Computatton of Shallow Water Shtp Mottons 
Surge is rather different, in that the mooring provides the 
only restoring force (4.6). If we assume, as is clear from the re- 
sults of the previous section, that all hydrodynamic effects on surge 
are negligible, we can analyse linear surging as a simple one-degree- 
of-freedom undamped spring-mass system, with the result that the 
surging amplitude is 
Fi 
Ge 2 Sa (4. 9) 
1 Zi 
k - Mo 
or (neglecting diffraction) 
AL: 
10 
Sel ke ga (4. 10) 
] 0 2 
k - Mo 
where aan = Eu is obtained from (AIII. 5) . 
In fact the surging amplitude in the presence of a mooring is 
simply equal to the factor 
k | 
0 1 
| (4, 11) 
2. 2 
Mo -k 1-0, /o | 
times the free surging amplitude, where Cia k/M is the resonant 
frequency. Figure 4.2 shows this factor as a function of frequency 
o . Note that unless the wave frequency is less than 70% of the 
resonant frequency 7p = k/M , the effect of the mooring is to in- 
crease the motions. For large ships, conceivable values of op cor- 
respond to periods of minutes or more, so that typical sea or swell 
gives frequencies well above resonance ; however (Wilson 1959) long 
period range action in harbors can produce resonance, with disas- 
trous effects. The condition « <70%o, is in general met only by 
tides and currents, and indeed the purpose of the moorings must be 
to overcome these very low frequency excitations. 
On the other hand if o >>op, itis clear that the mooring is 
having very little effect on the surge motion of the ship, which moves 
as if free. The force exerted on the mooring by the ship is then of 
prime interest, and this may simply be computed by assuming given 
free ship motions. This also applies of course to motions in other 
modes (e.g. sway), so long as the wave frequency is again well above 
1S59 
