The coupled-amplitude technique for solving the Helmholtz equation has been developed in the context of coupled normal modes by researchers working in a variety of wave propagation problems. In this article it is shown that this approach is not dependent on modal expansions and first-order differential equations in range for the coupled amplitudes are derived without invoking normal-mode expansions. The relationship of this exact transformation to the parabolic approximation is analyzed and numerical methods for solving the coupled-amplitude equations are discussed. The usual range-stepping algorithms used to obtain an approximate solution to the Helmholtz equation are based on the parabolic approximation and restricted to the forward propagating component of the solution. A complete solution of the Helmholtz equation in an inhomogeneous medium must also include backpropagating waves, that is, waves scattered towards the source by inhomogeneities. The inclusion of such effects in a numerically feasible full-wave approach to acoustic propagation is a problem of continual interest in the acoustics of inhomogeneous media and in ocean acoustics. The method discussed in this article addresses this difficult problem.