Eigenproblem for an ocean acoustic waveguide with random depth dependent sound speed

A recently developed deterministic FE model for range and depth dependent acoustic waveguides (Vendhan et al, J. Acoust. Am., 126, 3319-3326, 2010) may be extended to a medium with random properties. Such a model would require the eigensolution for a depth dependent waveguide at the far field of the FE domain. The aim of the present paper is to study the depth eigenproblem with random sound speed, which may be written as d²Z/dz² + (1/k¹_z)Z = 0 (1) where z denotes the depth coordinate, Z(z) the depth function and k_z the depth wavenumber given by k²_z = (ω²/c²(z)-k¹_r) (2) In Eq.2, r k denotes the radial wavenumber of a cylindrically symmetric waveguide and c(z) the sound speed which is assumed to be random variable in the form c(z) = c (z) (1+a) where a denotes a small random fluctuation of the sound speed with c(z)as the mean value. The depth eigenmodes of a deterministic isovelocity waveguide are adopted to set up a Rayleigh-Ritz approximation for the depth eigenproblem (see Eq.1) in the form [K]{ψ} = λ [M]{ψ} (3) Choosing a perturbation approach (Nakagiri and Hisada, Proc. Intl. Conf. on FEM, 206-211, 1982; Ghanem and Spanos, Stochastic Finite Elements: A Spectral Approach, Springer Verlag, 1991) an approximate solution may be written in the form of a Taylor series as [M] = [M] + α [M₁] + α²/2[M₂] (4a)λ = λ + αλ₁ + (α²/2)λ₂ (4b){ψ} = {ψ} + α{ψ₁} + α²/2{ψ₂} (4c) where an over bar denotes deterministic quantity.