Relationship between the Green's matrix of SVD and the Green's function matrix of SVE for exterior acoustics

Abstract

In this paper, the principal objective is to study the physical meaning of the singular value decomposition (SVD) in exterior acoustics. The degenerate kernel and image method are employed to derive the Green's function. The Green's function can be represented by the singular value expansion (SVE). The Green's matrix describes the field of acoustic pressure to the strengths of sources on the surface of a body, which radiates or scatters sound. The matrix decomposed by the SVD technique resulted in a set of singular values and two sets of orthogonal singular vectors. The singular value relates to the radiation efficiency and the two sets of orthogonal unitary vectors describe field mode shapes and source mode shapes, respectively. In addition, the relationship between the unitary vectors provided by the SVD and the basis function provided by SVE is constructed. The acoustic radiation mode shape of a circular cylinder is obtained by using the SVD technique and is compared with the analytical solution by using the SVE. 本文主要目的在探討奇異值分解法在外域聲場中的物理意義。我們採用退化核函數及映射法推導得到格林函數，而這個格林函數同時可以奇異值展開式表示出。而格林函數離散後的矩陣可描述出一個輻射或散射物體的聲壓場與物體表面聲 源強度的關聯。格林函數矩陣藉由奇異值分解法可得一組與輻射效率有關的奇異值，及兩組分別描述與場有關及與原點強度有關的正交的矩陣。此外，由奇異值分解法得到的酉向量與由奇異值展開式得到的基底函數之間的關聯將予以銜接。同時，藉由奇異值分解法得到的一圓形長柱體的輻射模態將與由奇異值展開式得到的解析解作一比較。