The dynamic response interaction of a vibrating offshore pipeline on a moving seabed is herein investigated where the pipeline is idealized as a beam vibrating on an elastic foundation. This problem is of relevance in offshore exploration where pipelines are laid either on or buried in the seabed. When such pipes carry oil and gas, the undulating topography of the sea floor and the internal motion of the fluid subject the entire structure to vibration due to bending forces and form the subject of our study. Our analysis revealed that in general, the seabed acts either as a damper or as a spring and in particular when we have sedimentation, the seabed geology permits the geomechanical property of the sediment cover to act only as a damper. As expected, external excitation will increase the response of these pipes for which an amplification factor has been derived. For soft beds, high transverse vibrations were dampened by increasing the internal fluid velocity whereas they became amplified for hard beds. These results are of contemporary interest in the oil/gas industry where deep sea exploration is now receiving significant attention.