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.