Bow and Stern Waves and Small-Wave-Ship Singularities 



DISCUSSION 



K. W. H. Eggers 



Institut fur Schiffbau der Universitdt Hamburg 



Hamburg, Germany 



Dr. Yim's approach is definitely unorthodox. But I feel that we should de- 

 viate from usual ideas still further on his line in order to have more simple 

 results. 



Dr. Yim tries to compensate for the ship's integrated wave system by in- 

 troduction of singularities at a peculiar position, namely, at vertical lines 

 through bow and stern only. He assumes a polynomial representation of the hull 

 form and then has to calculate derivatives at the stem, each higher derivative 

 giving rise to singularities of a corresponding high degree, producing even infi- 

 nite local wave height for the local component save for the first derivative. 

 Certainly any continuous hull form may be approximated arbitrarily close by 

 polynomials, but would such an approximation be adequate as well for the deriv- 

 atives in the sense under consideration here ? 



I propose the following modification of Dr. Yim's procedure: Let us as- 

 sume some spatial distribution of sources which, combined with its image above 

 the undisturbed free surface, will represent a double body composed of a ship 

 and its image in unbounded parallel flow, i.e., under zero Froude number. A 

 ship form generated like this by sources may be called an Inuid, even if the 

 sources are not necessarily on the centerplane. 



Now we should add to each source element a distribution of xx quadrupoles 

 on the vertical line extending from the source point Py down to infinity, and an 

 image system of negative intensity above. The intensity of this quadrupole dis- 

 tribution should be equal to the source strength times kg = g/c^ . This would 

 cancel the wave element generated by the source element. 



It is possible to perform closed form integration vertically. The expres- 

 sion for the potential G(P,Pi) at a point P due to a unit source element at Pj 

 then is 



G(P,Pi) = 1+ ^—- log 



z J - z + r 



XX 



This potential meets the linearized free surface condition; only the first two 

 terms determine the first-order wave elevation, which is entirely local. 



After integration over the assumed source distribution the contribution of 

 the third term will be singular only below some line marking the edge of the 

 horizontal extension of the source distribution. As quadrupoles do not contribute 

 to Kelvin-impulse, the volume of a body generated by the total system of singu- 

 larities should not differ much from that of the Inuid. The wave pattern due to 



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