On near-trapped modes and fictitious frequencies for water wave problems containing an array of circular cylinders using a null-field boundary integral equation

Abstract

To avoid using the addition theorem to translate the Bessel function, scattering of water waves by an array of circular cylinders is solved by using the null-field boundary integral equations, in conjunction with the adaptive observer system. Both the near-trapped modes (physics) and fictitious frequencies (mathematics) are observed. To alleviate the resonance problem of fictitious frequency for multiple cylinders, Combined Helmholtz Interior integral Equation Formulation (CHIEF) approach and Burton and Miller formulation are both considered. Regarding the Burton and Miller approach, hypersingular integrals can be easily calculated by using series summability without any difficulty owing to the introduction of degenerate kernels. The highly rank-deficient matrices for equal radius of cylinders are numerically examined and the rank is improved by adding valid CHIEF constraints. Besides, the selection of location and number for CHIEF points is addressed instead of trial and error. Parameter study of the incident angle on the resultant force is investigated. The effect of spacing and radius of cylinders on the near-trapped mode and fictitious frequency is also discussed. Several examples of water wave interaction by circular cylinders were demonstrated to see the validity of the present formulation.