Longitudinal motion of focused shear wave beams in soft elastic media

Abstract

Shear waves are employed in medical ultrasound imaging because they reveal variations in viscoelastic properties of soft tissue. Frequencies below 1 kHz are required due to the substantially higher attenuation and lower propagation speeds than for compressional waves. Shear waves exhibiting particle motion in the direction of propagation, referred to as longitudinal shear waves, can be generated with longitudinal motion of a circular disk on the surface of a soft elastic medium. This approach permits imaging of the longitudinal shear wave with a conventional ultrasound transducer that is coaxial with the source of the shear wave. Presented here is a mathematical model describing the complete wave field generated by displacement at the surface of an isotropic elastic half-space. Numerical simulations are shown for longitudinal, transverse, torsional, and radial source polarizations, with emphasis on focused longitudinal shear waves. Predictions are consistent with measurements of light beams revealing that the longitudinal electric field component produces a smaller focal spot than the transverse field component [Dorn, Quabis, and Leuchs, Phys. Rev. Lett. 91, 233901 (2003)]. Simulations are compared with preliminary measurements of a focused longitudinal shear wave beam generated in a soft tissue phantom by longitudinal motion of a spherically concave piston.