Whitham's Theory Research Articles

Comparative analysis of macroscopic traffic models for evaluating shockwaves under lane closures on an urban highway using the LWR model

ABSTRACT Lane closure due to traffic crashes is a complex phenomenon that negatively affects the operational performance of highways and leads to secondary lane closures. The present study investigates the effects of shockwaves caused by a heavy vehicle run-off-road crash on a three-lane urban highway in Isfahan, Iran. Additionally, the study simulates the effects of lane closures on shockwaves and density using the Lighthill, Whitham, and Richards (LWR) theory and macroscopic traffic flow based on field data. An analysis of traffic flow models revealed that one-lane and two-lane closures reduce maximum flow by 29% and 61%, respectively, compared to normal conditions. The results showed that the minimum relative error (RE) for lane closures is associated with shockwave models based on the Underwood speed-density model, expanded using the Taylor series. Therefore, the shockwave model based on the Underwood model with Taylor series expansion outperforms other models.

Optical undular bores in Riemann problem of photon fluid with quintic nonlinearity.

This work develops the Whitham theory to study the Riemann problem of the Gerdjikov-Ivanov equationthat describes the photon fluid with quintic nonlinearity. The one-phase periodic solution of the Gerdjikov-Ivanov equationand the corresponding Whitham equationare derived by the finite gap integration method. Subsequently, the main basic wave structures arising from the discontinuous initial-value conditions are found by distinguishing the distributions of the Riemann invariants. Some exotic optical undular bores are observed by classifying the solutions of the Riemann problem of the Gerdjikov-Ivanov equation. It is observed that the analytical results from Whitham theory are in excellent agreement with the numerical solutions.

The Whitham Modulation Solution of the Complex Modified KdV Equation

This paper primarily concerns the Whitham modulation equation of the complex modified Korteweg–de Vries (cmKdV) equation with a step-like initial value. By utilizing the Lax pair, we derive the N-genus Whitham equations via the averaging method. The Whitham equation can be integrated using the hodograph transformation. We investigate Krichever’s algebro-geometric scheme to propose the averaging method for the cmKdV integrable hierarchy and obtain the Whitham velocities of the integrable hierarchy and the hodograph transformation. The connection between the equations of the Euler–Poisson–Darboux type linear overdetermined system, which determines the solutions of the hodograph transformation, is constructed through Riemann integration, which demonstrates that the Whitham equation can be solved. Finally, a step-like initial value problem is solved and an exotic wave pattern is discovered. The results of direct numerical simulation agree well with the Whitham theory solution, which shows the validity of the theory.

Effect of longitudinal lift distribution on sonic boom of a canard-wing-stabilator-body configuration

After the last flight of the Concorde in 2003, sonic boom has been one of the obstacles to the return of a supersonic transport aircraft to service. To reduce the sonic boom intensity to an acceptable level, it is of great significance to study the effect of lift distribution on far-field sonic boom, since lift is one of the most important contributors to an intense sonic boom. Existing studies on the longitudinal lift distribution used low-fidelity methods, such as Whitham theory, and in turn, only preliminary conclusions were obtained, such as that extending the lift distribution can reduce sonic boom. This paper uses a newly developed high-fidelity prediction method to quantitatively study the effect of longitudinal lift distribution on the sonic boom of a Canard-Wing-Stabilator-Body (CWSB) configuration. This high-fidelity prediction method combines near-field CFD simulation with far-field propagation by solving the augmented Burgers equation. A multipole analysis method is employed for the extraction of near-field waveform in order to reduce computational cost. Seven configurations with the same total lift but different distributions are studied, and the quantitative relationship between the longitudinal lift distribution and far-field sonic boom intensity is investigated. It is observed that a small lift generated by the stabilator can prevent aft-stabilator and aft-fuselage shocks from merging, while the balanced lift generated by the canard and wing can effectively keep the corresponding shocks further apart, which is beneficial for reducing both the on-track and off-track sonic boom. In turn, the acoustic level perceived at the ground can be reduced by 5.9 PLdB on-track and 5.4 PLdB off-track, on average.

Shock motion inside a varying cross-section channel and consequences on the downstream flow

Using numerical simulations shock wave physics in a constant area channel behind a convergent or divergent channel is investigated. It is found that shock wave propagation in the downstream uniform area region is influenced by the post-shock flow unsteadinesses. A detailed flow description is provided and a quasi-steady model for determining the waves intensity at large times is proposed. Moreover, this study points out the limits of the use of Whitham's theory in a variable area channel.

Formation of dispersive shock waves in evolution of a two-temperature collisionless plasma

The nonlinear dynamics of pulses in a two-temperature collisionless plasma with the formation of dispersion shock waves is studied. An analytical description is given for an arbitrary form of an initial disturbance with a smooth enough density profile on a uniform density background. For large time after the wave breaking moment, dispersive shock waves are formed. Motion of their edges is studied in the framework of Gurevich–Pitaevskii theory and Whitham theory of modulations. The analytical results are compared with the numerical solution.

Whitham equations and phase shifts for the Korteweg-de Vries equation.

The semi-classical Korteweg-de Vries equation for step-like data is considered with a small parameter in front of the highest derivative. Using perturbation analysis, Whitham theory is constructed to the higher order. This allows the order one phase and the complete leading-order solution to be obtained; the results are confirmed by extensive numerical calculations.

Modulational Instability of Viscous Fluid Conduit Periodic Waves

In this paper, we are interested in studying the modulational dynamics of interfacial waves rising buoyantly along a conduit of a viscous liquid. Formally, the behavior of modulated periodic waves on large space and time scales may be described through the use of Whitham modulation theory. The application of Whitham theory, however, is based on formal asymptotic (WKB) methods, thus removing a layer of rigor that would otherwise support their predictions. In this study, we aim at rigorously verifying the predictions of the Whitham theory, as it pertains to the modulational stability of periodic waves, in the context of the so-called conduit equation, a nonlinear dispersive PDE governing the evolution of the circular interface separating a light, viscous fluid rising buoyantly through a heavy, more viscous, miscible fluid at small Reynolds numbers. In particular, using rigorous spectral perturbation theory, we connect the predictions of Whitham theory to the rigorous spectral (in particular, modulational) stability of the underlying wave trains. This makes rigorous recent formal results on the conduit equation obtained by Maiden and Hoefer.

Evolution of initial discontinuity for the defocusing complex modified KdV equation

The complete classification of solutions to the defocusing complex modified Korteweg-de Vries (cmKdV) equation with the step-like initial condition is given by Whitham theory. The process of studying the solution of cmKdV equation can be reduced to explore four quasi-linear equations, which predicts the evolution of dispersive shock wave. The results obtained here are quite different from the defocusing nonlinear Schrodinger equation: the bidirectionality of defocusing nonlinear Schrodinger equation determines that there are two basic rarefaction and shock structures while in the cmKdV case three basic rarefaction structures and four basic dispersive shock structures are constructed which lead to more complicated classification of step-like initial condition, and wave patterns even consisted of six different regions while each of wave patterns is consisted of five regions in the defocusing nonlinear Schrodinger equation. Direct numerical simulations of cmKdV equation are agreed well with the solutions corresponding to Whitham theory.

A model for the trajectory of the transverse detonation resulting from re-initiation of a diffracted detonation

A semi-analytical model is presented for predicting the trajectory of the transverse detonation during the re-initiation process of a diffracted cellular detonation wave from a channel. Numerical simulations based on the two-dimensional reactive Euler equations with a detailed hydrogen/oxygen chemistry model were first performed to observe key characteristics of cellular detonation wave diffraction and to obtain required input parameters for the model construction. The present numerical observations indicate that the transverse detonation stems from a location on the expansion wave front, and the horizontal distance from this initial location to the channel exit can be scaled by a constant multiplied by the detonation cell width for large deviation angles of the channel. The velocity of the transverse detonation basically equals the Chapman–Jouguet detonation wave velocity consisting of two orthogonal components: the expansion velocity of the diffracted wave front and the relative velocity to the diffracted wave front. The shape of the decoupled wave front is not affected by local explosion and thus can be predicted by the Chester–Chisnell–Whitham theory. Based on these numerical observations and the Chester–Chisnell–Whitham theory, a semi-analytical model is constructed to predict the wave trajectories as well as the distances of the wall reflection point for various deviation angles and initial pressures. The model prediction agrees with the corresponding numerical results. The model result shows that the distance of the wall reflection point varies from 15 to 20 multiples of the cell width with a minimal dependence on deviation angle, independent of the initial pressure. The trajectory calculated by the model is self-similar and determined by the horizontal distance of the initial location. The dimensionless trajectories divided by the horizontal distance are coincident for different initial pressures.

Evolution of wave pulses in fully nonlinear shallow-water theory

We consider evolution of wave pulses with formation of dispersive shock waves in framework of fully nonlinear shallow-water equations. Situations of initial elevations or initial dips on the water surface are treated, and motion of the dispersive shock edges is studied within the Whitham theory of modulations. Simple analytical formulas are obtained for asymptotic stage of evolution of initially localized pulses. Analytical results are confirmed by exact numerical solutions of the fully nonlinear shallow-water equations.

On the Whitham system for the radial nonlinear Schrödinger equation

AbstractDispersive shock waves (DSWs) of the defocusing radial nonlinear Schrödinger (rNLS) equation in two spatial dimensions are studied. This equation arises naturally in Bose‐Einstein condensates, water waves, and nonlinear optics. A unified nonlinear WKB approach, equally applicable to integrable or nonintegrable partial differential equations, is used to find the rNLS Whitham modulation equation system in both physical and hydrodynamic type variables. The description of DSWs obtained via Whitham theory is compared with direct rNLS numerics; the results demonstrate very good quantitative agreement. On the other hand, as expected, comparison with the corresponding DSW solutions of the one‐dimensional NLS equation exhibits significant qualitative and quantitative differences.

Nonlinear modulation near the Lighthill instability threshold in 2+1 Whitham theory.

The dispersionless Whitham modulation equations in 2+1 (two space dimensions and time) are reviewed and the instabilities identified. The modulation theory is then reformulated, near the Lighthill instability threshold, with a slow phase, moving frame and different scalings. The resulting nonlinear phase modulation equation near the Lighthill surfaces is a geometric form of the 2+1 two-way Boussinesq equation. This equation is universal in the same sense as Whitham theory. Moreover, it is dispersive, and it has a wide range of interesting multi-periodic, quasi-periodic and multi-pulse localized solutions. For illustration the theory is applied to a complex nonlinear 2+1 Klein-Gordon equation which has two Lighthill surfaces in the manifold of periodic travelling waves.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.

Whitham modulation theory for the Kadomtsev- Petviashvili equation.

The genus-1 Kadomtsev-Petviashvili (KP)-Whitham system is derived for both variants of the KP equation; namely the KPI and KPII equations. The basic properties of the KP-Whitham system, including symmetries, exact reductions and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg-de Vries equation are discussed. Finally, the KP-Whitham system is used to study the linear stability properties of the genus-1 solutions of the KPI and KPII equations; it is shown that all genus-1 solutions of KPI are linearly unstable, while all genus-1 solutions of KPII are linearly stable within the context of Whitham theory.

Modulations of viscous fluid conduit periodic waves

Modulated periodic interfacial waves along a conduit of viscous liquid are explored using nonlinear wave modulation theory and numerical methods. Large-amplitude periodic-wave modulation (Whitham) theory does not require integrability of the underlying model equation, yet often either integrable equations are studied or the full extent of Whitham theory is not developed. Periodic wave solutions of the nonlinear, dispersive, non-integrable conduit equation are characterized by their wavenumber and amplitude. In the weakly nonlinear regime, both the defocusing and focusing variants of the nonlinear Schrödinger (NLS) equation are derived, depending on the carrier wavenumber. Dark and bright envelope solitons are found to persist in long-time numerical solutions of the conduit equation, providing numerical evidence for the existence of strongly nonlinear, large-amplitude envelope solitons. Due to non-convex dispersion, modulational instability for periodic waves above a critical wavenumber is predicted and observed. In the large-amplitude regime, structural properties of the Whitham modulation equations are computed, including strict hyperbolicity, genuine nonlinearity and linear degeneracy. Bifurcating from the NLS critical wavenumber at zero amplitude is an amplitude-dependent elliptic region for the Whitham equations within which a maximally unstable periodic wave is identified. The viscous fluid conduit system is a mathematically tractable, experimentally viable model system for wide-ranging nonlinear, dispersive wave dynamics.

No-hair theorems for analogue black holes

We show that transonic one dimensional flows which are analogous to black holes obey no-hair theorems both at the level of linear perturbations and in non-linear regimes. Considering solutions of the Gross-Pitaevskii (or Korteweg-de Vries) equation, we show that stationary flows which are asymptotically uniform on both sides of the horizon are stable and act as attractors. Using Whitham's modulation theory, we analytically characterize the emitted waves when starting from uniform perturbations. Numerical simulations confirm the validity of this approximation and extend the results to more general perturbations and to the (non-integrable) cubic-quintic Gross-Pitaevskii equation. When considering time reversed flows that correspond to white holes, the asymptotically uniform flows are unstable to sufficiently large perturbations and emit either a macroscopic undulation in the supersonic side, or a non-linear superposition of soliton trains.

Semiclassical limit of the focusing NLS: Whitham equations and the Riemann–Hilbert Problem approach

The main goal of this paper is to put together: a) the Whitham theory applicable to slowly modulated N-phase nonlinear wave solutions to the focusing nonlinear Schrödinger (fNLS) equation, and b) the Riemann–Hilbert Problem approach to particular solutions of the fNLS in the semiclassical (small dispersion) limit that develop slowly modulated N-phase nonlinear wave in the process of evolution. Both approaches have their own merits and limitations. Understanding of the interrelations between them could prove beneficial for a broad range of problems involving the semiclassical fNLS.

Detonation Transition Around Cylindrical Reflector of Pulse Detonation Engine Initiator

To achieve reliable transmission of detonation waves to a pulse detonation engine combustor (detonation chamber), the authors propose a pulse detonation engine initiator that uses a cylindrical reflector downstream of a predetonator exit. The detonation wave propagates around the reflector to change the wave shape in three transition stages: from a planar detonation wave in the predetonator to an expanding cylindrical detonation wave, from the cylindrical wave to a planar toroidal detonation wave, and from the toroidal wave to a planar detonation wave in the detonation chamber. The cylindrical wave propagates along a cylindrical path between the reflector and front wall of the detonation chamber, and the toroidal wave propagates along an annular path between the reflector and sidewall of the detonation chamber. The purpose of this study was to examine the influence of the gap width of the annular path on the transition stages from cylindrical to toroidal and from toroidal to planar. A series of experiments that filled the entire test section with the driver gas mixture (stoichiometric hydrogen–oxygen mixture) showed that the expanding cylindrical detonation wave was sufficiently strong to survive the rarefaction waves from the corners of the reflector at all of the investigated annular gap widths (5, 10, 15, and 20 mm) and was transmitted to the planar toroidal wave successfully in all cases. When the strength of the cylindrical detonation wave was under a supercritical condition for diffraction at the reflector corner, the necessary filling distance for the driver gas was predicted well by the Whitham theory. A second series of experiments showed the influence of the annular gap width on the detonation transition from the planar toroidal detonation wave to the planar detonation wave. Two different types of detonation transitions termed “continuous transition” and “temporal quenching” were observed. The threshold value of for continuous transition is approximately four.

Calculation of shock-wave parameters far from origination by combined numerical-analytical methods

An algorithm is proposed for calculating the parameters of weak shock waves at large distances from their origination. In chosen meridional planes, the parameters of the near field of the three-dimensional flow are used to determine the streamwise coordinates of “phantom bodies” by linear relations. When the initial body is replaced by a system of “phantom bodies” for which discrete values of the Whitham function are found, the far-field parameters are calculated by the Whitham theory, independently in each meridional plane. Results calculated for a body with axial symmetry and for bodies with spatial symmetry are presented.

Dispersive dam-break and lock-exchange flows in a two-layer fluid

Dam-break and lock-exchange flows are considered in a Boussinesq two-layer fluid system in a uniform two-dimensional channel. The focus is on inviscid ‘weak’ dam breaks or lock exchanges, for which waves generated from the initial conditions do not break, but instead disperse in a so-called undular bore. The evolution of such flows can be described by the Miyata–Camassa–Choi (MCC) equations. Insight into solutions of the MCC equations is provided by the canonical form of their long wave limit, the two-layer shallow water equations, which can be related to their single-layer counterpart via a surjective map. The nature of this surjective map illustrates that whilst some Riemann-type initial-value problems (dam breaks) are analogous to those in the single-layer problem, others (lock exchanges) are not. Previous descriptions of MCC waves of permanent form (cnoidal and solitary waves) are generalised, including a description of the effects of a regularising surface tension. The wave solutions allow the application of a technique due to El's approach, based on Whitham's modulation theory, which is used to determine key features of the expanding undular bore as a function of the initial conditions. A typical dam-break flow consists of a leftwards-propagating simple rarefaction wave and a rightward-propagating simple undular bore. The leading and trailing edge speeds, leading edge solitary wave amplitude and trailing edge linear wavelength are determined for the undular bore. Lock-exchange flows, for which the initial interface shape crosses the mid-depth of the channel, by contrast, are found to be more complex, and depending on the value of the surface tension parameter may include ‘solibores’ or fronts connecting two distinct regimes of long-wave behaviour. All of the results presented are informed and verified by numerical solutions of the MCC equations.