  The displacement field for three dimensional dynamic elasticity problems in
the frequency domain can be decomposed into a sum of a longitudinal and a
transversal part known as a Helmholtz decomposition. The Cartesian components
of both the longitudinal and transverse fields satisfy scalar Helmholtz
equations that can be solved using a desingularized boundary element method
(BEM) framework. The curl free longitudinal and divergence free transversal
conditions can also be cast as additional scalar Helmholtz equations. When
compared to other BEM implementations, the current framework leads to smaller
matrix dimensions and a simpler conceptual approach. The numerical
implementation of this approach is benchmarked against the 3D elastic wave
field generated by a rigid vibrating sphere embedded in an infinite linear
elastic medium for which the analytical solution has been derived. Examples of
focussed 3D elastic waves generated by a vibrating bowl-shaped rigid object
with convex and concave surfaces are also considered. In the static zero
frequency limit, the Helmholtz decomposition becomes non-unique, and both the
longitudinal and transverse components contain divergent terms that are
proportional to the inverse square of the frequency. However, these divergences
are equal and opposite so that their sum, that is the displacement field that
reflects the physics of the problem, remains finite in the zero frequency
limit.
