A method of determining the propagating and the localized modes in an acoustic waveguide with continuously varying cross section is studied. For the rectangular and circular waveguides where the area of the cross section is dependent on the distance from the center of the waveguide, the wave equation with Dirichlet (pressure release) boundary condition can be transformed into an infinite set of coupled ordinary differential equations with variable coefficients. This set of differential equations has a first integral which is the flux of the wave and is constant along the length of the waveguide. If the cross section approaches a constant value at the two ends of the waveguide, then there are a finite number of propagating modes plus a number of localized modes. A technique for the numerical integration of the stiff ordinary differential equations resulting from the partial wave projection is discussed and applied to different examples.