Linear acoustics in fluids and solids and variational formulations

This chapter details the time-harmonic wave equation formulation for three types of linear media: fluids as well as elastic and poroelastic solids. Additionally, each material can attenuate the wave propagation via a visco-acoustic or visco-elastic extension. The solids can be isotropic and transversely isotropic. We also list the most extensively used types of acoustic pulses that can approximate typical transducers' behavior. Then we derive the weak formulations and detail the boundary and coupling conditions, which are fundamental to enable numerical modeling of complex systems. The numerical solution to the problem requires the truncation of the infinite domain, which we achieve by applying the Perfectly Matched Layer (PML) technique. We also describe the 2D-axisymmetrical formulation. Finally, we extend this axisymmetric description to the 2.5D axisymmetric formulation with a Fourier expansion in the azimuthal direction, enabling us to model borehole-eccentered tools.