In this paper we investigate the existence of embedded trapped modes for symmetric
obstacles which are placed on the centreline of a two-dimensional acoustic waveguide.
Modes are sought which are antisymmetric about the centreline of the channel but
which have frequencies that are above the first cut-off for antisymmetric wave propagation
down the guide. In the terminology of spectral theory this means that the
eigenvalue associated with the trapped mode is embedded in the continuous spectrum
of the relevant operator.
A numerical procedure based on a boundary integral technique is developed to
search for embedded trapped modes for bodies of general shape. In addition two
approximate solutions for trapped modes are found;the first is for long plates on the
centreline of the channel and the second is for slender bodies which are perturbations
of plates perpendicular to the guide walls. It is found that embedded trapped modes do
not exist for arbitrary symmetric bodies but if an obstacle is defined by two geometrical
parameters then branches of trapped modes may be obtained by varying both of these
parameters simultaneously. One such branch is found for a family of ellipses of varying
aspect ratio and size. The thin plates which are parallel and perpendicular to the guide
walls are found to correspond to the end points of this branch.