D'Alembert solution of point-source shear waves and the coupled displacement field of dilation and torsion

ABSTRACT The analytical representation and underlying mechanism of displacement fields induced by point source are of notable engineering significance and value. The displacement field generated by a point source consists of an irrotational longitudinal wave induced by a dilatational source and a rotational shear wave excited by a torsional source. The D'Alembert solution accounting for geometric attenuation is applied to the formulation of spherical longitudinal waves from point sources and 2-dimensional (2D) transverse waves from point sources. Moreover, the relationship between this solution and the governing differential equation for 2D point-source shear waves is elucidated. By employing the stream function for 2D shear waves and the potential function for spherical longitudinal waves, both derived from a D'Alembert-type solution, and integrating relevant results from fluid mechanics, the complex potentials corresponding to displacement flux sources and displacement vortex sources are obtained. Based on the commonality between point-source vector elastic wave fields and other point-source vector fields encountered in engineering, the displacement field generated by point source is expressed—according to Helmholtz's theorem—as a superposition of an irrotational field and a solenoidal field. The irrotational field is excited by a displacement flux source, while the solenoidal field is generated by a displacement vortex source. The steady-state results obtained from the finite element method are consistent with those derived from the analytical solution, validating the correctness of the theoretical derivation. Consequently, the analytical representation and physical mechanism of the displacement field induced by the point source are established.