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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

Published online by Cambridge University Press:  24 October 2008

P. M. Stocker  and
R. E. Meyer
P. M. Stocker
Affiliation:
Department of MathematicsThe UniversityManchester
R. E. Meyer
Affiliation:
Department of MathematicsThe UniversityManchester
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Abstract

The head-on interaction, in the one-dimensional, unsteady isentropic flow of a perfect gas, of a simple compression wave and a simple expansion wave is studied by considering typical examples. The physical aspect of the problem is discussed in (1); in this note the possibility of shock formation is ignored, and the correspondence defined by the complete mathematical solution of the equations of isentropic flow between the x, t-plane and the plane of the characteristic variables is elucidated.

The solution is distinguished by the appearance of two limit lines and a second-order limit point where they meet. It is found that the image of the characteristic plane in the x, t-plane is four-sheeted; all sheets overlap each other, but each covers only part of the plane, and the only point common to all sheets is the second-order limit point, where both limit lines are cusped (§ 3·1).

The solution also contains an edge of regression, and a discussion of the properties of this type of singularity will be found in §§ 2 and 2·1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 47 , Issue 3 , July 1951 , pp. 518 - 527
Copyright
Copyright © Cambridge Philosophical Society 1951

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References

REFERENCES

(1)Stocker, P. M. On a problem of interaction of plane waves of finite amplitude involving the retardation of shock-formation by an expansion wave. Quart. J. Mech. Appl. Math.Google Scholar
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