  The objective of the present study is to develop a basic understanding of the
interaction of shock waves with density inhomogeneities. We consider here the
particular instance of a planar air shock impinging on a spherical
water-droplet and discuss to what extent this interaction can lead to the
inception of cavitation inside the droplet. The effort centers on the early
phases of the interaction process during which the geometry and amplitude of
the propagating wavefront is modified by refraction and the subsequent internal
reflections at the droplet interface. The problem is analysed using both simple
ray theory and a 2-D multiphase, compressible hydrodynamic code (ECOGEN).
Within the context of ray theory, the occurrence of focusing is examined in
details and parametric equations are derived for the transmitted wavefront and
its multiple internal reflections. It is found that wave patterns predicted by
ray calculations compare extremely well with the more accurate numerical
solutions from simulations. In particular, it is shown that the internal
wavefront assumes a complex time-dependent shape whose dominant feature is the
existence of cusp singularities. These singular points are shown to trace out
surfaces that are the caustics of the associated system of rays. From the
singularities of the energy flux density of the refracted wave, the parametric
equations of the caustic surface associated to the $k$-th reflected wavefront
are deduced. As a consequence of the focusing process, the simulations show the
formation of negative-pressure regions in the internal flow field. These
low-pressure zones are identified as possible spots where cavitation may occur,
depending on the magnitude of the pressure reached. Finally, the numerical
results provide quantitative information on the dependence of negative pressure
peak upon incident-shock-wave strength.
