Some simple formulae are derived, which are useful in investigating wave propagation in a variable stratified medium. It is shown that the rays in a two-dimensional medium with slowly varying thickness and structure may be described by a characteristic time. From this time the ray angles at any horizontal position may be calculated. The characteristic time formula is also derived from considerations of normal modes, and then from considerations of energy flux. The formula still applies when there are slow bends or twists in the medium, with no limits on final amplitude. The extension to three-dimensional guides is discussed. Some formulae for intensity in slowly varying media are also obtained. The ideas are applied to the propagation of sound (especially underwater), shear wave, and electromagnetic radiation, and the present limitations of the method indicated.