Determination of the group and phase velocities from time–frequency representation of Wigner–Ville

Modeling the Cut-off Frequency of Acoustic Signal with an Adaptative Neuro-Fuzzy Inference System (ANFIS)

“…The analysis of acoustic signals scattered by an air-filled cylindrical shell immersed in water is a topic that has received large attention for several years [1][2][3][4][5][6][7][8]. In previous studies, the characterization of the scattering problem is mainly performed in the frequency domain.…”

“…The general form of the backscattered pressure field at normal incidence can be expressed as [7][8]23] (1)…”

“…This function is obtained by the relation [7][8]23] where is the Neumann factor ( n = 1, if n= 0;  n = 2, if n> 0), k=/c is the incident wave number and c is the phase velocity in water.…”

“…The calculation of the resonance frequencies of these waves have been made by the cancellation of the determinant D given by [8]: (5) where the 16 non-vanishing elements d ij can all be determined from the boundary conditions of the problem, and they have all been listed elsewhere [2]. The resolution of the equation D=0 allows to determine the different proper modes for each type of the symmetric and the anti-symmetric circumferential waves (S i and A i , i=0, 1, 2,…).…”

“…Once frequencies of resonances are determined, we calculate the difference ka between two successive resonance frequencies. The group velocity of the symmetric and the anti-symmetric circumferential waves for each frequency is given by [8]: ka c c g   (6) where ka the gap between two successive resonances.…”