Equivalent Wavenumber Research Articles

A Comparison of Linear Inverse Scattering Models for Contactless GPR Imaging

Ground-penetrating radars operating at a stand-off distance from the probed domain are attractive diagnostic tools in several contexts. Despite notable advantages offered in terms of noninvasiveness and flexibility, the usage of stand-off configurations makes the imaging process more difficult. Indeed, an accurate reconstruction of the subsurface/hidden scene requires accounting for the presence of the air–soil interface into the scattering model. This article deals with a comparison of three different linear models for describing the scattering phenomenon under a stand-off configuration, where the presence of the layered medium is explicitly accounted for in the scattering model. The first model is rigorous and based on the spectral domain representation of the incident field and Green’s function for the half-space scenario. The second model exploits a ray representation of the wave propagation from the radar to the target and vice versa by accounting for the refraction phenomenon at the interface. The third model is an approximated one, where the layered nature of the scenario is taken into account by means of an equivalent wavenumber, which allows a straightforward evaluation of the incident field and Green’s function. The resolution performance of each inversion approach is assessed by means of a numerical analysis. Moreover, data referred to an experiment carried out in controlled conditions are processed and reconstructions are provided to support the comparative analysis among the three imaging approaches.

One-step compensation downward continuation method free of iteration in wave number domain

As a kind of geomagnetic data processing, downward continuation of potential field plays an important role in geologic interpretation and geomagnetic localization. The inherent instability of larger distance downward continuation restricts its practical applications. In this paper, we report a kind of one-step compensation downward continuation method free of iteration using the equivalent wave number domain continuation operator, which will speed downward continuation compared to iterative compensation, and prove theoretically the convergence of the downward continuation filtering factor about one-step immune-iterative compensation. The frequency responses of the low-pass filter factor are also discussed by different damping factors, iterative numbers and continuation depths. The regularity analysis of one-step compensation method is also discussed. We use the theoretical model of magnetic bodies and real data to test experimentally the immune-iterative compensation algorithm based on the equivalent wave number domain continuation factor, respectively. The results all show high accuracy and good stability of the one-step compensation downward continuation algorithm. The performances of downward continuation including three kinds of algorithms are demonstrated by three parameters, and comparisons prove the reported algorithm has less calculated error than the generalized inverse method, and less elapsed time than the iterative compensation method.

Improved physical model in layered media for via structures

In this paper, the physical model is extended for vias embedded in layered media between two parallel-plate pairs. The effect of this inhomogeneity on signal integrity analysis is crucial, but is rarely studied especially by using the equivalent circuit model. The model presented in this paper is constituted by upper and lower via-to-plane capacitances which are different from each other due to the inhomogeneous dielectrics and equivalent impedance matrix of the parallel-plate cavity with equivalent wave number. The via-to-plate capacitance with layered media is calculated by using cascaded Green’s function with image theory and integral equation method based on partial capacitance, which is the main contribution in this paper. The parallel-plate impedance is calculated by using the cavity resonator method. Transverse resonance method as well as an equivalent approximation method is used for wave number evaluation. For the first time, the circuit model presented in this paper is used for the issue of layered media, which is much simpler with sufficient accuracy up to 20 GHz when compared with analytical method. The model is validated with full-wave simulations.

Elastoplastic Instability of the Double-Arc Ring-Stiffened Cylindrical Shell

The elastoplastic instability of the symmetrical double-arc ring-stiffened cylindrical shell was analyzed. The critical pressure was obtained for small deflection elastoplastic buckling of the cylindrical shell under hydrostatic pressure. The critical pressure for large deflection buckling of the cylindrical shell with initial geometric imperfections was determined under hydrostatic pressure by using nonlinear large deformation theory. The effects of hoop equivalent wave number, initial geometric imperfection and the central angle of circular arc on buckling mode and critical pressure of double-arc ring-stiffened cylindrical shell were analyzed. The results showed that the hoop equivalent wave number of arc cylindrical shell had an main effect on the critical pressure of the structure, designed equivalent hoop wave number of double-arc cylindrical shell should not be approach to the hoop wave number corresponding to minimum critical pressure, and the central angle of circular arc had less effect on critical pressure of the structure.

Nearshore Bathymetry Estimation Using Video Coastal Monitoring System

The invention of new digital technology of images from video camera systems now can provide information of the shoreward propagation of wave using pixel intensity time series that collected at cross-shore array. From this video image of intensity data, we can measure the wave speed (or, equivalent to wave number). Then the local water depth is inferred from linear wave theory dispersion relation equation. This paper describes the cross-spectral correlation approach scheme for investigating wave speed or equivalent wave number from video image data sets. The approach based on a pixel array analysis that utilizes a nonlinear inverse method. The solution is tolerant to noise and other forms of sampling deficiency. The solution includes error predictions that can be used to evaluate sample designs (data gaps) and the signal coherence. The technique was tested using 30 days of hourly data collected at Egmond beach, Netherlands. The result showed that the method have capability to derive wave number prediction from pixel intensity time series to support bathymetry estimation in near shore area through inversion of an appropriate wave dispersion model.

Updraft/Downdraft Constraints for Moist Baroclinic Modes and Their Implications for the Short-Wave Cutoff and Maximum Growth Rate

Abstract This paper examines the dynamics of moist baroclinic modes, based on the idealized model of moist baroclinic instability devised by Emanuel et al. These authors found that the finite static stability along the downdraft prevents the explosive short-wave cyclogenesis of the zero stratification limit in the moist problem, and allows only moderate (order 2) changes in the growth rate and short-wave cutoff, even when the moist static stability vanishes. To understand the limiting role of the dry static stability, a constraint is derived in this paper that relates the updraft and downdraft structures. This constraint is based on continuity and implies that a bulk wavenumber (defined in the paper) scales as the relevant deformation radius in each region. Because neutral solutions are separable, the vertical structure can be encapsulated in terms of a single, equivalent wavenumber based on the downdraft width. This allows an interpretation of the results in terms of the equivalent dry mode. As the ratio between moist and dry static stability decreases, the downdraft width takes an increasingly larger fraction of the total wavelength. In the limit of moist neutrality all the wavelength is occupied by the downdraft, so that the short-wave cutoff is halved. The vertical phase tilt makes unstable solutions nonseparable, and prevents defining an equivalent wavenumber in that case. However, the constraint between the bulk wavenumbers still applies. As the moist stability is reduced, the updraft solution becomes more suboptimal; in the limit of moist neutrality, the updraft wavenumber equals the short-wave cutoff. This provides a bound to the maximum growth rate in the moist problem, which is in agreement with the results of Emanuel et al.

Fourier prestack migration by equivalent wavenumber

Fourier prestack migration is reformulated through a change of variables, from offset wavenumber to a new equivalent wavenumber, which makes the migration phase shift independent of horizontal wavenumber. After the change of variables, the inverse Fourier transform over horizontal wavenumber can be performed to create unmigrated, but fully horizontally positioned, gathers at each output location. A complete prestack migration then results by imaging each gather independently with a poststack migration algorithm. This equivalent wavenumber migration (EWM) is the Fourier analog of the space‐time domain method of equivalent offset migration (EOM). The latter is a Kirchhoff time‐migration technique which forms common scatterpoint (CSP) gathers for each migrated trace and then images those gathers with a Kirchhoff summation. These CSP gathers are formed by trace mappings at constant time, and migration velocity analysis is easily done after the gathers are formed. Both EWM and EOM are motivated by the algebraic combination of a double square root equation into a single square root. This result defines equivalent wavenumber or offset. EWM is shown to be an exact reformulation of prestack f-k migration. The EWM theory provides explicit Fourier integrals for the formation of CSP gathers from the prestack data volume and the imaging of those gathers to form the final prestack migrated result. The CSP gathers are given by a Fourier mapping, at constant frequency, of the unmigrated spectrum followed by an inverse Fourier transform. The mapping requires angle‐dependent weighting factors for full amplitude preservation. The imaging expression (for each CSP gather) is formally identical to poststack migration with the result retained only at zero equivalent offset. Through a numerical simulation, the impulse responses of EOM and EWM are shown to be kinematically identical. Amplitude scale factors, which are exact in the constant velocity EWM theory, are implemented approximately in variable velocity EOM.

Stability of a nonorthogonal stagnation flow to three-dimensional disturbances

A similarity solution for a low Mach number nonorthogonal flow impinging on a hot or cold plate is presented. For the constant density case, it is known that the stagnation point shifts in the direction of the incoming flow and that this shift increases as the angle of attack decreases. When the effects of density variations are included, a critical plate temperature exists; above this temperature the stagnation point shifts away from the incoming stream as the angle is decreased. This flow field is believed to have application to the reattachment zone of certain separated flows or to a lifting body at a high angle of attack. Finally, the stability of this nonorthogonal flow to self similar, 3-D disturbances is examined. Stability properties of the flow are given as a function of the parameters of this study; ratio of the plate temperature to that of the outer potential flow and angle of attack. In particular, it is shown that the angle of attack can be scaled out by a suitable definition of an equivalent wavenumber and temporal growth rate, and the stability problem for the nonorthogonal case is identical to the stability problem for the orthogonal case.

On the instability of the flow in a squeeze lubrication film

When two parallel plates move normal to each other with a slow time-dependent speed, the velocity field developed in the intervening film of fluid is approximately that of plane Poiseuille flow, except that the magnitude of the velocity is dependent on time and on the coordinate parallel to the planes. This fact is intrinsic to Reynolds’ lubrication theory, and can be shown to follow from the Navier-Stokes equations when both the modified Reynolds number ( Re M ) and an aspect ratio ( δ ) are small. The modified Reynolds number is the product of δ and an actual Reynolds number ( Re ), which is based on the gap between the planes and on a characteristic velocity. The occurrence of flow instability and of turbulence in the film depend on Re . Typical values of Re , which are known to be required for the linear instability of plane Poiseuille flow, are of order 6000. This condition can be achieved, even if Re M is of order 1, provided that δ is of order 10 -4 . Such parameter values are typical of lubrication problems. The Orr-Sommerfeld equation governing flow instability is derived in this paper by use of the WKBJ technique, δ being the approximate small parameter to represent the small length-scale of the disturbance oscillations compared with the larger scale of the basic laminar flow. However, the coefficients in the Orr-Sommerfeld equation depend on slow space and time variables. Consequently the eigenrelation, derivable from the Orr-Sommerfeld equation and the associated boundary conditions, constitutes a nonlinear first-order partial differential equation for a phase function. This equation is solved by use of Charpit’s method for certain special forms of the time-dependent gap between the planes, followed by detailed numerical calculations. The relation between time-dependence and flow instability is delineated by the calculated results. In detail the nature of the instability can be described as follows. We consider a disturbance wave at or near a particular station, the initial distribution of amplitude being gaussian in the slow coordinate parallel to the planes. In the context of the Orr-Sommerfeld equation and its eigenrelation, the particular station implies an equivalent Reynolds number, while the initial distribution of the disturbance wave implies an equivalent wavenumber. As time increases, the disturbance wave can be considered to move in the instability diagram of equivalent wavenumber against Reynolds number, in the sense that these parameters are time- and space-dependent for the evolution of the disturbance-wave system. For our detailed calculations we use a quadratic approximation to the eigenrelation, an approximation which is quite accurate. If the initial distribution implies a point within the neutral curve, when the plates are squeezed together the equivalent wavenumber falls while the equivalent Reynolds number rises, and amplification takes place until the lower branch of the neutral curve is nearly crossed. If the plates are pulled apart (dilatation) the equivalent wavenumber rises, while the Reynolds number drops, and amplification takes place until the upper branch of the neutral curve has been just crossed. In the case of dilatation the transition from amplification to damping takes place more quickly than for the case of squeezing, in part due to the geometry of the neutral curve.

A new criterion for a bulk type I or type II superconductor

A new criterion for a bulk type I or type II superconductor is found by solving the first Landau-Ginsburg(LG) equation, including the cubic term. The functions obtained for the upper or lower critical field H (parallel or perpendicular to the surface of a sample) near Tc are: B=Dt′w2±wD2t′24+D32t′32 ky with t′=1− TTc, w=n2e, and B = μoH; D is an undetermined constant, ky is the equivalent wave number near Tc. When ky ≥ 0, it is a type I superconductor. When ky < 0, it is a type II superconductor. A similar criterion for a tin superconductor is also obtained. The agreement between the theory and experimental results is quite good.

Theory of the proximity effect in [formula omitted] magnetic field I

The Proximity Effect phenomenon in non-zero parallel magnetic field can be well explained by solving the complete L-G equation, including the cubic term. The L-G equation becomes the equation of the one-dimensional harmonic-oscillator. The energy eigenvalue equation naturally produces an explicit functional relation between the upper critical magnetic field H and the transition temperature T over a wide range of T near T c: T=T c{1− 4ed sξ 2 sB C n [ (k y− 2ed sB n ) 2+ 2eB n +(k y− 2ed sB n )]} Where B=μ oH, T c = T s for a single superconductor layer, T c = T csn for a S-N lamina, T c=T css for a S-S lamina, k y is the equivalent wave number, and C is a constant. The agreement between the theory and the experimental results is very good.

A demonstration of the suppression of turbulent cascades by coherent vortices in two-dimensional turbulence

The role of coherent vortices in suppressing the rates of cascade (i.e., spectrum evolution through scale transfer and dissipation) is demonstrated in numerical solutions of decaying two-dimensional turbulence at high Reynolds number. The demonstration is made by comparing the evolution of a primary solution that has well-developed vortices with other solutions that initially have equivalent wavenumber spectra but lack vortices because of an imposed phase scrambling. The solutions without vortices have substantially greater cascade rates. Furthermore, when the primary solution is compared with other solutions whose initial conditions are constructed only from isolated, axisymmetric vortices, their spectrum evolution is qualitatively similar. This demonstrates the control of cascade rates by coherent vortices.

Natural Convection in Horizontal Thin-Walled Honeycomb Panels

This paper presents an experimental study of the stability of and natural convection heat transfer through a horizontal fluid layer heated from below and constrained internally by a honeycomb. Examination of the types of boundary conditions exacted on the fluid at the cell side-walls has shown that there are three limiting cases: (1) perfectly conducting side-walls; (2) perfectly adiabatic side-walls; and (3) side-walls having zero thickness. Experiments described in this paper approach the latter category. The fluid used is air and the honeycomb used is square-celled. Measured critical Rayleigh numbers are found to be intermediate between those applying to cases (1) and (2), and consistent with an “equivalent wave number” of approximately 0.95 times that for case (1). The measured natural convective heat transfer after instability is found to be significantly less than that predicted by the Malkus-Veronis power integral technique. However, it is found to approach asymptotically the heat transfer which would take place through a similar fluid layer unconstrained by a honeycomb. A general correlation equation for the heat transfer is given.

Effect of Side Walls on Natural Convection Between Horizontal Plates Heated From Below

Experimental results are presented giving the Rayleigh number at which convection initiates in a closed cell and the Nusselt number versus Rayleigh number relationship which prevails after convection is initiated. Over 700 data points were obtained for two types of cells heated from below, one a phenolic-fiberglass hexcell of low thermal conductivity and the other aluminum hexcell of high thermal conductivity. Relations based upon the Malkus-Veronis power integral technique and a concept of an equivalent wave number are shown to give good correlation of the experimental data for both the low and high conductivity side walls.

The Crossed-orbit Model of Helium, its Ionisation Potential, and Lyman Series

TAKING for granted the dynamical legitimacy of the crossed-orbit model as originally proposed by Bohr (Zeits. für Physik, ix., 1922, p. 1) for normal helium, I find for its negatived total energy, with the usual Coulomb law of force, and treating the two orbits as “circular” in the literal sense of the word, where F is the complete elliptic integral of the first kind, i the inclination of the planes of the two one-quantic orbits, in Bohr's case 120°, and the remaining letters are the usual symbols of the universal constants. In accordance with symmetry, the electrons are assumed simultaneously to pass the nodes, that is, the opposite ends of the common diameter. (Details of deduction of (1) are given in a paper to be published shortly.) If one of the electrons be removed to “infinity”, the energy of the remaining ionised atom He+ is 4Nch, where N = N: (1 + m/M). The small difference N–N being irrelevant for our purpose, the ionisation work thus becomes, by (1), W = Nch(3–7F/4), or, the equivalent wave-number (of the flash emitted at the return of the removed electron),