Research PaperNo Access

Cross Orthogonality Between Eigenfunctions and Conjugate Eigenfunctions of a Class of Modified Helmholtz Operator for Pekeris Waveguide

Jianxin Zhu

Department of Mathematics, Jinan University, Guangzhou 510632, P. R. China

E-mail Address: zjxmath@163.com

Search for more papers by this author

and

Ying Zhang

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, P. R. China

E-mail Address: zhangyingmath@zju.edu.cn

Search for more papers by this author

https://doi.org/10.1142/S2591728518500482Cited by:2 (Source: Crossref)

Abstract

The eigenfunctions of the modified Helmholtz operator have no orthogonality in a bounded domain with a perfectly matched layer (PML), which makes it difficult to calculate the coordinates under two different local bases when the marching algorithm is applied. In this paper, we derive the conjugate eigenfunctions of the operator and discuss the cross orthogonality between the eigenfunctions and their conjugate eigenfunctions. On the other hand, we derive a simple formula for calculating the coordinates under the local base. The numerical results indicate that this method is effective.

Keywords:

References

1. F. Jensen, W. Kuperman, M. Porter and H. Schmidt, Computational Ocean Acoustics, 2nd edn. (Springer-Verlag, New York, 2011). Crossref, Google Scholar
2. J. Zhu and Y. Y. Lu, Asymptotic solutions of the leaky modes and PML modes in a Pekeris waveguide, Wave Motion 45(3) (2008) 207–216. Crossref, Google Scholar
3. Y. Y. Lu and J. Zhu, Perfectly matched layer for acoustic waveguide modeling benchmark calculations and pertubation analysis, Comput. Model. Eng. Sci. 22(3) (2007) 235–247. Google Scholar
4. J. Zhu and G. Wang, High-precision computation of optical propagation in inhomogeneous waveguides, J. Opt. Soc. Am. A 32(9) (2015) 1653–1660. Crossref, Google Scholar
5. Y. Y. Lu and J. Zhu, A local orthogonal transform for acoustic waveguides with an internal interface, J. Comput. Acoust. 12(1) (2004) 37–53. Link, Google Scholar
6. M. A. Ainslie, D. D. Ellis and C. H. Harrison, Low frequency bottom reverberation in a Pekeris waveguide with Lambert’s rule, J. Comput. Acoust. 24(2) (2016) 1650001. Link, Google Scholar
7. J. P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114(6) (1994) 185–200. Crossref, Google Scholar
8. F. Collino, Perfectly matched absorbing layers for the paraxial equations, J. Comput. Phys. 131(1) (1997) 164–180. Crossref, Google Scholar
9. C. C. Weng and W. H. Weedon, A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates, Microw. Opt. Technol. Lett. 7(13) (1994) 599–604. Crossref, Google Scholar
10. W. C. Chew, J. M. Jin and E. Michielssen, Complex coordinate stretching as a generalized absorbing boundary condition, Microw. Opt. Technol. Lett. 15(6) (1997) 363–369. Crossref, Google Scholar
11. F. Collino and P. Monk, The perfectly matched layer in curvilinear coordinates, SIAM J. Sci. Comput. 19(6) (1998) 2061–2090. Crossref, Google Scholar
12. M. Lassas and E. Somersalo, On the existence and convergence of the solution of PML equations, Computing 60(3) (1998) 229–241. Crossref, Google Scholar
13. E. Turkel and A. Yefet, Absorbing PML boundary layers for wave-like equations, Applied Numer. Math. 27(4) (1998) 533–557. Crossref, Google Scholar
14. C. X. Xu, S. C. Piao, S. E. Yang, H. G. Zhang and L. Li, This submission is for special issue on underwater acoustics: Perfectly matched layer technique for parabolic equation models in ocean acoustics, J. Comput. Acoust. 25(1) (2017) 1650021. Link, Google Scholar
15. L. Moheit and S. Marburg, Infinite elements and their influence on normal and radiation modes in exterior acoustics, J. Comput. Acoust. 25(4) (2017) 1650020. Link, Google Scholar
16. M. Aslan, K. B. Smith and G. Moss, Sensitivity analysis of hybrid split-step Fourier/finite difference parabolic equation models, J. Theor. Comput. Acoust. 26(2) (2018) 1850006. Link, Google Scholar
17. J. Wang, X. Meng, H. Liu, W. Zheng and Z. Liu, GPU elastic modeling using an optimal staggered-grid finite-difference operator, J. Theor. Comput. Acoust. 26(2) (2018) 1850005. Link, Google Scholar
18. L. Chen, S. Marburg, H. Chen, H. Zhang and H. Gao, An adjoint operator approach for sensitivity analysis of radiated sound power in fully coupled structural-acoustic systems, J. Comput. Acoust. 25(1) (2017) 1750003. Link, Google Scholar
19. J. A. Desanto, Scalar Wave Theory: Green’s Functions and Applications (Springer-Verlag, Berlin, 1992). Crossref, Google Scholar
20. D. C. Stickler and E. Ammicht, Uniform asymptotic evaluation of the continuous spectrum contribution for the Pekeris model, J. Acoust. Soc. Am. 67(6) (1980) 2018–2024. Crossref, Google Scholar
21. E. Ammicht and D. C. Stickler, Uniform asymptotic evaluation of the continuous spectrum contribution for a stratified ocean, J. Acoust. Soc. Am. 76(1) (1984) 186–191. Crossref, Google Scholar
22. V. V. Shevchenko, The expansion of the fields of open waveguides in proper and improper modes, Radiophys. Quantum Electronics 14(8) (1971) 972–977. Crossref, Google Scholar
23. J. Hu and C. R. Menyuk, Understanding leaky modes: Slab waveguide revisited, Adv. Opt. Photon. 1 (2009) 58–106. Crossref, Google Scholar