Sec. 50.6 



CALCULATION OF WAVEMAKING RESISTANCE 



215 



vortex theory, one might say that any method 

 actually used: 



". . . seems not to appeal readily to minds not thoroughly 

 trained mathematically, and gives rise to confusion 

 among practical men rather than serving to enlighten 

 them" [Proc. First Int. Cong. Appl. Mech., Delft, 1924, 

 p. 435). 



50.6 The Calculation of Wavemaking Resist- 

 ance. Having achieved a suitable expression for 

 the velocity potential which meets the continuity 

 and boundary conditions, much easier said than 

 done, there are several ways of using it to obtain 

 the answers desired. In the case of the resistance 

 due to wavemaking, at least six lines of approach 

 are open, defined briefly as follows: 



(a) The most direct and obvious method, con- 

 sidering that the pressure drag due to wavemaking 

 is indeed developed as a pressure directly on the 

 ship, is to integrate the longitudinal component 

 of the resultant liquid pressure over the hull. 

 This is equivalent, as T. H. Havelock says 

 [INA, 1934, p. 439], to obtaining "the combined 

 backward resultant of the fluid pressures taken 

 over the hull of the ship; but this is by no means 

 the simplest method for purposes of calculation." 



(b) Introducing artificial viscosity as a damping 

 effect, listed in assumption (h) of Sec. 50.4, or in 

 fact emplo3dng any artificial kind of liquid 

 resistance, means that the energy put into the 

 wave system has to be derived from the rate of 

 dissipation of energy in the liquid around the 

 body. This method, in the words of J. K. Lunde, 

 "... has certain important analytical advantages; 

 nevertheless, it is highly artificial ..." 



(c) By a direct application of the method of 

 energy and work, it is possible to calculate the 

 pressure drag due to wavemaking if the wave 

 pattern at a great distance to the rear of the ship 

 is known. As the hquid has no viscosity, all the 

 energy put into the wave system remains there, 

 and the wave pattern at that distance is free of 

 all the local disturbances produced by the ship. 

 This is, according to Havelock [INA, 1934, p. 

 440], the "most natural method" under the 

 circumstances. 



(d) By utilization of the Lagally Theorem and the 

 forces exerted between the boundaries of bodies 

 enclosing pairs of sources (or sinks) of equal 

 strength. J. K. Lunde explains this method as 

 follows ["On the Theory of Wave Resistance and 

 Wave Profile," Norwegian Model Basin Rep. 10, 

 Apr 1952, p. 17]: 



"It is known that the total force between two sources 

 and two sinlcs of strengths m and m', respectively, is given 

 by iiwpinm.' /r'^ where r is the distance between them 

 [Lamb, H., "Hydrodynamics," 5th ed., Art. 144, p. 138. 

 This result is not given in the later edition]. This force is 

 an attraction if m and m' are of the same sign, and a 

 repulsion when of opposite sign, that is, if one is a source 

 and one a sink. If we suppose a body to be at rest in a 

 uniform stream, we know that the resultant motion is 

 due to the stream itself together with the given sources 

 and sinks in the region outside the body whilst the effect 

 of the body is equivalent to an internal distribution of 

 sources and sinks. 



"Now Havelock has shown that the resultant forces and 

 couples on the body may be calculated from the forces on 

 the internal sources and sinks due to attraction and 

 repulsion between external and internal sources and 

 sinks taken in pairs" [Havelock, T. H., "The Vertical 

 Force on a Cylinder Submerged in a Uniform Stream," 

 Proc. Roy. Soc, Series A, 1928, Vol. 122, p. 387ff]. 



G. P. Weinblum adds the following comment: 



"When the velocity potential corresponding to a 

 source-sink distribution is known, the horizontal velocity 

 is also known, and the resistance X (the total resistance 

 Rt in standard notation) can be written down as the 

 integral of the product of the distribution and the hori- 

 zontal velocity over the region of the distribution" [TMB 

 Rep. 710, Sep 1950, p. 16]. 



(e) By the method of G. P. Weinblum, in which 

 the shape of the hull is expressed mathematically, 

 as described in Sees. 49.4 and 49.5. In general, 

 Weinblum's method differs from the others 

 mentioned here in that the velocity potential is 

 developed on the basis of equations expressing 

 the hull shape in mathematic terms, rather than 

 upon an array of sources and sinks, or upon the 

 slope of the hull surface with respect to the direc- 

 tion of motion. 



(f) By the method of R. Guilloton, in which the 

 hull is represented as a summation of simple 

 geometric bodies, in the form of wedges, and the 

 IVIichell velocity potential is used to calculate the 

 pressure disturbance of an elementary wedge. 

 The total velocity potential, obtained as the sum 

 of the component potentials, is then expressed 

 directly as a function of the hull shape as defined 

 by a table of offsets [Guilloton, R., INA, 1940, 

 Vol. 82, p. 69ff; 1946, Vol. 88, p. 308ff; 1948, 

 Vol. 90, p. 48ff; SNAME, I95I, Vol. 59, pp. 

 86-128; Korvin-Kroukovsky, B. V., and Jacobs, 

 W. R., ETT Stevens Rep. 541, Aug 1954, p. 4]. 



(g) IMore recently, Guilloton has developed a 

 new approach to the calculation of wavemaking 

 resistance utilizing the measurements of wave 

 profiles taken on towing models [Guilloton, R., 

 SNAME, Tech. and Res. Bull. I-I5, Dec 1953; 



