  We develop a well-posedness theory for second order systems in bounded
domains where boundary phenomena like glancing and surface waves play an
important role. Attempts have previously been made to write a second order
system consisting of n equations as a larger first order system. Unfortunately,
the resulting first order system consists, in general, of more than 2n
equations which leads to many complications, such as side conditions which must
be satisfied by the solution of the larger first order system. Here we will use
the theory of pseudo-differential operators combined with mode analysis. There
are many desirable properties of this approach: 1) The reduction to first order
systems of pseudo-differential equations poses no difficulty and always gives a
system of 2n equations. 2) We can localize the problem, i.e., it is only
necessary to study the Cauchy problem and halfplane problems with constant
coefficients. 3) The class of problems we can treat is much larger than
previous approaches based on "integration by parts". 4) The relation between
boundary conditions and boundary phenomena becomes transparent.
