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Twenty-First Symposium on NAVAL HYDRODYNAMICS
dynamic sinkage force and trimming moment on the vessel. Calculations have been completed for a standard Wigley hull. Once again, it is observed from the numerical results that the influence of the mud layer on these force components is such as to suggest that the equivalent overall water depth lies somewhere between that of the water layer alone and that of the total mud-water domain, depending on the physical properties of the mud.
During the last few years, there has been a renewed effort to develop a better hydrodynamic understanding of marine vehicles in water of finite depth. Such cases of interest include coastal regions, harbors, rivers, lakes, and ocean inlets. A particular case of concern is the effect of the overlaying mud layer in rivers on the hydrodynamics of fast moving craft (that is, at a depth Froude number greater than unity). Such an example is that of Australian river catamarans with a low keel clearance traveling in rivers and narrow channels with a movable muddy bed, with an average (length) Froude number of 0.65, and reported by Doctors, Renilson, Parker, and Hornsby .
The new physical feature of this problem, the subject of this paper, is the important effect of the underlying movable muddy sea bottom on the ship hydrodynamics. An example of recent work is that of Zilman and Miloh , which is applicable to slow ships (that is, moving in restricted water with a relatively low Froude number). They found that the compliance of the sea bottom has a profound effect on the hydrodynamic performance of such vessels when navigating in shallow water. In that work, the zero-Froude-number approximation was made; this resulted in the sea-surface being simplified and replaced by a rigid surface. Of course, the waves generated on the mud-water interface were modeled appropriately.
In the approximation used in that research, the mud was treated as a Newtonian viscous fluid. This approach had also been applied by Zilman, Miloh, and Kagan  to calculate the effect of mud viscosity on the added-mass and damping coefficients of two-dimensional ship cross sections undergoing periodic oscillations in the upper fluid layer.
This theoretical work on ship hydrodynamics in a two-layer environment was an outgrowth of earlier research in which the viscosity was ignored in both layers, but the effects of different densities were included. The results of such calculations were reported by Miloh, Tulin, and Zilman , where wave-drag calculations in a laterally unbounded domain were done.
More recently, the theory has been developed to the stage, where it is now analogous to the classical work of Michell  for a thin ship traveling in deep and inviscid water and that of Sretensky  for steady motion in a channel. That is to say, the traditional linearized free-surface conditions were used on both the sea surface and the mud-water interface by Zilman, Doctors, and Miloh . The mud was modeled as a linear viscoelastic substance. The numerical test cases presented in that publication were applicable to an air-cushion vehicle (ACV) traveling over the sea with a muddy bottom, an ACV traveling over mud alone, and a ship traveling in a sea with a muddy bottom. The computations demonstrated interesting effects of the compliance of the mud with anticipated results in limiting cases, such as when the mud had either extremely low or high viscosity or stiffness.
One of the outcomes of the current research, naturally, is an ability to predict the sinkage and trim of a vessel. In reference, firstly, to purely inviscid hydrodynamics, we should acknowledge the pioneering theoretical work of Tuck , in which the shallow-water approximation was employed, and Tuck , where the influence of the finite-width of the channel was included in the analysis. Other work, of an experimental or empirical nature, was reported by Tuck , Dand and Ferguson , Barrass [12 and 13], Ferguson, Seren, and McGregor , and Ferguson .
Returning more to the subject of the current investigation, the significance of a silt-covered sea bed on the maneuvering characteristics of large vessels has been demonstrated by Sellmeijer and Oortmerssen  in a towing tank. Further work on this subject was reported by Vantorre , while a survey paper by d'Angremond, Deelen, and Vantorre  also described a number of experiments in which the mud properties, such as density, viscosity, and its rheology in general, were shown to be important.