An extension of general identities for 3D water-wave diffraction with application to the Diffraction Transfer Matrix |
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| Institution: | 1. Imperial College London, Department of Civil and Environmental Engineering, Fluids Section, London, United Kingdom;2. School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia;1. University of L’Aquila, Department of Civil, Construction-Architectural and Environmental Engineering (DICEAA), Environmental and Maritime Hydraulic Laboratory (LIam), P.le Pontieri, 1, 67040 Monteluco di Roio, L’Aquila, Italy;2. Technical University of Bari, Department of Civil, Environmental, Building Engineering and Chemistry (DICATECh), Coastal Engineering Laboratory, Area Universitaria di Valenzano S.P. Valenzano Casamassima, Km.3, 70010 Valenzano, Bari, Italy;1. College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, UK;2. Faculty of Engineering and Physical Sciences, University of Southampton, Southampton, UK;1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, 149 Yanchang Road, Shanghai 200072, China;2. Shanghai Key Laboratory of Mechanics in Energy Engineering, 149 Yanchang Road, Shanghai 200072, China |
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| Abstract: | Interaction theories are used in numerous branches of physics to efficiently evaluate wave scattering by multiple obstacles. An example of these interaction theories is the direct matrix method introduced by Kagemoto and Yue 1], which enables fast computation of three-dimensional water-wave multiple-scattering problems. The building block of interaction theories is a mathematical operator that encapsulates the mapping between incident and scattered waves. This operator is generally referred to as T-matrix and satisfies both reciprocity and energy identities. In some branches of physics, such as acoustics and electromagnetism, these identities are well established; in hydrodynamics, however, they have only been derived for a T-matrix that maps two-dimensional incident and scattered water waves. In three dimensions, water waves can be represented as a series expansion of cylindrical eigenfunctions. In this paper, we use this representation of water waves to derive the reciprocity and energy identities satisfied by the T-matrix of the direct matrix method, known as Diffraction Transfer Matrix (dtm). The identities derived herein represent an extension of existing general relations between two diffraction solutions. We show that this extension can be applied to verify the accuracy of the dtm entries, thereby increasing the reliability of existing schemes for computing the dtm. We present results for the dtm of two geometrically different isolated obstacles, as well as for the dtm of an asymmetric array. Finally, we demonstrate that the results presented herein can be extended to floating bodies found in a wide range of ocean engineering problems. |
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| Keywords: | Multiple scattering Interaction theory Direct matrix method Diffraction Transfer Matrix Boundary element method |
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