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：本文研究了声波波导中在单一传播模式下的多频率反源问题。目标是为采用采样方法成像扩展源提供理论依据与高效算法。与现有基于空间变量积分的远/近场算子不同，本文引入了一种基于频率变量积分的多频远场算子。该远场算子的定义方式考虑了波导中特有的非线性色散关系。通过应用分解方法，建立了对源在波传播方向支撑集的严格表征。同时讨论了一种相关的基于分解的采样方法。研究表明，这些采样方法能够对源的纵向支撑进行成像。文中通过数值算例展示了采样方法的性能，包括一个对完整软声障碍物成像的示例。