1933年, 荷兰科学家Marcel Minnaert发表的论文《气泡的声音及流水的响声来源》标志着Minnaert谐振理论研究的开始. Minnaert谐振不仅解释了流水声音的真正来源, 而且解释了气泡对声波的谐振特性. 对于水中的单个气泡, 其声波谐振的波长$\lambda$与其半径$r$的关系为$\frac{\lambda}{r}=460$. 于是人们可以利用声波的谐振特性达到以小尺寸控制大波长声波的目的. 而在传统的固体声学研究中往往要求固体的尺寸必须与声波波长相当. 因此气泡的这种谐振特点使得它作为最简单的一种谐振材料得到了广泛研究. Ammari, Davies, Yu[1]利用双球坐标研究了一对球形谐振子相互靠近时的谐振模式, 证明了谐振模式的主项由区域的电容量系数来决定.

本文利用$Li$[2]的能量迭代技巧研究谐振子为一般$C^{2,\alpha}$凸区域的谐振模式, 通过计算外区域上的电容量系数, 获得了谐振频率对内含物的凸性和曲率等几何参数与材料的密度和体积模量等参数的具体依赖关系. 本文从本质上克服了[1]中双球坐标的局限性, 所证明的结论对任意维数任意形状的内含物都成立.

In 1933, scientist Marcel Minnaert published the article $"On musical air-bubbles and the sounds of the running water"$ and set up the Minneart resonance theorey, which explains the sounds of the running water. For a single bubble in liquid, ts acoustic wavelength $\lambda$ and radius $r$ satisfies $\frac{\lambda}{r}=460$. The Minneart resonance is a low frequency resonance in which the wavelength is much larger than the size of the bubble which is different from solid acoustic. $Ammari, Davies, Yu$[1]examine how the resonant modes of a pair of spherical resonators behave as they are brought close together. [1] shows that the leading-order behavior of the resonant modes is determined by capacitance coefficients. We will use the iteration technique in [2] to research the resonant modes of a pair of $C^{2,\alpha}$ resonators. Then we can express the resonant frequency by convexity, curvature and volume of resonators and material contrast.