This paper gives applications of the enclosure method introduced by the
author to typical inverse obstacle and crack scattering problems in two dimensions.
Explicit extraction formulae of the convex hull of unknown polygonal sound-hard
obstacles and piecewise linear cracks from the far field pattern of the scattered field at
a fixed wave number and at most two incident directions are given. The main new
points of this paper are: a combination of the enclosure method and the Herglotz wave
function; explicit construction of the density in the Herglotz wave function by using the
idea of the Vekua transform. By virtue of the construction, one can avoid any restriction
on the wave number in the extraction formulae. An attempt for the case when
the far field pattern is given on limited angles is also given.