V 

 WAVE RESISTANCE OF THIN SHIPS 



John V. Wehausen 

 University of California 



The theory of the interaction between a ship or similar body and the undulating 

 sea through which it moves has been developed by various approximations. In par- 

 ticular, in developing the theory of ship waves and wave resistance one neglects vis- 

 cosity in the general analytical formulation, later trying to take account of it in a patch- 

 work fashion. Even with this simplification, mathematical progress has been made only 

 in cases where one may linearize the boundary conditions on the water surface. The 

 resulting theory is valid only when the disturbance of the surface is small. In situa- 

 tions where one is dealing with waves generated by a moving body, this means that 

 some aspect of the body or its motion must be restricted in a suitable way. A 

 dimensional analysis will usually show one or more parameters whose vanishing leads 

 automatically to vanishing of the surface disturbance. In the case of steadily moving 

 ships and related submerged bodies, the restrictions inherent in the problems treated 

 analytically seem to fall into one of the following classes: 1. "thin" ships, approxi- 

 mating a vertical disc; 2. "flat" ships, approximating a horizontal disc on the free sur- 

 face; 3. bodies submerged so deeply that their influence on the surface is small; 4. sub- 

 merged "thin" wings of small angle of attack (in this case approach to the surface leads 

 to a further complication). In the case of oscillating bodies some statement about 

 smallness of amplitude is necessary. 



The theory of perturbations has been extended in recent years to surface- 

 wave problems by F. John [1], Stoker and Peters [1], Sekerzh-Zenkovich [1] and others. 

 It was our original intention to make the appropriate expansions in each of the cases 

 listed above, as a means of formulating the proper linearized problem as well as 

 approximations of higher order, and then to survey the existing status of theory and 

 experiment in each case. In the present paper, however, only the first part of this 

 program has been attempted, the rest being left for a later time. Even so, much of the 

 material has been gone over lightly because of the existence of several available sur- 

 veys. We mention specifically papers by Hogner [2], Wigley [1, 2, 3], Weinblum 

 [10], Havelock [10, 11], and particularly the comprehensive surveys by Lunde 

 [1, 2]. Inui [1] has recently outlined the achievements of Japanese scientists in this 

 field. It seems appropriate to remark that since Michell's original contribution [1], the 

 development of the "thin"-ship theory has been carried on until recent times almost 

 exclusively by T. H. Havelock and L. Sretenskii, the bridging of theory with experi- 

 ment and design almost exclusively by W. C. S. Wigley and G. Weinblum. 



I. Linearization by Perturbation 



Let us suppose that the y-axis is directed vertically upwards, the jc-axis to the 

 right and the z-axis coming out of the plane of the paper. We assume that the fluid 

 is bounded partly by a free surface and possibly also by solid boundaries. When the 

 fluid is at rest, the plane y zz will be taken to coincide with the free surface. Let 

 the equation of the free surface be represented by y = rj(x, z, t) and the surfaces of 



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