PMC

A two-dimensional slice of the three-dimensional wave field for the homogeneous velocity field. The wave front of the solution is compared to the analytical solution of the eikonal equation (black). The analytic solution of the isotropic eikonal equation aligns with the wave front of the solution of the wave equation .

Three two-dimensional slices of the three-dimensional wave field of a homogeneous velocity field in the metric shown in . The arrows show the longest axis of the velocity metric. The wave front represents a circle in the metric given by . Note, that the wave front resembles an ellipsoid in the Euclidean metric with the semi-principal axes given by the tensor .

Cross-sectional view of atomic structure, energy propagation, stress, and particle velocity of porous NiTi samples after impact compression: (a) 3.0 ps; (b) 4.5 ps (the black chain line indicates the position of 95% maximum velocity, the black dotted line indicates the position of the shock wave front, the red solid line indicates the elastic–plastic transition point, the red dotted line indicates the shape of the front edge of the porous sample wave front, and the black solid line indicates the range of the front edge when the multi-space sample shock wave passes through the hole); (c) schematic diagram of the propagation path of the shock wave; (d) three-dimensional view of the wave proof shape at position a-a in (b) (the blue solid line in (d) is a meridian of the wave front shape).

Shear wave velocity is the rate of wave energy propagation as a function of time and distance, while the shear wave attenuation is the rate of wave energy dissipation as a function of distance. The shear wave velocity is unaffected by the shape of the wave front and the plane harmonic, cylindrical harmonic and cylindrical impulse excitations ((a)–(c)) result in the same shear wave velocity at a given frequency. The shear wave attenuation estimate, however, does depend on the nature of excitation since attenuation depends on the shape of the wave front of energy dissipation ((d)–(f)). Appropriate correction factors must be considered to correct for the cylindrical and spherical wave propagation. Shear wave ultrasound vibrometry methods use focused radiation force to move tissue and the geometry of such excitation can be approximated by a cylindrical source.