The Influence of Tank Walls and Bottom Upon Low-Frequency Damping
Authors: Hearn G.E. , Liou S.Y. 
Published: October 2008

ABSTRACT
Open-water and in-tank theoretical predictions of the low frequency damping of four distinct structures are presented to demonstrate the sensitivity of the damping coefficients to tank wall influences and/or finite water depth effects. To include these effects, and to solve the open-water situation, an eigenfunction expansion matching technique is used to predict the required first order velocity potentials. Added resistance force predictions are performed using the near-field pressure integration analysis. The added resistance gradient (ARC) method is then used to predict the low frequency damping coefficients of the structures in open-water and in in-tank situations. The results presented are discussed in some detail.

NOMENCLATURE
Alj complex expansion coefficients D hemisphere and cylinder diameter Hl Hankel function of the first kind H Vertical distance from sea-bed / tank bottom to lowest underside point of structure h water depth J highest evanescent mode KI modified Bessel function of the second kind k superscript, k=1,2,.., 6 denoting surge, sway, heave, roll, pitch & yaw L highest harmonic propagating mode, or barge waterline or semi-submersible pontoon length mo real root of the dispersion equation mj imaginary roots of the dispersion equation _n inward normal vector on mean surface S Nk direction cosine in kth direction associated with _n p, q field point, source point R radius of matching(radiation) boundary r hemisphere and cylinder radius, or distance between generic points p and q Sb sea-bed surface Se side wall surface of wave tank Sf free-surface Sr matching boundary between inner and outer domain Sw wetted surface of structure T draught of barge U forward speed of structure W width of wave tank ?k complex amplitude of first order displacement in kth direction ? wavelength of incident wave ?I, ?O inner, outer fluid domains ? wave frequency ?e encounter wave frequency ?I, ?D incident, diffraction wave velocity potential ?k, ?k first order radiation velocity potentials

INTRODUCTION
Previously (1,2,3)* the surge related low frequency or wave drift damping coefficients have been calculated assuming infinite water depth and open-water conditions. Therefore, when comparing the theoretical predictions with the experimentally measured values it has been implicitly assumed that model scale and tank dimensions were such that tank influences were not important. Kaplan et al (4) suggested, in the presentation of the paper, that the difference between their predictions and Wichers' (5)experimental measurements of the low frequency damping of a 200,000 dwt tanker, was due to finite water depth effects at the lower frequencies. Alternatively, it is possible to attribute the differences to the use of the far-field radiation energy analysis approach to calculating the second order forces used to predict the low frequency damping. Certainly the apparent shift of the whole predicted low frequency damping transfer function to the right in Figure 2 of reference 4, which effectively attributes calculated damping values to higher frequencies, is consistent with our own experiences(2) of using a full 3D far-field based method.