Normal Mode Method Research Articles

A new method for the computation of global viscoelastic post-seismic deformation in a realistic earth model (II)-horizontal displacement

SUMMARY Post-seismic deformation caused by the 2004 Sumatra-Andaman Earthquake (M= 9.3) has been observed by space geodetic techniques such as the Global Positioning System (GPS) and the Gravity Recovery and Climate Experiment (GRACE). To estimate such deformation with a spatial scale exceeding 100 km, we can use theories of global post-seismic deformation in which the Earth is treated as a self-gravitating viscoelastic sphere. Previous theories have imposed limitations on employed earth models, such as neglecting compressibility and time variation of self-gravitation or simplifying the radial profile of the viscosity. This is because the normal mode method on which these theories are based cannot evaluate a denumerably infinite set of eigenmodes that will appear when those limitations are removed in the practical computation. To bypass this difficulty, in our previous paper, we showed that the numerical inverse Laplace integration with a rectangular path enabled us to obtain the vertical deformation without using those approximations. In this paper, we apply the same method to the horizontal deformation for a complete set of the fault mechanisms (strike-slip, dip-slip and vertical and horizontal tensile). Next, we reveal error sources in our approach. The largest error arises in different selections of the integration path. This error is less than 1 per cent in the lower-degree deformations (n∼ 10) and 0.1–0.5 per cent in the higher-degree ones (100 < n < 1000), relative to the result obtained with an independent method. Finally, we elucidate differences in far-field deformations obtained for the earth model employed in this study and those used in the previous studies for a large dip-slip event (Mw= 8.3). The result shows that (1) differences of 6–50 per cent are seen in the coseismic offset and (2) horizontal rate differences of 1–5 mm yr−1 (5–25 per cent of the rates) are observed at the epicentral distance of 100 km during the first thirty years after the event. These differences are detectable with GPS. Therefore, for theory to meet observational accuracy, effects of compressibility and stratification on global post-seismic deformation should be considered.

Rayleigh-Taylor instability of an interface in a nonwettable porous medium

The diffusion of vapor through the roof of an underground structure located beneath an aquifer is considered. In the process of evaporation, an interface between the upper water-saturated layer and the lower layer containing an air-vapor mixture is formed. A mathematical model of the evaporation process is proposed and a solution of the steady-state problem is found. It is shown that in the presence of capillary forces in the case of a nonwettable medium the solution is not unique. Using the normal mode method, it is shown that Rayleigh-Taylor instability of the interface can develop in the nonwettable porous medium. It is found that there are two scenarios of loss of stability corresponding to the occurrence of the most unstable wavenumber at zero and at infinity, respectively. It is shown that for zero wavenumber the stability limit is reached at the same time as the solution of the steady-state problem disappears.

Perturbation dynamics in unsteady pipe flows

This paper deals with perturbed unsteady laminar flows in a pipe. Three types of flows are considered: a flow accelerated from rest; a flow in a pipe generated by the controlled motion of a piston; and a water hammer flow where the transient is generated by the instantaneous closure of a valve. Methods of linear stability theory are used to analyse the behaviour of small perturbations in the flow. Since the base flow is unsteady, the linearized problem is formulated as an initial-value problem. This allows us to consider arbitrary initial conditions and describe both short-time and long-time evolution of the flow. The role of initial conditions on short-time transients is investigated. It is shown that the phenomenon of transient growth is not associated with a certain type of initial conditions. Perturbation dynamics is also studied for long times. In addition, optimal perturbations, i.e. initial perturbations that maximize the energy growth, are determined for all three types of flow discussed. Despite the fact that these optimal perturbations, most probably, will not occur in practice, they do provide an upper bound for energy growth and can be used as a point of reference. Results of numerical simulation are compared with previous experimental data. The comparison with data for accelerated flows shows that the instability cannot be explained by long-time asymptotics. In particular, the method of normal modes applied with the quasi-steady assumption will fail to predict the flow instability. In contrast, the transient growth mechanism may be used to explain transition since experimental transition time is found to be in the interval where the energy of perturbation experiences substantial growth. Instability of rapidly decelerated flows is found to be associated with asymptotic growth mechanism. Energy growth of perturbations is used in an attempt to explain previous experimental results. Numerical results show satisfactory agreement with the experimental features such as the wavelength of the most unstable mode and the structure of the most unstable disturbance. The validity of the quasi-steady assumption for stability studies of unsteady non-periodic laminar flows is discussed.

A comparison of experimental and theoretical forward scattering by the Kermit‐Roosevelt Seamount

The SPICEX‐LOAPEX‐BASSEX experiments measured forward scattering from the Kermit‐Roosevelt seamounts. LFM and M‐sequence signals from 250‐Hz moored sources and a 75‐Hz ship‐deployed source were received with Penn State University’s 162‐element FORA array. Receptions were obtained from these sources at ranges of hundreds to 1600 km. Previous work [J. Acoust. Soc. Am. 117, (2549)] discussed the BASSEX forward shadow and patterns of horizontal refraction around the seamount. Additional work [J. Acoust. Soc. Am. 118, (1936)] compared experimental results with theory using a conical section model of a seamount. This presentation concerns the pulse compressed and array processed time series of M‐sequence signals in the forward scattered field of the seamounts. Both travel times and angles of arrival are reconciled with 2‐D ray‐trace models. Results are comparable to refracted rays, which mainly propagate over the seamount. These, however, do not explain additional echoes, which appear in the experiment, possibly due to reflection and/or diffraction. Normal mode and PE methods are used to confirm ray models and identify additional echoes.

Interpreting Correlated Motions Using Normal Mode Analysis

With the increased popularity of normal mode analyses in structural biology, it is important to carefully consider how to best utilize the results for gaining biological insights without over interpretation. The discussion in this article argues that for the purpose of identifying correlated motions in biomolecules, a case separate from concomitant conformational changes of structural motifs, it is generally important to use a large number of normal modes. This is illustrated through three increasingly complex examples. The simplest case includes two bilinearly coupled harmonic oscillators and serves as a straightforward problem where the important considerations are explicit and transparent. The argument is then generalized to include a system of N-coupled harmonic oscillators and finally to a realistic biomolecule. Although a small number of normal modes are useful for probing structural flexibility, it is clear that a much larger number of modes are required for properly investigating correlated motions in biomolecules.

Variation in Cutoff Effect and Sound Field Caused by Geometrical Structures near the Coast

The mysterious mass stranding of whales often arises in specific areas in the world. Moreover, vibrational noise, suspected to originate from large-scale construction in coastal areas, is problem. These problems could be considered to be related to sound propagation near coastal areas with a special geometrical structure and an ambient noise environment. We set up a propagation model in a coastal area, and simulated low-frequency sound propagation near the coast using the parabolic equation method (PE method) and the normal mode method. It has been shown that low-frequency sound cannot propagate from the ocean to the coast owing to the cutoff effect of shallow water that occurs when there is a water layer with a basement half space. However, when the water layer has a sediment layer, mode coupling occurs between the water layer and the sediment layer and propagation attenuation rapidly decreases. Sound can propagate to the coast in this case.

Regular wave impact onto an elastic plate

A computational analysis of an elastic plate dropped against regular long waves is presented. The problem is considered within the linear potential-flow theory. The liquid flow is two-dimensional and the plate is modelled by an Euler beam. The analysis is based on the normal-mode method with hydroelastic behavior of the plate being of main interest. Different impact conditions are considered to study the dependence of the total energy of the plate–liquid system on impact geometry and plate properties. The contributions of kinetic and potential energies to the total energy are analyzed. It is shown that the kinetic part of the system energy is small at the instant of time when bending stresses in the beam approach their maximum values. Estimations of both the total energy and the maximum of bending stresses are presented. Most of the calculations are performed for the conditions of experiments carried out in MARINTEK. A range of the problem parameters is also considered, to reveal peculiarities of the unsteady interaction between a falling elastic plate and surface waves.

Time-Reversal Communication with Moving Source-Receiver in Shallow Water

The applications of time-reversal waves to underwater acoustic technologies have been researched by the authors, particularly for long horizontal communication with autonomous underwater vehicle (AUV). In this study, the influence of a moving source-receiver to time-reversal communication is investigated because AUV is desired to maintain its speed. The simulations based on the normal mode method including such effect are executed. As for the results, it is clarified that the convergency of time reversal is not so affected and that the proposed method of combining time reversal and an adaptive filter can compensate the influences of the moving source-receiver adequately whose speed is not so high.

Scattering of a Small Angle Incident Acoustic Wave from a Submerged Piezoelectric Cylinder

This paper is concerned with developing a mathematical model to describe the scattering of a plane wave obliquely incident on a piezoelectric hexagonal 6mm cylinder in a fluid. The mode is based on the normal-mode expansion method. Far-field form functions are obtained. The sensitivities of Rayleigh, Whispering Gallery, and guided wave resonances to perturbations in elastic and piezoelectric constants are discussed. The resonance scattering theory can be used for characterizing the sample in both transverse and axial directions.

Using the normal-mode method of probing the infrasonic propagation for purposes of the comprehensive nuclear-test-ban treaty

We examine the problem of assessing the state of Atmospheric Acoustic Channels, that is, of the possible propagation paths of acoustic signals, based on using a priori information about atmospheric conditions. It is concluded that this can only be best accomplished through the use of global atmospheric models. Based on the normal-mode method, an analysis is made of the generalized characteristics of atmospheric waveguide such as the location of the waveguide boundaries, the mode composition, and the transmissivity of the waveguide upper and lower boundaries. The method can be used to analyze the particular paths as well as the overall situation around a given point. Furthermore, all the surrounding space (extending as far as the whole of the geosphere) is divided into regions that are accessible and inaccessible for a given mode. To determine the particular height distributions of physical characteristics over the entire path, the NRLMSISE-2000 atmosphere model and the HWM-93 wind model are used. Some of the calculated paths are compared with the known source and receiver positions and with observational results. It is concluded that the method can be used in a general assessment of the accessibility of a given region for acoustic monitoring; however, it is not sufficiently reliable to permit real-time predictions.

Nonlinear stability analysis of thin condensate falling film inside a rotating vertical cylinder

This paper investigates the stability of thin condensate film flowing down on the inner surface of a rotating vertical cylinder by means of long wave perturbation method in a two-step procedure. In the first step, the normal mode method is used to characterize the linear behavior. In the second step, an elaborated nonlinear film flow model is solved by using the method of multiple scales to characterize flow behavior at various states of sub-critical stability, sub-critical instability, supercritical stability, and supercritical explosion. The procedure follows the previous research [C.I. Chen, C.K. Chen, Y.T. Yang, Int. J. Heat Mass Transfer 47 (2004) 1937–1951] which concerns with the thin condensate falling film on the outer surface of a rotating vertical cylinder. The modeling results indicate that by increasing the rotation speed, Ω, and the radius of cylinder, R, the condensate film becomes more stable, which is totally opposite to the previous study.

A theoretical description of the polarization dependence of the sum frequency generation spectroscopy of the water/vapor interface

An improved time correlation function (TCF) description of sum frequency generation (SFG) spectroscopy was developed and applied to theoretically describing the spectroscopy of the ambient water/vapor interface. A more general TCF expression than was published previously is presented-it is valid over the entire vibrational spectrum for both the real and imaginary parts of the signal. Computationally, earlier time correlation function approaches were limited to short correlation times that made signal processing challenging. Here, this limitation is overcome, and well-averaged spectra are presented for the three independent polarization conditions that are possible for electronically nonresonant SFG. The theoretical spectra compare quite favorably in shape and relative magnitude to extant experimental results in the O-H stretching region of water for all polarization geometries. The methodological improvements also allow the calculation of intermolecular SFG spectra. While the intermolecular spectrum of bulk water shows relatively little structure, the interfacial spectra (for polarizations that are sensitive to dipole derivatives normal to the interface--SSP and PPP) show a well-defined intermolecular mode at 875 cm(-1) that is comparable in intensity to the rest of the intermolecular structure, and has an intensity that is approximately one-sixth of the magnitude of the intense free O-H stretching peak. Using instantaneous normal mode methods, the resonance is shown to be due to a wagging mode localized on a single water molecule, almost parallel to the interface, with two hydrogens displaced normal to the interface, and the oxygen anchored in the interface. We have also uncovered the origin of another intermolecular mode at 95 cm(-1) for the SSP and PPP spectra, and at 220 cm(-1) for the SPS spectra. These resonances are due to hindered translations perpendicular to the interface for the SSP and PPP spectra, and translations parallel to the interface for the SPS spectra. Further, by examining the real and imaginary parts of the SFG signal, several resonances are shown to be due to a single spectroscopic species while the "donor" O-H region is shown to consist of three distinct species-consistent with an earlier experimental analysis.

Nonlinear Stability Analysis of Thin Viscoelastic Film Flowing Down on the Inner Surface of a Rotating Vertical Cylinder

This paper investigates the stability of thin viscoelastic liquid film flowing down on the inner surface of a rotating vertical cylinder by means of the long wave perturbation. After proving the insufficiency of the linear model in characterization of certain flow behaviors, a generalized nonlinear kinematic model is then derived to represent the physical system. This model is solved through the following procedure. In the first step, the normal mode method is used to characterize the linear behaviors. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. In the second step, a nonlinear film flow model is solved by using the method of multiple scales to characterize flow behaviors at various states of sub-critical stability, sub-critical instability, supercritical stability, and supercritical explosion. The modeling results indicate that with the increase in the rotation speed Ω and the radius of cylinder R, the film flow system will be more stable.

Nonlinear stability analysis of thin Newtonian film flowing down on the inner surface of a rotating vertical cylinder

The paper presents both of the linear and nonlinear stability theories for characterization of Newtonian film flows down on the inner surface of a rotating in.nite vertical cylinder. After showing the insufficiency of the linear model in characterizing certain flow behaviors, a generalized nonlinear kinematic model is then derived to represent the physical system. The model is solved by the long wave perturbation method in a two-step procedure. In the first step, the normal mode method is used to characterize the linear behaviors. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. In the second step, an elaborated nonlinear film flow model is solved by using the method of multiple scales to characterize flow behaviors at various states of sub-critical stability, sub-critical instability, supercritical stability, and supercritical explosion. The modeling results indicate that by increasing the rotation speed, X, and the radius of cylinder, R, the film flow will make the flow system more stable.

Non-Linear Stability Analysis of a Thin Newtonian Film Flowing Down an Infinite Vertical Rotating Cylinder

Non-linear stability theories for the characterization of Newtonian film flow down an infinite vertical rotating cylinder is given. A generalized non-linear kinematic model is derived to represent the physical system and is solved by the long-wave perturbation method in a two-step procedure. In the first step, the normal mode method is used to characterize the linear behaviours. In the second step, an elaborated non-linear film flow model is solved by using the method of multiple scales to characterize flow behaviours at various states of subcritical stability, subcritical instability, supercritical stability, and supercritical explosion. The modelling results indicate that by increasing the rotation speed, ω, and decreasing the radius of cylinder, R, the film flow becomes less stable, generally.

Snap-through truss as an absorber of forced oscillations

The possibility of absorption of linear elastic system forced oscillations by means of an essentially nonlinear absorber (snap-through truss), which is attached to the linear subsystem, is analyzed. The simplest mass–spring linear subsystem is chosen to study the problem of forced oscillations absorption. The combination of the nonlinear normal vibrations modes method, the Rauscher approach and the asymptotic analysis is used to study forced oscillations of the two-dof system. The absorption vibrations mode in the form of nonlinear normal vibrations mode with small oscillations amplitudes of the linear subsystem and large amplitudes of the snap-through truss is studied. It is shown that this mode is stable over the wide range of the excitation frequency.

Non-linear stability characterization of the thin micropolar liquid film flowing down the inner surface of a rotating vertical cylinder

Linear and non-linear stability analysis for characterization of micropolar film flowing down the inner surface of a rotating infinite vertical cylinder is given. A generalized non-linear kinematic model is derived to represent the physical system and is solved by the long wave perturbation method in the following procedure. First, the normal mode method is used to characterize the linear behaviors. Then, an elaborated non-linear film flow model is solved by using the method of multiple scales to characterize flow behaviors at various states of sub-critical stability, sub-critical instability, supercritical stability, and supercritical explosion. The modeling results indicate that by increasing the rotation speed, Ω, and the radius of cylinder, R, the film flow will generally stabilize the flow system.

Range Step Size Analysis of Parabolic Equation Method in Megarange Propagation

It is important to check the calculation accuracy of the simulation model when calculating sound propagation loss in the ocean. Conventionally, simulation of propagation loss has often been calculated for ranges shorter than several hundred km. In such cases, the result has been compared with a benchmark or standard result, for example those obtained by the normal mode method. However, a benchmark result is not available for long-range propagations exceeding 1000 km. Furthermore, it is difficult to make the range interval of calculation into a wavelength using the normal mode method. The parabolic equation (PE) method must have the calculation interval set shorter than the wavelength. Therefore, it is difficult to compare the results of the normal mode method and the PE method in the same calculation environment. On the other hand, with the PE method it is known that the error of division decreases when the calculation step size is set to small. However, the calculation time increases according to the reduction of calculation step size. Thus, we propose a calculation step size that is most suitable for megarange propagation simulation using the PE method.

Model of wind-generated ambient noise in stratified shallow water

In order to build a rapid ocean ambient noise model adapted for a stratified shallow water, a hybrid model of normal mode method (for far field) and ray method (for near field) is suggested which combines the advantages of both methods. Since the near field of wind-generated noise is not sensitive to the sound speed profile, the sound speed profile is regarded as a constant; which makes the model rapid and accurate. The simulation results are in agreement with those of the wave model.

Two-dimensional problem of periodic loading of an elastic plate floating on the surface of an infinitely deep fluid

The action of external periodic pressure on an elastic plate floating on the surface of a fluid assumed to be ideal and incompressible is examined by the method of normal modes in the linear formulation. The behavior of the matrix of coefficients of the hydrodynamic load on the plate is considered in detail for different frequencies. The behavior of the plate under localized periodic loading is compared for the cases of a heavy fluid with a finite or infinite depth and for a weightless infinite-depth fluid.