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Simple progressive solutions of the wave equation

Published online by Cambridge University Press:  24 October 2008

F. G. Friedlander
F. G. Friedlander
Affiliation:
Trinity CollegeCambridge
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Extract

The object of this paper is to determine all the solutions of the wave equation

which are of the simple form

where F denotes an arbitrary function. It will be shown that, in addition to the obvious cases of plane or spherical progressive waves, such solutions exist only when the wave fronts

are certain algebraic surfaces of the fourth order, the cyclides of Dupin. These include, as degenerate cases, the sphere, the plane, the cylinder, the cone, and the torus.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 43 , Issue 3 , July 1947 , pp. 360 - 373
Copyright
Copyright © Cambridge Philosophical Society 1947

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References

REFERENCES

(1)Friedlander, F. G.On the solutions of the wave equation with discontinuous derivatives. Proc. Cambridge Phil. Soc. 38 (1942), 378–82.CrossRefGoogle Scholar
(2) See, for example, Blaschke, , Differentialgeometrie, 1 (3rd ed., Berlin, 1930), 95.Google Scholar
(3)Blaschke, , loc. cit. p. 196.Google Scholar
(4)Darboux, . Surfaces, 2 (2nd ed., Paris, 1915), 280.Google Scholar
(5)Lamb, . Diffraction of a solitary wave. Proc. London Math. Soc. (2), 8 (1910), 422–37.CrossRefGoogle Scholar
(6)Darboux, . Leçons sur lea systemes orthogonaux (2nd ed., Paris, 1910), p. 495.Google Scholar
(7) See, for example, Lamb, , Hydrodynamics (6th ed.), p. 521.Google Scholar