Trapped modes in a wave guide with a circular cylinder

Trapped mode in three-dimensional waveguide with parallel vertical walls a distance 2d apart, when there is a circular cylinder of radius placed symmetrically between them, is investigated. The potential is the solution of the three-dimensional Laplace equation in the fluid region between the cylinder and the parallel walls and the normal derivatives of potential are zero on both the walls and the cylinder. It is shown that for a cylinder of sufficiently small radius there exists a trapped mode, having a frequency close to the cutoff frequency, which is antisymmetric about the centerline of the guide and symmetric about a line through the center of the cylinder perpendicular to the centerline. The method used in this paper is to utilize the Fourier transform of complex form to the formula derived in the Appendix which can easily transform the solution in polar coordinates into the form useful for the rectangular coordinates. Thus, it is possible to express the unknown function in the solution in rectangular coordinates in terms of the coefficients of the solution in polar coordinates.