This work deals with the ultrasonic wave propagation in the cortical layer of long bones which is known as being a functionally-graded anisotropic material coupled with fluids. The motivation arises from mechanical modeling of the ultrasound axial transmission technique in vivo for cortical long
bone which is known as being a functionally-graded anisotropic material. The proposed method is based on a combined Laplace-Fourier transform which substitutes a problem defined by partial differential equations into a system of differential equations established in the frequency-wavenumber domain. In the spectral domain, as radiation conditions may be exactly introduced in the infinite fluid halfspaces, only the heterogeneous solid layer needs to be analyzed using finite element method. Several numerical tests are presented
showing very good performance of the proposed approach.

Keywords:  Spectral finite element,  transient wave,  ultrasound,  anisotropic,  vibroacoustic,   cortical bone,  axial transmission.