Modal acoustic impedance of a finite-size spherical source immersed in an unbounded viscoelastic liquid

A modal impedance matrix is formulated for a spherical source undergoing axisymmetric oscillations in a viscoelastic acoustic medium. The formulation utilizes the governing equations of Verdier and Piau [1] and the analysis method of Hasheminejad and Geers [2]. A matrix based method is used to generate dilatational and translational surfaceforce curves. At first governing equations of the acoustic fields in the viscoelastic fluid are manipulated to yield the well-known Helmholtz equations and corresponding wave numbers. Equations of radial and meridional velocities and the excess temperature are also derived. In addition, the classical relations for radial and meridional stresses and the heat flux are written as potential based equations. Velocities and the excess temperature as well as the stresses and the heat flux at the surface of the source are expressed as Legendre series with unknown coefficients. The generalized Fourier series solution for the field equations with unknown coefficients are substituted in the potential based equations to find the matrix relations between the coefficients. Frequency dependant modulus and it’s relation with the viscosities for the single relaxation time fluid are adopted from Meister et al. [3]. Eventually the modal impedances for the two first modes are calculated and plotted as a function of distance from the source surface. Numerical results for immersion in glycerol as an example are obtained. As it will be explained in the paper, in a limiting case, results of the viscoelastic fluid lead to those of the viscous fluid, hence, as the validation, modal impedance results of the viscoelastic fluid are compared with the results of the viscous fluid of Hasheminejad and Geers [2] and good agreement between the results is observed. Considering the viscoelastic effects (i.e. frequency dependent moduli) can be regarded as the main contribution of the present work to the previous ones (e.g. [2]).