Asymptotic solutions of the leaky modes and PML modes in a Pekeris waveguide

Leaky modes are useful to partially represent the continuous spectrum in open waveguides. The perfectly matched layer (PML) is a widely used technique for truncating unbounded domains in numerical simulations of wave propagation problems. When a Pekeris waveguide is terminated by a finite PML, it gives rise to three classes of modes corresponding to the trapped modes, the leaky modes and the modes that approximate the branch line integral along the Pekeris cut. In this paper, we derive high order asymptotic solutions for these modes in a Pekeris waveguide and a Pekeris waveguide terminated by a finite PML.