The computation of acoustic normal modes in oceanic or atmospheric waveguides is a delicate task, which involves determining the complex eigenvalues and eigenvectors of a variable-coefficient Helmholtz equation. The efficiency of an algorithm to perform such a calculation relies upon the effectiveness of its root finder. When dealing with semi-infinite media, only iterative methods can be utilized. In this cases, a major difficulty is associated with the need for reasonable first guesses of the sought eigenmodes. Since a priori estimations are generally unavailable, many existing codes may in some cases fail to converge. In this paper, a novel method for the computation of acoustic normal modes, which does not require any initial guess, is proposed. It consists of four different steps: the Helmholtz equation is first rewritten exactly in terms of an offset eigenvalue; it is then discretized by a multi-domain Chebyshev collocation method, which allows handling multi-layer waveguides; the resulting eigenvalue problem is quadratic and is transformed into a generalized linear eigensystem through the so-called companion matrix method (for instance, see "Differential eigenvalue problems in which the parameter appears nonlinearly", J Comp Phys 55 (1984), by T. J. Bridges and P. J. Morris); the latter problem is finally solved by the QZ algorithm. To illustrate the capabilities of the present technique, the results of three test cases are described and discussed.
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Sabatini et al. (2019) studied this question.