Shallow Water Effects On Low-Frequency Wave Excitation of Moored Ships
Authors: Grant Robert , Holboke Monica 
Published: October 2008

Abstract
This paper demonstrates how the bound wave, the spectrum of the second-order difference frequency force on an LNG carrier, and the moored ship's dynamic response increase with increasing period of the fundamental wave spectrum. The paper demonstrates the key engineering issues and concepts with analysis results for a conventionally moored LNG carrier. These issues are important because, unless they are understood, ship motions and mooring loads can be significantly underestimated.

Introduction
Irregular waves in deepwater contain small, low-frequency bound waves associated with the fundamental wave groups. As the fundamental waves encounter shallow water, the amplitude of the bound wave increases. This bound wave is sometimes called the "setdown wave" because it reduces a ship's under-keel clearance. Figure 1 shows a short timehistory of an irregular fundamental wave along with its associated low-frequency bound wave. The bound wave amplitude is generally positive when the fundamental wave amplitude is low, and is generally negative when the fundamental wave amplitude is high. The bound waves, and the associated low-frequency dynamic wave loads that excite the resonant surge, sway and yaw modes of moored, floating systems, are produced by non-linear interactions between the frequency components of the irregular fundamental wave.
This paper demonstrates that the bound wave amplitude, he spectrum of the second-order difference frequency (lowfrequency) forces on a typical LNG carrier, and the moored ship's dynamic response increase with increasing wave period. The paper demonstrates key concepts by presenting and interpreting the results of 2nd-order diffraction/radiation analysis, and non-linear, time-domain, ship motions/mooring analysis for a typical carrier with an LNG capacity of 138,000 cu m.
The authors encountered these issues while assessing the dynamic performance of alternative designs for an LNG marine terminal located offshore a coast that is exposed to persistent long-period (low-frequency) swells. The relatively shallow water and the long wave periods combined to produce low-frequency dynamic wave forces on the moored ship that produced resonant surge, sway and yaw motions, and mooring responses, that were remarkably high. We found that industryaccepted ynamic analysis tools that are routinely applied to moored structures in deepwater are not universally applicable to these shallow water conditions, and that they can lead to calculated responses that are very unconservative.

Ship & Mooring Characteristics
Figure 2 shows the ship in a conventional mooring consisting of 6 mooring dolphins outfitted with quick-release mooring hooks, and 4 breasting dolphins outfitted with rubber fenders. The mooring line stiffness is controlled by the standard 11m long nylon "tail" that connects each steel line segment to the hook. These nylon segments are flexible relative to the steel line that runs through the ship chocks to the on-board winches. Each mooring line is pre-tensioned to about 10% of its minimum breaking strength to pre-load the ship against the fenders.
Figure 3 shows the force (vs) offset curves for the fenders and for a typical mooring line. Both are highly non-linear. Rather than resist tension (positive displacement on Figure 3), the fenders gap, and rather than resist compression (negative displacement on Figure 3), the mooring lines go slack.