A theory is presented that permits the extension of the method of normal modes to guided‐wave propagation in a medium with properties varying slowly with horizontal coordinates in addition to varying with the vertical coordinate. The principal assumption is the neglect of coupling between normal modes. It is predicted that different frequencies in different modes follow different horizontal paths. A ray‐tracing method is described for computing these paths. The theory is then applied to the study of the dispersion of waves from an explosive source in shallow water with slowly varying depth.