The Exciting Forces on Fixed Bodies in Waves 



By J. N. Newman^ 



General expressions, originally given by Haskind, are derived for the exciting forces on 

 an arbitrary fixed body in waves. These give the exciting forces and moments in terms 

 of the far-field Velocity potentials for forced oscillations in calm water and do not de- 

 pend on the diffraction potential, or the disturbance of the incident wave by the body. 

 These expressions are then used to compute the exciting forces on a submerged ellipsoid, 

 and on fk)ating two-dimensional ellipses. For the ellipsoid, the problem is solved using 

 the far-field potentials, and detailed results and calculations are given for the roll moment. 

 The other forces agree, for the special case of a spheroid, with earlier results obtained by 

 Havelock. In the case of two-dimensional motion the exciting forces are related to the 

 wave amplitude ratio A for forced oscillations in calm water, and this relation is used to 

 compute the heave exciting force for several elliptic cylinders. Expressions are also given 

 relating the damping coefficients and the exciting forces. 



Nomenclature 



A = wave amplitude 



A = wave-height ratio for forced oscillations 



(oi, 05, at) = semi-axis of ellipsoid 



B„ = damping coefficients 



C< = nondimensional roll exciting-force coefficient 



D, = virtual-mass coefficients, defined by equations (18) 

 and (19) 



g = gravitational acceleration 



h = depth of submergence 



i = y-1 



j = index referring to direction of force or motion 



i.(z) = spherical Bessel function, j„(2) = (^) J-h-'/M 



K = wave number, K = w'/g 



P, = functions defined following equation (17) 



R = polar coordinate 



», = velocity components 



(x, y, z) = Cartesian coordinates 



a, = Green's integrals, defined by equation (20) 



(3 = angle of incidence of wave system 



fi = polar coordinate 



P = fiuid density 



<Pj = velocity potentials 



01 = circular frequency of encounter 



Introduction 



In order to determine the exciting forces on a ship in 

 waves, it is necessary to know not only the hydrody- 

 namic pressure in the incident wave system, but also the 

 effects on this pressure field due to the presence of the 

 ship. In the linearized theory the undisturbed pressure 

 of the incident-wave system is well known for a given 

 plane progressive wave system, but the diffraction or dis- 

 turbance of this incident system due to the ship is gen- 

 erally very difficult to evaluate, and in fact it is neglected 

 in the so-called "Froude-Krylov" hypothesis. 



Recently Haskind [1]^ has derived expressions for the 



' David Taylor Model Basin, Washington, D. C. 



• Numbers in brackets designate References at the end of the 



exciting forces and moments on a fixed body, which do 

 not require a knowledge of the diffraction effects men- 

 tioned in the foregoing, but depwnd instead on the 

 velocity potential for forced oscillations of the body in 

 cahn water. Moreover, it is easily shown that the 

 as3maptotic characteristics of this velocity potential for 

 large distance from the body is sufficient to determine 

 the exciting forces for a given incident-wave system. 

 For many problems this as3Tnptotic potential is rela- 

 tively easy to obtain, compared to either the near field 

 forced-oscUlation potential or the diffraction potential, 

 and thus Haskind's relations are extremely v^uable. 

 For example, it is known that for a submerged body, 

 such as an ellipsoid, the potential in the far field can be 

 obtained, to first order of approximation, in terms of the 

 singularity distribution for the same body in an infinite 

 fluid, but the near-field potential requires the "image" of 

 the free surface inside the body. With this in mind, 

 asymptotic far-field potentials were recently used [2] to 

 study the damping of an oscillating submerged eUipsoid ; 

 to study the near field potential for the same problem 

 would probably require expansion in Lam6 functions, 

 and would certainly be extremely difficult. Thus it is 

 apparent that Haskind's relations jjermit the determina- 

 tion of exciting forces for bodies which would otherwise 

 be highly untractable. 



Since Haskind's relations are not well known, we shall 

 first present an outline of their derivation. As one il- 

 lustration of their use, we shall utilize the far-field ve- 

 locity potential of a submerged oscillating ellipsoid, as 

 derived in [2], to obtain the six forces and moments 

 acting on a fixed submerged ellipsoid in oblique regular 

 waves. For all but the roll moment, these results reduce 

 to expressions obtained by Havelock [3] for the special 

 case of a spheroid, or an ellipsoid of revolution. The 

 roll moment, which of course cannot be obtained for the 



