Computatton of Shallow Water Shtp Mottons 
where a;:(7) is a real added mass and bi;(¢) a real damping coef- 
ficient, both frequency-dependent. 
Fj; is.the force in the ith mode due to the incident wave. If 
the latter is a pure sine wave of amplitude [, atanangle f to the 
x .axis, 1.e.. has equation 
phere oik(x cos B+y sin®B ) - ict (2. 6) 
where k = 27/wavelength, we can write (Tuck, 1970) 
(2: %) 
where T;, is the Froude-Krylov force per unit wave amplitude i.e. 
that obtained by integrating the incident pressure field over the equi- 
librium hull position, and T;_, is the correction due to diffraction of 
the incident wave around the (acce dh hull, This notation for the excit- 
ing forces is convenient in allowing us to display all hydrodynamic 
effects in the form of a 6x 8 matrix ee) Nene I ig ees (5 
Hes Os Ee, kite 0 Be 
The foregoing applies toa very general class of ship motion 
problems, and in particular is not yet subject to restrictions on the 
nature of the sea floor. However, bottom topography determines the 
dispersion relationship between k and o , and in the present work 
we assume the shallow water approximation in uniform depth h, 
namely 
o = ghk , (2. 8) 
which is valid only so long as kh <«< 1, a very restrictive condition, 
as we shall see. 
In addition, of course, the bottom topography has a profound 
effect on the numerical values of the frequency-dependent transfer 
functions Tj; . The whole difficulty in any ship motions calculation 
is in the computation of Tj; , since once these quantities are known, 
(2.1) is trivially solvable. In following sections of this paper we dis- 
cuss in detail various specifications of Tij and solve some of the 
resulting motions problems. 
First, however, it is of interest to provide a general summary 
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