Abstract: The physical property of the material, the space structure and the boundary condition are three important parts of the guided-wave system. The effect of the boundary condition in elastic guided-wave systemis discussed. The author finds that the cardinality of the boundary condition set equals the cardinality of the real number set; the boundary equation and the governing equation construct the linear differential expression in Hilbert space, and the spectrums have affine law between two structures which are affine transformation each other. So that if the spectrumand the steady-state response of one structure of the set about affine structures can be solved, the spectrum and the steady-state response of the whole set are obtained simultaneously.

Key words: boundary condition set, Hilbert space, linear differential expression, affine transformation, affine structure

摘要： 材料的物理属性、结构的空间表现和系统的边界状态是波导系统的3个组成要素。讨论了弹性波导系统的边界状态在整个系统的作用。发现边界条件的集合是与实数集等势的集合; 边界条件方程与控制方程共同构成Hilbert函数空间中的线性微分表达式, 并且其具备的线性性质使得结构经过仿射变换后, 频谱之间具有仿射变换的规律。这些优异性质使得求解出仿射结构集合中某一结构的频谱, 就能够完全得到该集合中所有结构的频谱和稳态响应。

关键词: 边界条件集合, Hilbert函数空间, 线性微分表达式, 仿射变换, 仿射结构