Matthew A. Dzieciuch
Scripps Inst. of Oceanogr., IGPP-0225, UCSD, La Jolla, CA 92093
A new method employing Chebyshev polynomials to calculate the underwater acoustic normal mode equation was developed. The method is a spectral approach using Chebyshev polynomials as basis functions. This expansion has the advantage of being a particularly efficient and accurate representation of the normal modes (especially of the lower order) since they give an exceedingly good representation of narrow boundary layers such as the sound-speed profile that can undergo rapid changes near the surface. This representation reduces the size of the eigenvalue problem to be solved. Since the CPU time scales with N[sup 3], where N is the size of the matrix, any size reduction is an advantage computationally. This approach has a significant speed advantage over finite difference methods without sacrificing accuracy. A Chebyshev representation is usually remarkably close to the minimax polynomial that minimizes the maximum error implying high accuracy. Chebyshev polynomials are also useful for the calculation of quantities involving the integral of the mode function, such as modal group velocity, loss, and coupling coefficient matrices in non-adiabatic propagation environments. These quantities can be calculated easily and accurately given their spectral representations without introducing any further numerical error.