A novel continuous adjoint-based acoustic propagation method is proposed for low-noise turbofan duct design. A fan bypass duct tonal noise propagation model that is verified by comparison with an analytical solution of the modal radiation from a semi-infinite duct with the shear layer is enhanced with its continuous adjoint formulation, having been applied to design the bypass duct. First, this article presents the complete formulation of the time-dependent optimal design problem. Second, a continuous adjoint-based acoustic propagation method for two-dimensional bypass duct configurations is derived and presented. This article aims at describing the potential of the adjoint technique for aeroacoustic shape optimization. The implementation of the unsteady aeroacoustic adjoint method is validated by comparing the sensitivity derivative with that obtained by finite differences. Using a continuous adjoint formulation, the necessary aerodynamic gradient information is obtained with large computational savings over traditional finite-difference methods. The examples presented demonstrate that the combination of a continuous-adjoint algorithm with a noise prediction method can be an efficient design tool in the bypass duct noise design problem.