The Perfectly Matched Layer (PML) has become a powerful tool in computational underwater acoustics and elastodynamics. By employing complex coordinate stretching in the wavenumber–frequency domain, PMLs effectively attenuate outgoing waves from the physical domain and thereby provide an efficient means to truncate the computational domain. Although PMLs have been widely adopted in the finite element and finite difference communities, their use in semi-analytical solutions remains limited. A major challenge is that, when modal analysis is applied to the acousto-elastic domain with PML in a semi-analytical framework, the found modes are not orthogonal. This challenge formulates the main motivation of this research. On the other hand, the modes obtained from the discrete solution of the elastic layer with PML, based on the thin-layer method, do preserve orthogonality. Therefore, this thesis aims to understand the differences between the modal solutions of the semi-analytical and thin-layer methods in the elastic domains with PMLs, which may provide insights into the reasons why modes in the semi-analytical solution are not orthogonal to each other.

The main storyline of this thesis is developed through four cases with increasing system complexity. In the first case, the modes of the elastic domain are computed using both the semi-analytical approach and the thin-layer method (TLM), and the comparison demonstrates the equivalence of the two methods in the absence of PMLs. In the second case, the acoustic domain with PML is investigated using the semi-analytical approach, with emphasis on the polynomial order of the complex-stretching function. Mathematical derivations show that a zero-order polynomial induces discontinuities at the interface, leading to uneliminated boundary terms and perturbing modal orthogonality, while numerical results confirm that higher-order polynomials preserve the cross-orthogonality of modes, as well as the continuous slopes of the potential functions at the interface. In the third case, a quadratic complex-stretching function is employed, and the elastic domain with PML is analyzed using both approaches. The comparison reveals differences in eigenvalues and eigenvectors; finer TLM discretization yields increased matches between the two methods, but excessive discretization results in orthogonality violations. Finally, in the fourth case, the semi-analytical modes of the acousto-elastic domain with PML are studied. Propagating, evanescent, and Bérenger modes are identified, with cross-orthogonality preserved given sufficient integration points. Bérenger modes consistently arise in PML formulations and exhibit anomalous dispersion characteristics.

The main contribution of this thesis lies in revealing the influence of the polynomial order of the complex stretching function on the modes of the acoustic domain with PML. When a quadratic complex-stretching functions are employed, the numerical results suggest that the semi-analytical modes of the elastic or acousto-elastic domains with PML are orthogonal. Therefore, it is suggested that a positive value of polynomial order is recommended when computing normal modes of the acousto-elastic domain with PML. However, in the future, a systematic study on the influence of the polynomial order should be conducted for the elastic layer or acousto-elastic domain with PML.
Furthermore, the comparative study of modal solutions highlights the differences between the semi-analytical approach and the thin-layer method. The nature of modal solutions comes from the different formulations of the eigenvalue problem, leading to different eigenvalues and eigenmodes. For TLM, the over-discretization of the PML domain is not suggested due to the violated orthogonality, although the reasons behind that require further investigations.

Overall, this thesis advances the fundamental understanding of the modal basis of acoustic, elastic, and acousto-elastic layers with PML formulations, providing a foundation for future research in two main directions: (i) the study on modes of the acoustic layers coupled with multiple elastic layers with PML, which better represent realistic ocean environments with geological strata; and (ii) the computation of forced responses of structures in acousto-elastic layers with PMLs to model the pile-water-soil interactions using modal matching techniques.