Aspects of a damped surface wave in the Fourier diamond spaces. New surface wave analysis method (SWAM)

Abstract

A damped surface wave (complex wave number K=K′+jK″) is described by its space-frequency representation S(x,ω)=exp(jK(ω)x). The wave-number-frequency representation Ksi(k,ω) is the spatial Fourier transform of S(x,ω). A Ksi-cut versus k is a k-Breit Wigner form (square modulus maximum for k=K′,K″ half-width). A Ksi-cut versus ω is then shown to be an ω-Breit Wigner form (square modulus maximum for ω=Ω′, with an Ω″ half-width). New interesting properties are found: the phase velocity is Cp=Ω′/K′ and the group velocity is Cg=Ω″/K″. The wave-number-time representation N(k,t) is the inverse frequency Fourier transform of Ksi(k,ω). N(k,t) is then shown to be N(k,t)=exp(jΩ(k)t). The space-time representation s(x,t) is obtained by inverse frequency (respectively, wave number) Fourier transform of S(x,ω) [respectively of N(k,t)]. The four spaces s, S, Ksi, and N are the four Fourier diamond spaces. Used on s(x,t) experimental datas, the two cuts properties are the basis of the surface wave analysis methods (SWAM). Those methods identify the complex K (respectively, Ω) using k-ARMA (respectively ω-ARMA) models. Using SWAM on a cylindrical shell, the A-wave is experimentaly fully characterized. The agreement with theoretical results is excellent.