Trapped modes in a waveguide with a long obstacle

Consider an infinite two-dimensional acoustic waveguide containing a long r ectangular obstacle placed symmetrically with respect to the centreline. We search for trapped modes, i.e. modes of oscillation at particular frequenc ies which decay down the waveguide. We provide analytic estimates for trapp ed mode frequencies and prove that the number of trapped modes is asymptoti cally proportional to the length of the obstacle.