A note on the correspondence between the <i>x</i>, <i>t</i>-plane and the characteristic plane in a problem of interaction of plane waves of finite amplitude

Stocker, Paul; Meyer, Robert B.

doi:10.1017/s030500410002692x

Cited by 3 publications

(4 citation statements)

References 2 publications

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“…and an expression for the general term of the series, e.g. t(k) will be shown to be (9) and where (10) q=O q…”

Section: Expansion Procedures and Shock Wave Developmentmentioning

confidence: 99%

“…_ As(a) £a rP+il dp. ] • 137 (15) (16) (17) This expression is of the form (8) and upon proper treatment of indices yields the recurrence formula, (10). From (10) of course one finds that the rule for symmetry in the indices (by pairs, (A, p) and (p., v» holds at the (n + l)st stage if it holds at the nth.…”

Section: Expansion Procedures and Shock Wave Developmentmentioning

confidence: 99%

“…General descriptions of the geometry of the situation are to be found in Craggs [9] and in Stocker and Meyer [10].…”

Section: Shock Developmentmentioning

confidence: 99%

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Perturbation Theory of Wave Propagation Based on the Method of Characteristics

Fox

1955

Journal of Mathematics and Physics

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“…and an expression for the general term of the series, e.g. t(k) will be shown to be (9) and where (10) q=O q…”

Section: Expansion Procedures and Shock Wave Developmentmentioning

confidence: 99%

Section: Expansion Procedures and Shock Wave Developmentmentioning

confidence: 99%

See 1 more Smart Citation

Perturbation Theory of Wave Propagation Based on the Method of Characteristics

Fox

1955

Journal of Mathematics and Physics

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“…In order to prove these results, it is necessary to have an expression for the Jacobian of the transformation in terms of derivatives of t. Thus from (5) and hence breakdown of the solution will occur when either t r or t s is zero along some curve in the speedgraph plane.…”

Section: The Initial Value Problem Of the First Hind The Normal Formmentioning

confidence: 99%

On an extension of Riemann's method of integration, with applications to one-dimensional gas dynamics

Ludford

1952

Math. Proc. Camb. Phil. Soc.

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The object of this paper is to investigate the influence of the initial values on the behaviour of the one-dimensional unsteady isentropic motion of a perfect gas, and in particular the occurrence of singularities. For this purpose it is expedient to develop an unfolding procedure in the theory of partial differential equations of hyperbolic type. This not only resolves a difficulty in the classical treatment of the initial value problem associated with this type of equation, but also simplifies the presentation.The geometrical properties of the singularities (branch and limit lines, edges of regression, etc.) occurring in two-dimensional steady flow have been well discussed already, and their counterparts in one-dimensional unsteady motion have been indicated by Stocker and Meyer(5). However, here a method for finding the explicit analytic connexion between the occurrence of such singularities and the prescribed initial conditions is developed.Assuming quite general initial conditions, two examples are considered. In the first, a conjecture of Riemann is verified; in the second a special type of periodic solution is discussed, and it is shown that this solution cannot be continued to all values of t > 0. Finally, this latter result is shown to be true for arbitrary periodic initial conditions.

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On the structure of supersonic flow

Meyer

1958

Journal of Applied Mathematics and Physics (ZAMP)

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If viscosity and heat conduction be neglected, the supersonic motion of a perfect gas is a radiation process, that is, the equations representing the mechanism of motion are of the hyperbolic type. A feature of such equations is that proper boundary or initial conditions prescribed on surfaces of finite extent determine the solution only in a finite portion of space, beyond which it can be continued in an arbitrary manner. This seems to suggest that the study of the structure of supersonic flow is limited to the consideration of such restricted portions of space, much in contrast to the study of the structure of incompressible flow, for example. The limitation is not entirely necessary, however. It is the purpose of this note to unify some of the known facts regarding the structure in the large of steady, two-dimensional, shockfree supersonic flow and to present new results and points of view.Closely analogous results hold for the one-dimensional, unsteady motion of a perfect, inviscid gas, and the considerations outlined in the following ma x" also be extended to steady, axially symmetrical, supersonic flow and other hyperbolic problems in two independent variables. UniquenessThe differential equations governing the steady, two-dimensional, irrotational, homentropic, supersonic flow of a perfect gas possess two families of characteristics with the respective slopes .dy_ : tan (0 +/~) (' minus' Mach lines)

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A note on the correspondence between the x, t-plane and the characteristic plane in a problem of interaction of plane waves of finite amplitude

Cited by 3 publications

References 2 publications

Perturbation Theory of Wave Propagation Based on the Method of Characteristics

Perturbation Theory of Wave Propagation Based on the Method of Characteristics

On an extension of Riemann's method of integration, with applications to one-dimensional gas dynamics

On the structure of supersonic flow

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