Abstract: In two previous papers the author presented a generalized theory of local normal modes by introducing the method of slowly varying coefficients-a mathematical method to solve systems of first order linear differential equations with slowly varying coefficients, which are associated with problems of multi-coupled modes in irregular waveguides. However, the theory presented therein was not in a form to adapt itself to problems involving abrupt Changes of the waveguide parameters, which are inevitable in any practical transmission system. The purpose of the present paper is to consider such abrupt changes in the light of the above theory, so that the concept of local normal modes may be used to describe a broader class of problems.