  We show that converging spherical and cylindrical shock waves may ignite a
detonation wave in a combustible medium, provided the radius at which the
shocks become strong exceeds a critical radius, R_c. An approximate analytic
expression for R_c is derived for an ideal gas equation of state and a simple
(power-law-Arrhenius) reaction law, and shown to reproduce the results of
numerical solutions. For typical acetylene-air experiments we find R_c~0.1 mm
(spherical) and R_c~1 mm (cylindrical). We suggest that the deflagration to
detonation transition (DDT) observed in these systems may be due to converging
shocks produced by the turbulent deflagration flow, which reaches sub (but
near) sonic velocities on scales >>R_c. Our suggested mechanism differs from
that proposed by Zel'dovich et al., in which a fine-tuned spatial gradient in
the chemical induction time is required to be maintained within the turbulent
deflagration flow. Our analysis may be readily extended to more complicated
equations of state and reaction laws. An order of magnitude estimate of R_c
within a white dwarf at the pre-detonation conditions believed to lead to Type
Ia supernova explosions is 0.1 km, suggesting that our proposed mechanism may
be relevant for DDT initiation in these systems. The relevance of our proposed
ignition mechanism to DDT initiation may be tested by both experiments and
numerical simulations.
