Unstable thermonuclear burning on accreting white dwarfs (WDs) can lead to a wide variety of outcomes, and induce shock waves in several contexts. In classical and recurrent novae, a WD accreting hydrogen-rich material from a binary companion can experience thermonuclear runaways, ejecting mass into the interstellar/circumbinary environment at ~1000 km/s. This highly supersonic ejecta drives shock waves into the interstellar gas which may be relevant for sweeping out gas from globular clusters or forming circumstellar absorption regions in interacting supernovae. While runaway nuclear burning in novae releases enough energy for these objects to brighten by a factor of ~10 4 over roughly a weeklong outburst, it does not become dynamically unstable. In contrast, certain helium accretion scenarios may allow for dynamical burning modes, in part due to the higher temperature sensitivity of helium burning reactions and larger accreted envelopes.

The majority of this thesis involves such dynamical burning modes, specifically detonations - shock waves sustained by nuclear energy release behind the shock front. We investigate when steady-state detonations are realizable in accreted helium layers on WDs, and model their strength and burning products using both semi-analytic and numerical models. We find the minimum helium layer thickness that will sustain a steady laterally propagating detonation and show that it depends on the density and composition of the helium layer, specifically 12 C and 16O. Though gravitationally unbound, the ashes still have unburned helium (~80% in the thinnest cases) and only reach up to heavy elements such as 40Ca, 44Ti, 48Cr, and 52Fe. It is rare for these thin shells to generate large amounts of radioactive isotopes necessary to power light curves, such as 56Ni.

This has important implications on whether the unbound helium burning ashes may create faint and fast peculiar supernovae or events with virtually no radioactivity, as well as on off-center ignition of the underlying WD in the double detonation scenario for Type Ia supernovae.