Ogilvie 



these potentials far away from the ship (effectively at infinity) and from the re- 

 sulting simplified functions determine some of the damping coefficients. From 

 the same asymptotic forms of the potentials we can also find the forces on a 

 ship due to sinusoidal incident waves from any direction, without having to solve 

 the problem of determining the diffracted waves arovind the ship. In both prob- 

 lems we avoid the necessity of integrating the pressure over the ship hull. It is 

 only necessary to integrate over a simplified mathematical surface far away 

 from the ship. Finally, in any case for which we know the damping coefficients, 

 we can find the corresponding added mass coefficients. 



These relationships all depend on our use of a linear model to describe the 

 ship and fluid motions, but they do not depend on a specific mathematical repre- 

 sentation of the ship. In general, we shall be talking about the frequency-domain 

 equations of motion; the concept of "damping coefficient" has no meaning in the 

 time-domain equations which were developed in the last chapter. 



In order to use any of the relations proved in this chapter, we must be able 

 to find the velocity potentials for the oscillating ship problems, or at least the 

 far-field asymptotic forms of these potentials. Finding these functions requires 

 the assumption of a particular mathematical model for the ship, for the velocity 

 potentials cannot be found until we have formulated appropriate boundary condi- 

 tions for the whole problem, and this obviously requires some statements about 

 the flow near the ship. Two general methods of finding the velocity potentials 

 will be discussed in the following chapters. 



It may perhaps be argued that all of these relations are academic, for there 

 are several important gaps. To fill these gaps requires the integration of the 

 pressure over the hull, and thus the complete potential, including the compli- 

 cated local flow, must be considered. It then follows that perhaps one may as 

 well solve the whole problem by evaluating the local potential and integrating 

 pressure over the hull to find the forces. This may turn out to be true, but the 

 simplicity of using the far-field potentials is so attractive that I have considered 

 it desirable to present these partial results, hoping that someone may be able to 

 fill the gaps in an equally simple manner. 



Relation Between Added Mass and Damping Coefficients 



It was pointed out previously that the frequency-dependent parts of the added 

 mass and damping coefficients, as defined in (16), are proportional to the sine 

 and cosine transforms of a single function, K.^(t). From the theory of Fourier 

 transforms, it is well-known that, if Kj^(t) is well-enough behaved, either of 

 these transforms uniquely determines the inverse transform. Therefore, if 

 either transform is known, the function K.^(t) can be found, and from this the 

 other transform can be determined. 



In the language of the ship motions problem, this means that if we know 

 Mjk(^) fo^ ^^y single frequency and the damping coefficients for all frequencies, 

 we can obtain the added mass for any frequency. This result is sufficiently im- 

 portant that it deserves to be stated explicitly in formulas. 



40 



