A new approximation method for scattering by long finite arrays

Thompson, Ian; Linton, C. M.; Porter, Robert

doi:10.1093/qjmam/hbn006

Cited by 36 publications

(49 citation statements)

References 25 publications

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“…We also show how the interaction theory for defects can be combined with the infinite array subtraction technique developed in [11], and the large array approximation used in [24] to validate results, and extend the range of applicability. Particular attention is paid to the determination of the amplitude with which RB waves are excited by the defects.…”

Section: Illustrative Resultsmentioning

confidence: 96%

“…This must be chosen to be large enough to yield accurate results, but not so large as to unnecessarily increase program execution time or generate near singular linear systems. The truncation levels used by our numerical codes are the same as those reported in [24]. Unless otherwise stated, Neumann boundary conditions are applied on the surface of the regular scatterers.…”

Section: Illustrative Resultsmentioning

confidence: 99%

“…This approximation was used in generating the data for Figure 5.6; it amounts to assuming that significant interactions between widely spaced defects are caused solely by the RB modes and has been shown to work well in practice in the related case of a long, finite array [24].…”

Section: Widely Spaced Defectsmentioning

confidence: 99%

“…This was later extended in [5] to account for situations where more than one defect is present. Our formulation, which is a generalization of earlier work in [22], is rather different and automatically includes the case of multiple defects. Indeed, by first reducing to canonical problems, we obtain an "interaction theory for defects" by means of which the solutions for a wide variety of cases can be constructed at very little computational expense.…”

mentioning

confidence: 99%

“…We also demonstrate how the methods in sections 2-4 can be combined with other approaches, such as infinite array subtraction [11] and the large array approximation method used in [24], to widen the class of problems that can be considered.…”

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An Interaction Theory for Scattering by Defects in Arrays

Thompson¹,

Linton²

2008

SIAM J. Appl. Math.

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Abstract.Wave scattering by an array of bodies that is periodic except for a finite number of missing or irregular elements is considered. The field is decomposed into contributions from a set of canonical problems, which are solved using a modified array scanning method. The resulting interaction theory for defects is very efficient and can be used to construct the field in a large number of different situations. Numerical results are presented for several cases, and particular attention is paid to the amplitude with which surface waves are excited along the array. We also show how other approaches can be incorporated into the theory so as to increase the range of problems that can be solved.Key words. arrays, defects, scattering, surface waves AMS subject classifications. 78A45, 78A50, 42A99 DOI. 10.1137/0707031441. Introduction. Wave scattering by arrays of bodies is of fundamental importance in numerous engineering and physics applications. Here we are concerned with the effect of one or more defects in an infinite, periodic array. This problem is of significant current interest in several fields, including elastodynamics [19] and phononic [7,28] and photonic [1,26,5] crystals. The presence of defects leads to a significant increase in difficulty in determining the scattered field, because the geometry is no longer periodic. In particular, Rayleigh-Bloch (RB) surface waves (also known as array guided surface waves) are excited if the array geometry and physical parameters are such that these modes can exist. RB waves propagate without loss along an array, and are evanescent in other directions. They are known to occur in a wide variety of situations [15,21,18,10]. One of the key goals of this article is to develop an efficient and accurate method for the determination of the amplitude with which they are excited. The theory is presented in a form that can be directly interpreted in a number of different physical contexts. These include the acoustic case, in which the wavenumber k is the ratio of the angular frequency ω to the speed of sound c, and the interaction of linear water waves with bottom mounted, surface penetrating cylinders, in which case k is the positive solution to the dispersion relation k tanh kh = ω 2 /g, g being the acceleration due to gravity and h the quiescent fluid depth. For acoustics, Dirichlet and Neumann boundary conditions are used to model sound hard and sound soft bodies, respectively, whereas Neumann conditions are appropriate for solid bodies immersed in water. The method is also applicable in the electromagnetic and elastodynamic cases, provided that the overall vector wave problem decouples into separate scalar components.Our first step in obtaining the field scattered by a defective array is to decompose the solution into contributions arising from a set of simpler, canonical problems. This is achieved by modifying the field generated when a wave interacts with a periodic

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Section: Illustrative Resultsmentioning

confidence: 96%

Section: Illustrative Resultsmentioning

confidence: 99%

Section: Widely Spaced Defectsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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