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The adiabatic range variation method of Pierce (1965) and Milder (1969) is reformulated, the approximate mode separation of the wave equation with gradual range dependence is carried out, and the first-order mode coupling terms are derived including the effect of range-dependent boundaries. The latter had not been taken into account by Milder. In order to compare the method with existing exact solutions, it is applied to a wedge-shaped isovelocity ocean with rigid ocean floor. Good agreement is obtained with the solution of Bradley and Hudimac (1970) for moderately large wedge angles, even when mode coupling is neglected. A computer programme is to be published later for solving the system of coupled range equations for realistic sound velocity profiles and arbitrary variations in range and ocean floor.