This paper deals with a wave process induced by an acoustic impulse
in a vertically
inhomogeneous half-space. The solvability of the corresponding direct initial
boundary value
problem is proved. We also consider a class of one-dimensional inverse
problems including,
in particular, the classical inverse problem of the scattering theory,
the transmission and
reflection problems of seismology, etc. It is shown that
all these inverse problems are equivalent
in a certain sense, and therefore can be solved by identical methods. Theoretical
results are
illustrated by numerical examples.