DALRYMPLE, R.A. AND LIU, P.L.-F.: “Waves over Soft Muds: A Two-Layer Fluid Model”, J. Physical Oceanography, Vol. 8, No. 6, pp 1121–1131 ( November 1978)
 MACPHERSON, H.: “The Attenuation of Water Waves over a Non-Rigid Bed” , J. Fluid Mechanics, Vol. 97, Part 4, pp 721–742 ( April 1980)
 HSIAO, S.V. AND SHEMDIN, O.H.: “Interaction of Ocean Waves with a Soft Bottom”, J. Physical Oceanography, Vol. 10, No. 4, pp 605–610 ( April 1980)
 MAA, J.P.-Y. AND MEHTA, A.J.: “Soft Mud Response to Water Waves”, J. Waterway, Port, Coastal, and Ocean Engineering, Vol. 116, No. 5, pp 634–650 ( September/October 1990)
 FODA, M.A., HUNT, J.R., AND CHOU, H.-T.: “A Nonlinear Model for the Fluidization of Marine Mud by Waves”, J. Geophysical Research, Vol. 98, No. C4, pp 7039–7047 ( April 1993)
 LUNDE, J.K.: “On the Linearized Theory of Wave Resistance for Displacement Ships in Steady and Accelerated Motion”, Trans. Society of Naval Architects and Marine Engineers, Vol. 59, pp 25–76, Discussion: 76–85 ( December 1951)
 WIGLEY, W.C.S.: “A Comparison of Experiment and Calculated Wave-Profiles and Wave-Resistances for a Form Having Parabolic Waterlines”, Proc. Royal Society of London, Series A, Vol. 144, No. 851, pp 144–159+4 plates ( March 1934)
University of Michigan, USA
Your potential flow upper layer ensures zero shear stress on the mud interface. How would you expect your conclusion to change if you added a boundary-layer at the interface to get the more realistic condition of high stresses?
We are grateful for Professor Schultz's discussion of our paper, as well as the other verbal discussion that took place after the presentation.
In summarizing our work, the mud layer is modeled as a viscoelastic substance and the flow is considered to be laminar. The concept of linearization of the problem, on the assumption of a small input disturbance, is quite realistic since the particle motions in both the mud and the water are small.
This smallness of motion implies that the boundary-layer effects at the mud-water interface would be unimportant. We therefore believe that the boundary layer in the water on top of the interface would not greatly affect the forces on the vessel—in the same way that the boundary layer in the bottom of a towing tank or a river (even in the case of relatively shallow water) is unlikely to have a significant influence on the predictions for the forces.
An interesting extension of this work would be to analyze a multilayer problem, so that the case of an indistinct interface could be studied. The principles are identical to those expounded in the paper, but the algebra and the corresponding computer program would be far more complicated.