Upstream propagating curved shock in a steady transonic flow

We derive a new set of equations to give successive positions of slowly moving and slowly turning curved shock front in a transonic flow. The equations are in conservation form - two of these are kinematical conservation laws

Fermat's and Huygens' principles, and hyperbolic equations and their equivalence in wavefront construction

Consider propagation of a wavefront in a medium. According to Fermat's principle a ray, travelling from one point P0 to another point P1 in space, chooses a path such that the time of transit is stationary. Given initial position of a wavefront Ω0, we can use rays to construct the wavefront Ωt at any time t. Huygens' method states that all points of a wavefront Ω0 at t = 0 can be considered as point sources of spherical secondary wavelets and after time t the new position Ωt of the wavefront is an envelope of these secondary wavelets. The equivalence of the two methods of construction of a wavefront Ωt in a medium governed by a general hyperbolic system of equations does not seem to have been proved. Hyperbolic equations have their own method of construction of a wavefront. We shall discuss this still open (as far as I know) problem for a general hyperbolic system and briefly sketch the relation between the three methods for a particular case when the medium is governed by Euler equations of a polytropic gas in free space [16]

An asymptotic derivation of weakly nonlinear ray theory

Using a method of expansion similar to Chapman-Enskog expansion, a new formal perturbation scheme based on high frequency approximation has been constructed. The scheme leads to an eikonal equation in which the leading order amplitude appears. The transport equation for the amplitude has been deduced with an error 0(ε2) where ε is the small parameter appearing in the high frequency approximation. On a length scale over which Choquet-Bruhat's theory is valid, this theory reduces to the former. The theory is valid on a much larger length scale and the leading order terms give the weakly nonlinear ray theory (WNLRT) of Prasad, which has been very successful in giving physically realistic results and also in showing that the caustic of a linear theory is resolved when nonlinear effects are included. The weak shock ray theory with infinite system of compatibility conditions also follows from this theory

Ray theories for hyperbolic waves, Kinematical conservation laws (KCL) and applications

Ray theory, for the construction of the successive positions of a wavefront governed by linear hyperbolic equations, is a method which had its origin from the work of Fermat (and is related to Huygen's method). However, for a nonlinear wavefront governed by a hyperbolic system of quasilinear equations, the ray equations are coupled to a transport equation for an amplitude of the intensity of the wave on the wavefront and some progress has been made by us in its derivation and use. We have also derived some purely differential geometric results on a moving curve in a plane (surface ∊IR3), these kinematical conservation laws are intimately related to the ray theory. In this article, we review these recent results, derive same new results and highlight their applications, specially to a challenging problem: sonic boom produced by a maneuvering aerofoil

Geometrical features of a nonlinear wavefront

We use the equations of weakly nonlinear ray theory (WNLRT), developed by us over a number of years, to study all possible shapes which a nonlinear wavefront in a polytropic gas can have. As seen in experiments, a converging nonlinear wavefront avoids folding itself in a caustic region of a linear theory and emerges unfolded with a pair of kinks. We review the work of Baskar, Potadar and Szeftel showing the way in which the solution of a Riemann problem of the conservation form of the equations of WNLRT can be used to study the formation of new shapes of a nonlinear wavefront from a single singularity on it. We also study the ultimate result of interactions of elementary shapes on the front

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

Canonical form of a quasilinear hyperbolic system of First order equations

A canonical form of the compatability condition along a characteristic surface for a quasilinear hyperbolic system of first order equations in m+ 1 independent variables is derived. This form of the compatibility condition is distinguished by the fact that special emphasis is given to the interior derivative in the bicharacteristic direction, which alone contains derivatives of the type ∂/∂t, whereas all other interior derivatives present are spatial in nature

Propagation of a curved weak shock

Propagation of a curved shock is governed by a system of shock ray equations which is coupled to an infinite system of transport equations along these rays. For a two-dimensional weak shock, it has been suggested that this system can be approximated by a hyperbolic system of four partial differential equations in a ray coordinate system, which consists of two independent variables (ζ, t) where the curves t = constant give successive positions of the shock and ζ = constant give rays. The equations show that shock rays not only stretch longitudinally due to finite amplitude on a shock front but also turn due to a non-uniform distribution of the shock strength on it. These changes finally lead to a modification of the amplitude of the shock strength. Since discontinuities in the form of kinks appear on the shock, it is necessary to study the problem by using the correct conservation form of these equations. We use such a system of equations in conservation form to construct a total-variation-bounded finite difference scheme. The numerical solution captures converging shock fronts with a pair of kinks on them-the shock front emerges without the usual folds in the caustic region. The shock strength, even when the shock passes through the caustic region, remains so small that the small-amplitude theory remains valid. The shock strength ultimately decays with a well-defined geometrical shape of the shock front-a pair of kinks which separate a central disc from a pair of wings on the two sides. We also study the ultimate shape and decay of shocks of initially periodic shapes and plane shocks with a dent and a bulge