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Water Waves Over a Channel of Finite Depth With a Submerged Plane Barrier

Published online by Cambridge University Press:  20 November 2018

Albert E. Heins
Albert E. Heins*
Affiliation:
The Carnegie Institute of Technology
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This is the third in a series of problems in the study of surface waves which have been disturbed by the presence of a plane barrier and to which a solution may be provided. We assume as in part I, that the fluid is incompressible and non-viscous, and that motion is irrotational.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 2 , 1950 , pp. 210 - 222
Copyright
Copyright © Canadian Mathematical Society 1950

References

1 Albert E. Heins, “Water waves over a channel of finite depth with a dock”, American Journal of Mathematics, vol. 70 (1948), pp. 730-748. Henceforth we shall refer to this paper as I. The second part of this series is entitled “Water waves over a channel of infinite depth with a dock” and is to be submitted for publication shortly. Reference to the physical background of this problem discussed in this paper may be found in I.

2 The constant β is defined in Sec. 5 of this paper.

3 In order to formulate the integral equation, this asymptotic form need not be specified so definitely. Indeed from the Green's function which we employ, we shall find that ϕ(x,y) need only grow less rapidly than exp . With the solution of the integral equation we shall find that ϕ(x,y) has the prescribed asymptotic form for .

4 For further details see Sommerfeld, A.,“Die Greensche Funktion der Schwingungsgleichung”, Deutsche Mathematiker Vereinigung, vol. 21 (1912), pp. 309–353.Google Scholar In particular, for a discussion of the logarithmic character of the Green's function employed here, see I, p. 735.