On the representation of two‐dimensional scalar wave fields in the complex plane

Abstract Experimental results of internal tide generation by two-dimensional topography are presented. The synthetic Schlieren technique is used to study the wave fields generated by a Gaussian bump and a knife edge. The data compare well to theoretical predictions, supporting the use of these models to predict tidal conversion rates. In the experiments, viscosity plays an important role in smoothing the wave fields, which heals the singularities that can appear in inviscid theory and suppresses secondary instabilities of the experimental wave field.

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Path-linking interpretation of Kirchhoff diffraction

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1995.0071 ◽

1995 ◽

Vol 450 (1938) ◽

pp. 51-65 ◽

Cited By ~ 10

Keyword(s):

Exact Solution ◽

Wave Equation ◽

Wave Field ◽

Diffraction Field ◽

Scalar Wave ◽

Wave Fields ◽

Linking Number ◽

Diffraction Integral ◽

Pass Through ◽

Kirchhoff Diffraction Integral

The Kirchhoff-diffraction integral is often used to describe the (scalar) wave field from a monochromatic point source in the presence of ‘opaque’ screens. Despite criticisms that can be made of its ‘derivation’, the Kirchhoff field is an exact solution of the wave equation, and exactly obeys definite, though unusual, boundary conditions (Kottler 1923, 1965). Here, the path-integral picture of wave fields is used to interpret the Kirchhoff-diffraction field in terms of all conceivable propagation paths, whether or not they pass through the opaque screens. Specifically, it is noted that the Kirchhoff field equals Ʃ(1 ─ m )ψ m , where the sum is over all integers m , and ψ m is the wave field due to all paths from the source to the field point for which the number of outward screen crossings minus the number of backwards screen crossings is m . Expressed more topologically, m is the total linking number of a path, when closed by any unobstructed path, with the screen edge lines. Other models of diffraction by screens are compared with Kirchhoff diffraction in the path interpretation.

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On the observability of finite-depth short-crested water waves

Journal of Fluid Mechanics ◽

10.1017/s0022112096002686 ◽

1996 ◽

Vol 322 ◽

pp. 1-19 ◽

Cited By ~ 10

Author(s):

M. Ioualalen ◽

A. J. Roberts ◽

C. Kharif

Keyword(s):

Standing Wave ◽

Water Waves ◽

Numerical Study ◽

Standing Waves ◽

Finite Depth ◽

Two Dimensional ◽

Wave Fields ◽

Progressive Waves ◽

Wave Limit ◽

Narrow Bands

A numerical study of the superharmonic instabilities of short-crested waves on water of finite depth is performed in order to measure their time scales. It is shown that these superharmonic instabilities can be significant-unlike the deep-water case-in parts of the parameter regime. New resonances associated with the standing wave limit are studied closely. These instabilities ‘contaminate’ most of the parameter space, excluding that near two-dimensional progressive waves; however, they are significant only near the standing wave limit. The main result is that very narrow bands of both short-crested waves ‘close’ to two-dimensional standing waves, and of well developed short-crested waves, perturbed by superharmonic instabilities, are unstable for depths shallower than approximately a non-dimensional depth d= 1; the study is performed down to depth d= 0.5 beyond which the computations do not converge sufficiently. As a corollary, the present study predicts that these very narrow sub-domains of short-crested wave fields will not be observable, although most of the short-crested wave fields will be.

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