This paper aims at introducing the concept of time-dependent polarization tensors (TDPTs) for the wave equation associated to a diametrically small acoustic inclusion, with constitutive parameters different from those of the background and size smaller than the operating wavelength. Firstly, the solution to the Helmholtz equation is considered, and a rigourous systematic derivation of a complete asymptotic expansion of the scattered field due to the presence of the inclusion is presented. Then, by applying the Fourier transform, the corresponding time-domain expansion is readily obtained after truncating the high frequencies. The new concept of TDPTs is shown to be promising for performing imaging. In particular, the optimization approach proposed by Ammari et al. (Ammari, H., Kang, H., Kim, & E. Lee, J.-Y. (2012)The generalized polarization tensors for resolved imaging. Part II: shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements. Math. Comp., 81, 839–860.) is extended to TDPTs. Numerical simulations are presented, showing that the TDPTs reconstructed from noisy measurements allow to image fine shape details of the inclusion.

中文翻译：

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通过与时间有关的极化张量重建小的声学夹杂物

本文旨在针对与直径较小的声学夹杂物有关的波动方程引入时变极化张量（TDPT）的概念，其本构参数不同于背景参数，并且尺寸小于工作波长。首先，考虑了亥姆霍兹方程的解，并提出了由于包含物的存在而对散射场的完全渐近展开的严格系统推导。然后，通过应用傅立叶变换，在高频被截断之后，很容易获得相应的时域扩展。TDPT的新概念显示出有望用于成像。特别是，Ammari等人提出的优化方法。（Ammari，H.，Kang，H.，Kim，＆E.Lee，J.-Y.（2012）用于分辨成像的广义极化张量。第二部分：从多静态测量中重建电磁夹杂物的形状和电磁参数。Math。Comp 。，第81卷，第839-860页）扩展到了TDPT。提出了数值模拟，表明从噪声测量中重建的TDPT可以对夹杂物的精细形状细节进行成像。