In this paper, we establish new results for the uniform far-field asymptotics of the two-layered Green function (together with its derivatives) in two dimensions in the frequency domain. To the best of our knowledge, our results are the sharpest yet obtained. The steepest descent method plays an important role in the proofs of our results. Further, as an application of our new results, we derive the uniform far-field asymptotics of the scattered field to the acoustic scattering problem by buried obstacles in a two-layered medium with a locally rough interface. The results obtained in this paper provide a theoretical foundation for our recent work, where direct imaging methods have been developed to image the locally rough interface from phaseless total-field data or phased far-field data at a fixed frequency. It is believed that the results obtained in this paper will also be useful on its own right.

Publication:

SIAM Journal on Mathematical Analysis, Vol. 56, Iss. 3 (2024)

http://dx.doi.org/10.1137/22M1525910

Author:

Long Li

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.

Jiansheng Yang

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China.

Bo Zhang

LSEC and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China.

Email: b.zhang@amt.ac.cn

Haiwen Zhang

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.