Extension of Haskind's relations to cylindrical wave fields in the context of an interaction theory

The Direct Matrix Method Interaction Theory (IT) proposed by Kagemoto and Yue [1] speeds up the computation of hydrodynamic coefficients for large arrays of bodies when compared to direct calculations using standard Boundary Element Method (BEM) solvers. One of the most computationally expensive parts of the matrix method is the calculation of two hydrodynamic operators, known as Diffraction Transfer Matrix (DTM) and Radiation Characteristics (RC), which describe the way an isolated geometry scatters and radiates waves, respectively. A third operator, called Force Transfer Matrix (FTM), was introduced by McNatt et al. [2] to facilitate the calculation of the forces exerted on the bodies. In this paper, a novel set of relations between the FTM and RC components is obtained using the Kochin functions specific to the cylindrical basis solutions. They extend the classical Haskind's relations, valid with incident plane waves, to the cylindrical components of the scattered and radiated fields. Moreover, an alternative demonstration of the identities is given, which does not rely on the far-field asymptotic representation of the potential. Additional expressions are provided that relate the hydrodynamic coefficients and the RC for isolated bodies as well as for arrays, and numerical checking of the derived mathematical expressions is presented. These new relations can be used to speed up calculation of the hydrodynamic operators required for the use of the IT and to test its accuracy.