Kinematical Conservation Laws Research Articles

System of kinematical conservation laws

System of kinematical conservation laws

Kinematical conservation laws in a space of arbitrary dimensions

In a large number of physical phenomena, we find propagating surfaces which need mathematical treatment. In this paper, we present the theory of kinematical conservation laws (KCL) in a space of arbitrary dimensions, i.e., d-D KCL, which are equations of evolution of a moving surface Ωt in d-dimensional x-space, where x = (x1, x2,..., xd) ∈ Rd. The KCL are derived in a specially defined ray coordinates (ξ = (ξ1, ξ2,..., ξd−1), t), where ξ1, ξ2,..., ξd−1 are surface coordinates on Ωt and t is time. KCL are the most general equations in conservation form, governing the evolution of Ωt with physically realistic singularities. A very special type of singularity is a kink, which is a point on Ωt when Ωt is a curve in R2 and is a curve on Ωt when it is a surface in R3. Across a kink the normal n to Ωt and normal velocity m on Ωt are discontinuous.

Eigenvalues of kinematical conservation laws (KCL) based 3-D weakly nonlinear ray theory (WNLRT)

System of kinematical conservation laws (KCL) govern evolution of a curve in a plane or a surface in space, even if the curve or the surface has singularities on it. In our recent publication [K.R. Arun, P. Prasad, 3-D kinematical conservation laws (KCL): evolution of a surface in R 3 -in particular propagation of a nonlinear wavefront, Wave Motion 46 (2009) 293–311] we have developed a mathematical theory to study the successive positions and geometry of a 3-D weakly nonlinear wavefront by adding an energy transport equation to KCL. The 7 × 7 system of equations of this KCL based 3-D weakly nonlinear ray theory (WNLRT) is quite complex and explicit expressions for its two nonzero eigenvalues could not be obtained before. In this short note, we use two different methods: (i) the equivalence of KCL and ray equations and (ii) the transformation of surface coordinates, to derive the same exact expressions for these eigenvalues. The explicit expressions for nonzero eigenvalues are important also for checking stability of any numerical scheme to solve 3-D WNLRT.

An Application of 3-D Kinematical Conservation Laws: Propagation of a 3-D Wavefront

Three-dimensional (3-D) kinematical conservation laws (KCL) are equations of evolution of a propagating surface $\Omega_t$ in three space dimensions. We start with a brief review of the 3-D KCL system and mention some of its properties relevant to this paper. The 3-D KCL, a system of six conservation laws, is an underdetermined system to which we add an energy transport equation for a small amplitude 3-D nonlinear wavefront propagating in a polytropic gas in a uniform state and at rest. We call the enlarged system of 3-D KCL with the energy transport equation equations of weakly nonlinear ray theory (WNLRT). We highlight some interesting properties of the eigenstructure of the equations of WNLRT, but the main aim of this paper is to test the numerical efficacy of this system of seven conservation laws. We take several initial shapes for a nonlinear wavefront with a suitable amplitude distribution on it and let it evolve according to the 3-D WNLRT. The 3-D WNLRT is a weakly hyperbolic $7\times7$ system that is highly nonlinear. Here we use the staggered Lax–Friedrichs and Nessyahu–Tadmor central schemes and have obtained some very interesting shapes of the wavefronts. We find the 3-D KCL to be suitable for solving many complex problems for which there presently seems to be no other method capable of giving such physically realistic features.

3-D kinematical conservation laws (KCL): Evolution of a surface in [formula omitted] – in particular propagation of a nonlinear wavefront

3-D KCL are equations of evolution of a propagating surface (or a wavefront) Ω t in 3-space dimensions and were first derived by Giles, Prasad and Ravindran in 1995 assuming the motion of the surface to be isotropic. Here we discuss various properties of these 3-D KCL. These are the most general equations in conservation form, governing the evolution of Ω t with singularities which we call kinks and which are curves across which the normal n to Ω t and amplitude w on Ω t are discontinuous. From KCL we derive a system of six differential equations and show that the KCL system is equivalent to the ray equations of Ω t . The six independent equations and an energy transport equation (for small amplitude waves in a polytropic gas) involving an amplitude w (which is related to the normal velocity m of Ω t ) form a completely determined system of seven equations. We have determined eigenvalues of the system by a very novel method and find that the system has two distinct nonzero eigenvalues and five zero eigenvalues and the dimension of the eigenspace associated with the multiple eigenvalue 0 is only 4. For an appropriately defined m, the two nonzero eigenvalues are real when m > 1 and pure imaginary when m < 1 . Finally we give some examples of evolution of weakly nonlinear wavefronts.

Riemann problem for kinematical conservation laws and geometrical features of nonlinear wavefronts

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.

Extension of relativistic kinematics

We give an interpretation of the standard coordinates on Minkowski space time that permits us to introduce an additional invariant time concept. We then consider an alternative set of coordinates and kinematical conservation laws on the resulting “space-time.” We obtain two conservation laws in addition to those of relativistic kinematics. The new conservation laws concern the quantities kinematical mass and mass defect, which we have introduced in order to have a more complete description of composite systems.