analytical method.

Propagation Model Accuracy for MFP

Journal of Computational AcousticsVol. 08, No. 03, pp. 389-399 (2000) No AccessPROPAGATION MODEL ACCURACY FOR MFPA. TOLSTOYA. TOLSTOYATolstoy Sciences, 8610 Battailles Ct., Annandale VA 22003, USA Search for more papers by this author https://doi.org/10.1142/S0218396X00000315Cited by:3 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractThe selection of a propagation model for use in many underwater acoustic applications has been understood to be highly important for quite some time. However, it has not previously been understood how various models might actually degrade Matched Field Processing (MFP) performance even when input parameters are known exactly. That is, acoustic propagation models have not previously been benchmarked within the context of MFP where acoustic amplitudes and phases need to be highly accurate depending on the nature of the processor of interest, such as for a high resolution Capon processor. This paper discusses the SCOOTER, ORCA, and KRAKEN models, and a high angle PE model within the context of MFP. These models are compared to the SAFARI data generated for the Workshop97 geoacoustic inversion CAL case. In general it seems that all the abovementioned models show excellent accuracy for use with the Linear Processor. However, it is found that KRAKEN may experience unexpected difficulties at unpredictable frequencies where these errors can affect high resolution processing (as demonstrated by geoacoustic inversions via the RIGS method),14 that SCOOTER and PE require relatively long CPU times for high accuracy, and that how accurately parameters are interpolated, e.g., sound-speed profiles, can also be important. It should be noted that all of the models discussed in this work may be considered to be very accurate in the context of most applications, particularly those involving data. FiguresReferencesRelatedDetailsCited By 3Developing a Method for Experimental Studies of Crustal Structure in Marine Areas in Different SeasonsG. I. Dolgikh, S. S. Budrin, S. G. Dolgikh, A. A. Pivovarov and A. N. Samchenko et al.7 August 2019 | Seismic Instruments, Vol. 55, No. 4ANALYTICAL TIME DOMAIN NORMAL MODE SOLUTION OF AN ACOUSTIC WAVEGUIDE WITH PERFECTLY REFLECTING WALLSHÜSEYIN ÖZKAN SERTLEK and SERKAN AKSOY29 April 2013 | Journal of Computational Acoustics, Vol. 21, No. 02WHAT ABOUT ADIABATIC NORMAL MODES?A. TOLSTOY21 November 2011 | Journal of Computational Acoustics, Vol. 09, No. 01 Recommended Vol. 08, No. 03 Metrics History Received 22 July 1999 Revised 21 September 1999 PDF download

Coupled mode perturbation theory of range dependence

The conventional coupled mode solution is combined with perturbation theory to give a fast, accurate range-dependent normal mode solution for deep water acoustic propagation. Perturbation theory is used to calculate the new normal modes at each range step. The new modes are obtained as a linear combination of the modes for the previous step without requiring a numerical solution of the depth-separated wave equation. The process may be repeated for many steps and yields normal modes and eigenvalues which are sufficiently accurate for solution of practical problems in deep water. The method is applied to long-range propagation through oceanic fronts.

Three-dimensional acoustic scattering from a penetrable layered cylindrical obstacle in a horizontally stratified ocean waveguide

In this work, a normal-mode solution is presented for the three-dimensional problem of acoustic scattering from a penetrable horizontally layered cylindrical obstacle in a shallow-water waveguide. The ocean environment around the obstacle is considered horizontally stratified and the bottom is assumed to be rigid. In the general case of depth-dependent sound-speed and density profiles (ssdp) the total acoustic field is calculated numerically, while an analytic solution is obtained in the special case of depth-independent ssdp. Numerical results concerning the transmission loss outside and inside single or double-layered cylindrical structures made of acoustic materials are given for a typical depth-dependent ocean environment. Comparisons with the cases of ideally soft and ideally hard cylindrical obstacles [J. Acoust. Soc. Am. 100, 206-218 (1996)] are also made, illustrating the effect of acoustic properties of the obstacle. An important feature, which clearly emerges from the theoretical analysis and the numerical results, is the necessity of including evanescent modes in the calculations, in order to obtain physically meaningful and numerically accurate results. Furthermore, analytical expressions for the scattering cross section of a (penetrable or impenetrable) cylindrical obstacle are derived in terms of the expansion coefficients of the pressure field, and their behavior as frequency increases is numerically investigated. The solution presented in this paper, although addressing a special geometry, provides a means for handling strong discontinuities in both vertical and horizontal directions, and can serve as a benchmark solution to a problem for which no general numerical model exists, i.e., modeling acoustic scattering from a 3D obstacle in a 3D shallow-water waveguide.

Love waves in multilayered media with irregular interfaces: I. Modal solutions and excitation formulation

Abstract In this article, we present a new formulation of Love waves in arbitrarily irregular multilayered media by using the global generalized reflection/transmission matrices method (abbreviated to GGRTM; Chen, 1990, 1995, 1996). From the basic principle that the modal solutions are the nontrivial solutions of the free elastodynamic equation under appropriate boundary conditions, we naturally derived the characteristic frequencies and the corresponding distorted modes of Love wave in irregular multilayered media. The basic principle used here for defining modal solutions is a general one. It is identical to that used for defining normal mode solutions in laterally homogeneous-layered media (Chen, 1993) and that used for determining the resonant modes in finite-scatters case (e.g., sedimentary basin structure; see Zhou and Dravinski, 1994) and is independent of any particular mathematical technique. We found that the distorted mode in 2D structure is nonseparable in (x, z) coordinates, that is, wn,v(x,z)=∑m{a(z,ωn,v)}meikmx; whereas the normal mode for 1D structure is separable in (x,z) coordinates: un,v(x,z) = l(z,ωn,v)eiknx. Based on the formulation of GGRTM and the modal solutions, we also analytically derive the excitation formula of Love waves in irregular multilayered media, that is, the formulation of synthetic Love waves due to an arbitrarily seismic point source in such lateral heterogeneous media. We found that the synthetic Love wave in time domain can be expressed as a superposition of a series of distorted modes that is similar to the excitation formula of classic Love waves. Since the structure model used here is quite arbitrary, the new formulation of Love waves derived in this article can be applied to study a variety of seismological problems ranging from resonated motion in a sedimentary basin structure to excitation of Love waves in irregular multilayered media. It offers an alternative mean to understand the nature of Love waves in laterally heterogeneous media.

Sound propagation over concave surfaces

Diffraction of sound by concave surfaces is investigated theoretically and experimentally. In an earlier study [J. Acoust. Soc. Am. 104, 2683–2691 (1998)], it has been demonstrated that a rigorous analogy exists for the sound field above a convex circular cylinder in an otherwise homogeneous medium. The predicted sound field corresponds to the situation where the sound speed of the medium decreases exponentially with height. Extending the previous work, this paper investigates of the sound field above a concave surface and explores the corresponding analogy. Normal mode solutions have been developed for a downward refracting medium with an exponential sound speed profile. The solutions are used to predict the sound fields diffracted by a cylindrical concave surface. A series of laboratory experiments is conducted using point monopole, horizontal dipole, and vertical dipole sources over cylindrical concave surfaces. The experimental measurements are compared with the normal mode predictions. For monopole and horizontal dipole sources, good agreement has been found between measurements and the normal mode predictions using an exponential profile. However, the agreement is less satisfactory where the sound field was due to vertical dipole sources.