Shock ray theory (SRT) has been found to be useful and computationally efficient in finding successive positions of a curved weak shock front. In this paper, we solve some piston problems and show that the shock ray theory with two compatibility conditions gives shock positions, which are very close to those obtained by solving the same problems by the numerical solution of Euler's equations (Euler solutions). Comparison of the results obtained by shock ray theory and geometrical shock dynamics (GSD) of Whitham (J. Fluid Mech. vol. 2, 1957, p. 146) with the Euler solution shows that the shock ray theory gives more accurate results for any piston motion. The aim of the work is not just this comparison, but also to investigate the role of the nonlinearity in accelerating the process of the evolution of a shock, produced by an explosion of a non-circular finite charge, into a circular shock front. We find that the nonlinear waves propagating on the shock front appreciably accelerate this process. We also discuss a situation, for shock Mach number very close to 1, when GSD and shock ray theory may fail to give any result.
We study the reflection of acoustic shock waves grazing at a small angle over a rigid surface. Depending on the incidence angle and the Mach number, the reflection patterns are mainly categorized into two types, namely regular reflection and irregular reflection. In the present work, using the nonlinear KZ equation, this reflection problem is investigated for extremely weak shocks as encountered in acoustics. A critical parameter, defined as the ratio of the sine of the incidence angle and the square root of the acoustic Mach number, is introduced in a natural way. For step shocks, we recover the self-similar (pseudo-steady) nature of the reflection, which is well known from von Neumann's work. Four types of reflection as a function of the critical parameter can be categorized. Thus, we describe the continuous but nonlinear and non-monotonic transition from linear reflection (according to the Snell–Descartes laws) to the weak von-Neumann-type reflection observed for almost perfectly grazing incidence. This last regime is a new, one-shock regime, in contrast with the other, already known, two-shock (regular reflection) or three-shock (von Neumann-type reflection) regimes. Hence, the transition also resolves another paradox on acoustic shock waves addressed by von Neumann in his classical paper. However, step shocks are quite unrealistic in acoustics. Therefore, we investigate the generalization of this transition for N-waves or periodic sawtooth waves, which are more appropriate for acoustics. Our results show an unsteady reflection effect necessarily associated with the energy decay of the incident wave. This effect is the counterpart of step-shock propagation over a concave surface. For a given value of the critical parameter, all the patterns categorized for the step shock may successively appear when the shock is propagating along the surface, starting from weak von-Neumann-type reflection, then gradually turning to von Neumann reflection and finally evolving into nonlinear regular reflection. This last one will asymptotically result in linear regular reflection (Snell–Descartes). The transition back to regular reflection is one of two types, depending on whether a secondary reflected shock is observed. The latter case, here described for the first time, appears to be related to the non-constant state behind the incident shock, which prevents secondary reflection.
A pair of kinematical conservation laws (KCL) in a ray coordinate system (ξ, t) are the basic equations governing the evolution of a moving curve in two space-dimensions. We first study elementary wave solutions and then the Riemann problem for KCL when the metric g, associated with the coordinate ξ designating different rays, is an arbitrary function of the velocity of propagation m of the moving curve. We assume that m > 1 (m is appropriately normalized), for which the system of KCL becomes hyperbolic. We interpret the images of the elementary wave solutions in the (ξ, t)plane to the (x, y)-plane as elementary shapes of the moving curve (or a nonlinear wavefront when interpreted in a physical system) and then describe their geometrical properties. Solutions of the Riemann problem with different initial data give the shapes of the nonlinear wavefront with different combinations of elementary shapes. Finally, we study all possible interactions of elementary shapes.
a b s t r a c tIn this article, we intend to use quadratic and cubic B-spline quasi-interpolants to develop higher order numerical methods for some Sobolev type equations in one space dimension. Our aim is also to compare the performance of the proposed methods in terms of the accuracy and the rate of convergence. We also discuss another approach to the cubic B-spline quasi-interpolation based method, where we achieve fourth order of accuracy in space. We theoretically establish the order of accuracy for the three proposed methods and also establish the L 2 -stability in the linear case using von Neumann analysis. As a particular case of the Sobolev type equations, we take the equal width and the Benjamin-Bona-Mahony-Burgers equations, and perform several numerical experiments to support our theoretical results.
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