This paper investigates the inverse source problem with a single propagating mode at multiple frequencies in an acoustic waveguide. The goal is to provide both theoretical justifications and efficient algorithms for imaging extended sources using the sampling methods. In contrast to the existing far/near field operator based on the integral over the space variable in the sampling methods, a multi-frequency far-field operator is introduced based on the integral over the frequency variable. This far-field operator is defined in a way to incorporate the possibly non-linear dispersion relation, a unique feature in waveguides. The factorization method is deployed to establish a rigorous characterization of the range support which is the support of source in the direction of wave propagation. A related factorization-based sampling method is also discussed. These sampling methods are shown to be capable of imaging the range support of the source. Numerical examples are provided to illustrate the performance of the sampling methods, including an example to image a complete sound-soft block.
翻译:本文调查了在声波制导的多个频率上单一传播模式的反源问题。目的是为使用取样方法的成像扩展源提供理论依据和高效算法。与取样方法中基于空间变量整体部分的现有远近实地操作员相比,根据频率变量整体部分引入了多频远地操作员。这个远地操作员的定义是将可能的非线性分散关系(波导中的一种独特特征)纳入其中。采用系数化方法是为了建立对范围支持的严格定性,这是源在波传播方向上的支持。也讨论了一种相关的因子化取样方法。这些取样方法已证明能够将源的射程支持成像。提供了数字示例,以说明取样方法的性能,包括一个将一个完整的声软块成像的示例。