  This chapter is an introduction to the Bogoliubov theory of dilute Bose
condensates as applied to the study of the spontaneous emission of phonons in a
stationary condensate flowing at supersonic speeds. This emission process is a
condensed-matter analog of Hawking radiation from astrophysical black holes but
is derived here from a microscopic quantum theory of the condensate without any
use of the analogy with gravitational systems. To facilitate physical
understanding of the basic concepts, a simple one-dimensional geometry with a
stepwise homogenous flow is considered which allows for a fully analytical
treatment.
