An approximate-analytic expression of the eigenvalues of“mode decomposition matrix”for a short vertical array

The characteristics of “mode decomposition matrix” for a short vertical array in an idea waveguide model with the reflective coefficients of both sea-surface and bottom to be equal -1 are discussed, and the approximate-analytic expression of the eigenvalues λ_n with the parameters D/H (D is the length of array, H is the depth of sea water) and n are obtained. The eigenvalues λ_n. for a sea-surface short array is proportional to the (4n-1) power of α(πD/H), while the eigenvalues λ_n. for a sea-middle short array is proportional to α_2n-1. The theory derivation we have done in this paper can be used to explain why the ability of decomposing mode and matched-field processing are poor, and to improve the quality of mode decomposition and the accuracy of source localization when the depth of array is selected reasonably.