Study of near-trapped modes and fictitious frequencies using null-field integral equation

Abstract

In this paper, we employ the null-field integral equations to solve the scattering of water waves by bottom-mounted vertical circular cylinders. Based on the null-field integral equations in conjunction with degenerate kernels and Fourier series, we can avoid using principal-values sense in calculating singular integral, even though a null-field point is exactly located on the real boundary. This gain is owing to the introduction of degenerate kernels for fundamental solutions. Two kinds of peaks for the resultant force on the cylinder versus the wavenumber are observed. First, the critical (physical) wavenumber for the near-trapped mode is numerically detected. Second, the peak occurs at the fictitious frequency (mathematics) due to the integral formulation for exterior Helmholtz problems. Both peaks of near-trapped mode and fictitious frequency are physically and mathematically realizable, respectively. By increasing the number of Fourier terms, peaks due to fictitious frequencies are suppressed to be smooth while the peak value due to the near-trapped mode still keeps a sharp constant. Besides, the effect of angle of incident wave on the near-trapped mode is studied in this paper. Furthermore, a disorder of one cylinder by changing the radius and the position of center can suppress the occurrence of near-trapped mode due to the destruction of the periodical pattern. Numerical examples were demonstrated to show the validity of the present formulation.本文採用零場積分方程求解多圓柱水波散射問題。透過零場積分方程並配合退化核與傅立葉級數，可將零場點直接放置到真實邊界上，並能免除以主值觀念計算奇異積分的困擾。文中在計算圓柱所受之合力對波數作圖時，觀察到兩種尖峰值現象。一個是數 值實驗觀察到陷阱模態(物理上的)之臨界波數。另一個是以積分方程求解外域Helmholtz 方程時，所出現的虛擬頻率(數學上的)。上述之陷阱模態和虛擬頻率之尖峰分別可用物理與數學觀點加以解釋。本文藉由增加傅立葉級數之項數，發現陷阱模態尖峰值仍保持不變，而虛擬頻率所引起的尖峰值則受抑制而趨於平滑。文中亦探討入射波之入射角度改變，對於陷阱模態所造成影響。此外，透過改變圓柱半徑或中心位置，來破壞圓柱週期性配置，證實可抑制陷阱模態的發生。最後本文用數值算例來證實本文方法之有效性。