Wentzel-Kramers-Brillouin Solution Research Articles

Novel Wentzel–Kramers–Brillouin Solutions to the Nonisentropic Helmholtz Equation in a Nonuniform Duct With Mean Temperature Gradient and Mean Flow

Abstract We derive the generalized Helmholtz equation (GHE) governing nonisentropic acoustic fluctuations in a quasi 1D duct with nonuniform cross section, mean temperature gradient, and nonuniform mean flow. Nonisentropic effects are included via heat conduction terms in the mean and fluctuating energy equations. To derive the Helmholtz equation exclusively in terms of the fluctuating pressure field, a relationship between density and pressure fluctuations is needed, which is shown to be a second-order differential equation for nonisentropic motions. Novel analytical solutions that are accurate for both low/high frequencies and small/large mean gradients are presented for the GHE based on the Wentzel–Kramers–Brillouin (WKB) method. WKB solutions are developed using the ansatz that the pressure fluctuation field has a travelling wave form, p^(x)=exp[∫0x(a+ib)dx], where x is the axial coordinate. Substituting this form into the Helmholtz equation yields coupled, nonlinear ordinary differential equations (ODEs) for a and b. Analytical solutions to the ODEs are obtained using the approximations of high frequency and slowly varying mean properties. This simplification allows us to obtain the lower order solutions b0 and a0. We then enhance solution accuracy by using a0 to solve for b1 without any approximations. Finally, b1 is employed to get a1, giving us the higher order solution. The p^ calculated from (a1, b1) is in good to excellent agreement with numerical solution of the GHE for both low and high frequencies and for a range of mean Mach numbers, including M¯≳1.

A simple estimation of OLED light extraction enhancement due to refractive modulation lattice diffraction

To estimate the light extraction enhancement of an organic light emitting diode due to a periodic refractive modulation lattice simulating a photonic or plasmonic lattice, a noniterative scheme was proposed based on the Wentzel–Kramers–Brillouin solution used to calculate the input/output efficiency of a refractive index grating coupler for optical circuits, treating the lattice as a two-dimensional Bravais lattice.

Analytical WKB theory for high-harmonic generation and its application to massive Dirac electrons

We propose an analytical approach to high-harmonic generation (HHG) for nonperturbative low-frequency and high-intensity fields based on the (Jeffreys-)Wentzel-Kramers-Brillouin (WKB) approximation. By properly taking into account Stokes phenomena of WKB solutions, we obtain wave functions that systematically include the repetitive dynamics of production and acceleration of electron-hole pairs and quantum interference due to phase accumulation between different pair production times (St\"uckelberg phase). Using the obtained wave functions without relying on any phenomenological assumptions, we explicitly compute electric current (including intra- and interband contributions) as the source of HHG for a massive Dirac system in $(1+1)$ dimensions under an ac electric field. We demonstrate that the WKB approximation agrees well with numerical results obtained by solving the time-dependent Schr\"odinger equation and point out that the quantum interference is important in HHG. We also predict in the deep nonperturbative regime that (1) harmonic intensities oscillate with respect to electric-field amplitude ${E}_{0}$ and frequency $\mathrm{\ensuremath{\Omega}}$, with a period determined by the St\"uckelberg phase, (2) the cutoff order of HHG is determined by $2e{E}_{0}/\ensuremath{\hbar}{\mathrm{\ensuremath{\Omega}}}^{2}$, with $e$ being the electron charge, and that (3) noninteger harmonics, controlled by the St\"uckelberg phase, appear as a transient effect. Our WKB theory is particularly suited for a parameter regime, where the Keldysh parameter $\ensuremath{\gamma}=(\mathrm{\ensuremath{\Delta}}/2)\mathrm{\ensuremath{\Omega}}/e{E}_{0}$, with $\mathrm{\ensuremath{\Delta}}$ being the gap size, is small. This parameter regime corresponds to intense lasers in the terahertz regime for realistic massive Dirac materials. Our analysis implies that the so-called HHG plateau can be observed at the terahertz frequency within the current technology.

A WKB derivation for internal waves generated by a horizontally moving body in a thermocline

An approximate solution for the steady linear internal wave field generated by a localized, horizontally moving source in a thermocline is derived using ray and WKB (Wentzel–Kramers–Brillouin) theory. The waves are assumed to be steady in a reference frame moving with the source velocity. This solution is shown to agree in the limit of propagating waves with an existing solution that is expressed in terms of an eigenfunction expansion. The WKB method is shown to reproduce all the factors in the eigenfunction solution, not just the eigenfunctions themselves. It also reveals the physical significance of the terms in the eigenfunction solution. Furthermore, the WKB solution suggests an alternate way to compute the wave field numerically that is computationally more efficient.

Recursive formulation of the WKB solution for linear time-varying dynamic systems

This work proposes a new calculation algorithm of the Wentzel–Kramers–Brillouin (WKB) solution for slow linear time-varying (LTV) systems based on the recursive formulation. For a linear dynamic system with time-varying (TV) mass, damping or stiffness, the recursive relations in real function form among the components of the WKB solution are obtained. The integral terms within the sampling intervals that cannot be solved analytically are approximated by simple algebraic calculations with high precision. Thus, the conventional expression of the WKB solution involving complex numerical integral terms is reduced to an analytical recursive formulation. An explicit function relationship between the TV system parameters and the dynamic response can be obtained when the external excitation is given. In the premise of similar accuracy, the recursive formulation has much higher efficiency than the conventional calculation process, which is verified by applying the proposed method to a particular LTV dynamic system.

Demographic stochasticity and extinction in populations with Allee effect.

We study simple stochastic scenarios, based on birth-and-death Markovian processes, that describe populations with the Allee effect, to account for the role of demographic stochasticity. In the mean-field deterministic limit we recover well-known deterministic evolution equations widely employed in population ecology. The mean time to extinction is in general obtained by the Wentzel-Kramers-Brillouin (WKB) approximation for populations with the strong and weak Allee effects. An exact solution for the mean time to extinction can be found via a recursive equation for special cases of the stochastic dynamics. We study the conditions for the validity of the WKB solution and analyze the boundary between the weak and strong Allee effect by comparing exact solutions with numerical simulations.

WKB solutions to the quasi 1-D acoustic wave equation in ducts with non-uniform cross-section and inhomogeneous mean flow properties – Acoustic field and combustion instability

Linear modal analysis based on the spatio-temporal harmonic representation of acoustic fluctuations is a well-known reduced order method to predict combustion instability frequencies. However, a limitation of this method is that it is only applicable to domains with homogeneous mean properties. The Wentzel-Kramers-Brillouin (WKB) method facilitates the development of approximate analytical solutions to the one-dimensional (1-D) acoustic wave equation in domains with inhomogeneous mean properties. The “classical” form of the WKB solution to the wave equation is based on the assumption of high frequency (or, short wavelength), and is applicable to uniform cross-section ducts with a mean temperature gradient, but no mean flow. Subsequently, the WKB method was extended to include uniform mean velocity as well. In the current study, we derive a WKB-type solution to the generalized Helmholtz equation in quasi 1-D ducts with non-uniform cross-sectional areas and inhomogeneities in mean flow properties such as the velocity, temperature, density, and pressure. The WKB solution is applicable under the assumptions of high frequency and slowly varying mean properties. In deriving this solution, a novel approach has been developed to relate the first and second spatial derivatives of the axial velocity fluctuations to the pressure fluctuations (and its derivatives).A second WKB-type solution (referred to as the WKB2 solution) is also obtained by relaxing the slowly-varying mean property assumption, but retaining the high frequency approximation. Thirdly, the Helmholtz equation for pressure is solved numerically without invoking the two WKB assumptions. The WKB, WKB2 and numerical solutions are compared for a number of cases involving combinations of non-uniform duct areas and axially varying mean properties. Finally, we apply the WKB solution to predict the longitudinal combustion instabilities in a 1-D combustor characterized by a sudden expansion, with uniform mean flow in each of the two ducts comprising the combustor, and a linear temperature profile downstream of a compact, planar flame.

Cut-off frequencies of circumferential horizontal shear waves in various functionally graded cylinder shells

In this study, cut-off frequencies of the circumferential SH waves in functionally graded piezoelectric-piezomagnetic material (FGPPM) cylinder shells with traction free, electrical and magnetic open boundary conditions are investigated analytically. The Wentzel-Kramers-Brillouin (WKB) method is employed for solving differential equations with variable coefficients for general cases. For comparison, Bessel functions and Kummer functions are used for solving cut-off frequency problems in homogenous and ideal FGPPM cylinder shells. It is shown that the WKB solution for the cut-off frequencies has good precise. The set of cut-off frequencies is a series of approximate arithmetic progressions, for which the difference is a function of the density and the effective elastic parameter. The relationship between the difference and the gradient coefficient is described. These results provide theoretical guidance for the non-destructive evaluation of curved shells based on the cut-off frequencies.

Tunnelling effects for acoustic waves in slowly varying axisymmetric flow ducts

The multiple-scales Wentzel–Kramers–Brillouin (WKB) approximation is used to model the propagation of acoustic waves in an axisymmetric duct with a constriction in the presence of mean flow. An analysis of the reflection/transmission process of modes tunnelling through the constriction is conducted, and the key mathematical feature is the presence of two turning points, located at either real axial locations or in the complex plane. The resulting asymptotic solution consists of WKB solutions in regions away from the constriction and an inner solution valid in the near vicinity of the constriction. A solution which is uniformly valid throughout the duct is also derived. A range of test cases are considered, and the importance of accounting for the inner region, even in cases in which the turning points lie away from the real axis, is demonstrated.

SAUSAGE WAVES IN TRANSVERSELY NONUNIFORM MONOLITHIC CORONAL TUBES

We investigate fast sausage waves in a monolithic coronal magnetic tube, modeled as a local density inhomogeneity with a continuous radial profile. This work is a natural extension of our previous results, obtained for a slab loop model for the case of cylindrical geometry. Using Kneser's oscillating theorem, we provided the criteria for the existence of trapped and leaky wave regimes as a function of the profile features. For a number of density profiles there are only trapped modes for the entire range of longitudinal wave numbers. The phase speed of these modes tends toward the external Alfven speed in the long wavelength limit. The generalized results were supported by the analytic solution of the wave equation for the specific density profiles. The approximate Wentzel–Kramers–Brillouin solutions allowed us to obtain the desired dispersion relations and to study their properties as a function of the profile parameters. The multicomponent quasi-periodic pulsations in flaring loops, observed on 2001 May 2 and 2002 July 3, are interpreted in terms of the transversely fundamental trapped fast sausage mode with several longitudinal harmonics in a smooth coronal waveguide.

Wentzel–Kramers–Brillouin approximation for atmospheric waves

Ray and Wentzel–Kramers–Brillouin (WKB) approximations have long been important tools in understanding and modelling propagation of atmospheric waves. However, contradictory claims regarding the applicability and uniqueness of the WKB approximation persist in the literature. Here, we consider linear acoustic–gravity waves (AGWs) in a layered atmosphere with horizontal winds. A self-consistent version of the WKB approximation is systematically derived from first principles and compared toad hocapproximations proposed earlier. The parameters of the problem are identified that need to be small to ensure the validity of the WKB approximation. Properties of low-order WKB approximations are discussed in some detail. Contrary to the better-studied cases of acoustic waves and internal gravity waves in the Boussinesq approximation, the WKB solution contains the geometric, or Berry, phase. The Berry phase is generally non-negligible for AGWs in a moving atmosphere. In other words, knowledge of the AGW dispersion relation is not sufficient for calculation of the wave phase.

Estimating the Error of an Asymptotic Solution Describing the Angular Oscillations of the Axis of Symmetry of a Rotating Rigid Body

The error of the Wentzel–Kramers–Brillouin solution of the equations describing the angular motion of the axis of symmetry of rotation of a rigid body (projectile) is estimated. It is established that order of this estimate does not depend on whether the low-frequency oscillations of the axis of symmetry are damped or not

Wentzel–Kramers–Brillouin solution of cut-off frequency for horizontal shear (SH) waves in various inhomogeneous thin films

The Wentzel–Kramers–Brillouin method is employed to study the cut-off frequencies of the horizontal shear waves in a freestanding functionally graded piezoelectric–piezomagnetic material film with the electrically and magnetically open boundary conditions. An analytical solution, which could be used in analyzing the problems of various functionally graded materials, is proven to have high precision by analytical analysis and a numerical example. The results reveal that the set of cut-off frequencies is a series of approximate arithmetic progressions. A theoretical foundation based on the relationship between the cut-off frequencies and the materials’ gradient property is established for nondestructive evaluation.

Zener tunneling: Congruence between semi-classical and quantum ballistic formalisms

We compare the results of self-consistent ballistic quantum transport simulation of Zener tunneling in InAs, a direct bandgap semiconductor, with corresponding semi-classical solutions using the well-known Kane’s model and the Wentzel-Kramers-Brillouin (WKB) approximation. We find a qualitative difference between solutions obtained from Kane’s formulation and rigorous quantum-mechanical formalism. However, the WKB solution, with evaluation of action integral along the tunneling paths determined from the nearest neighbor sp3s* Hamiltonian, is shown to provide qualitative agreement. We discuss the issues involved in fitting semi-classical solutions with their quantum counterpart and also present a brief comparison of our results with experimental data.

On the WKB-theoretic structure of a Schrödinger operator with a merging pair of a simple pole and a simple turning point

A Schrodinger equation with a merging pair of a simple pole and a simple turning point (called MPPT equation for short) is studied from the viewpoint of exact Wentzel-Kramers-Brillouin (WKB) analysis. In a way parallel to the case of merging-turning-points (MTP) equations, we construct a WKB-theoretic transformation that brings an MPPT equation to its canonical form (the ∞-Whittaker equation in this case). Combining this transformation with the explicit description of the Voros coefficient for the Whittaker equation in terms of the Bernoulli numbers found by Koike, we discuss analytic properties of Borel-transformed WKB solutions of an MPPT equation.

Analytical semiclassical theory for general non-adiabatic transition and tunneling

A semiclassical solution of general two-state non-adiabatic transition and tunneling is found analytically within the Wentzel–Kramers–Brillouin (WKB) semiclassical framework associated with the Stokes phenomenon in mathematics. The non-adiabatic scattering matrix is determined by a complex quantity called the Stokes constant, which can be directly connected to the complex transition points of the WKB solution. An accurate and compact analytical solution is found for this Stokes constant which is a function of three parameters, one of which corresponds to the diabatic-to-adiabatic transformation angle that is interpreted as a type of non-adiabatic transition. Numerical examples demonstrate that the present unified analytical semiclassical theory works very well for both non-adiabatic transition and non-adiabatic tunneling. The present analytical semiclassical method can be a very powerful tool for application to multidimensional non-adiabatic dynamic processes.

Multiple asymptotic solutions for axially travelling waves in porous channels

Travelling waves in confined enclosures, such as porous channels, develop boundary layers that evolve over varying spatial scales. The present analysis employs a technique that circumvents guessing of the inner coordinate transformations at the forefront of a multiple-scales expansion. The work extends a former study in which a two-dimensional oscillatory solution was derived for the rotational travelling wave in a porous channel. This asymptotic solution was based on a free coordinate that could be evaluated using Prandtl's principle of matching with supplementary expansions. Its derivation required matching the dominant term in the multiple-scales expansion to an available Wentzel-Kramers-Brillouin (WKB) solution. Presently, the principle of least singular behaviour is used. This approach leads to a multiple-scales approximation that can be obtained independently of supplementary expansions. Furthermore, a procedure that yields different types of WKB solutions is described and extended to arbitrary order in the viscous perturbation parameter. Among those, the WKB expansion of type I is shown to exhibit an alternating singularity at odd orders in the perturbation parameter. This singularity is identified and suppressed using matched asymptotic tools. In contrast, the WKB expansion of type II is found to be uniformly valid at any order. Additionally, matched asymptotic, WKB and multiple-scales expansions are developed for several test cases. These enable us to characterize the essential vortico-acoustic features of the axially travelling waves in a porous channel. All solutions are numerically verified, compared and discussed.

Analytical investigation on 3D non-Boussinesq mountain wave drag for wind profiles with vertical variations

A new analytical model was developed to predict the gravity wave drag (GWD) induced by an isolated 3-dimensional mountain, over which a stratified, nonrotating non-Boussinesq sheared flow is impinged. The model is confined to small amplitude motion and assumes the ambient velocity varying slowly with height. The modified Taylor-Goldstein equation with variable coefficients is solved with a Wentzel-Kramers-Brillouin (WKB) approximation, formally valid at high Richardson numbers. With this WKB solution, generic formulae of second order accuracy, for the GWD and surface pressure perturbation (both for hydrostatic and non-hydrostatic flow) are presented, enabling a rigorous treatment on the effects by vertical variations in wind profiles. In an ideal test to the circular bell-shaped mountain, it was found that when the wind is linearly sheared, that the GWD decreases as the Richardson number decreases. However, the GWD for a forward sheared wind (wind increases with height) decreases always faster than that for the backward sheared wind (wind deceases with height). This difference is evident whenever the model is hydrostatic or not.

Propagation of an electromagnetic wave in an atmospheric pressure plasma: Numerical solutions

The propagation of an electromagnetic wave in an atmospheric pressure plasma (APP) layer is described numerically with an integral-differential wave equation. When the wave passes through the APP layer, the amplitude and phase of the transmission wave electric field are obviously modulated by the electron density and the collision frequency between the electrons and neutrals in the APP. The dependences of the wave behaviors, such as the phase shift, the coefficient of the transmission, reflection and absorption, on these APP layer characteristics are presented. Appleton’s equation is derived from the Wentzel–Kramers–Brillouin solution of the integral-differential wave equation and is compared with the numerical solution.

An Analytical Model of Mountain Wave Drag for Wind Profiles withShear and Curvature

Abstract An analytical model is developed to predict the surface drag exerted by internal gravity waves on an isolated axisymmetric mountain over which there is a stratified flow with a velocity profile that varies relatively slowly with height. The model is linear with respect to the perturbations induced by the mountain, and solves the Taylor–Goldstein equation with variable coefficients using a Wentzel–Kramers–Brillouin (WKB) approximation, formally valid for high Richardson numbers, Ri. The WKB solution is extended to a higher order than in previous studies, enabling a rigorous treatment of the effects of shear and curvature of the wind profile on the surface drag. In the hydrostatic approximation, closed formulas for the drag are derived for generic wind profiles, where the relative magnitude of the corrections to the leading-order drag (valid for a constant wind profile) does not depend on the detailed shape of the orography. The drag is found to vary proportionally to Ri−1, decreasing as Ri decreas...