## Direction Of Propagation

Fig. 1-62 Variation of particle velocity with depth below sea surface.

2. Large Volume Structures

Slender elements like those just considered do not influence the velocity and acceleration field around the structure. A large body (e.g., oil storage tank) will, however, reflect the waves producing disturbances to the velocity and acceleration field. An increase in velocities and accelerations around the body may result.

Consider for instance the structure shown in Figure 1-64, The forces on the towers may be calculated by applying the Morison's equation, but the water particle kinematics should be corrected for the presence of the caisson. This is illustrated in Figure 1-65. This effect may be approximately accounted for by applying the water depth above the caisson as total water depth in the calculation. Another more exact method is to calculate the water particle kinematics in the

h. : water depth 7\ : wave length

Fig. 1-63 Water particle trajectories.

fluid by means of diffraction theory, which takes into account the reflection of waves from the caisson.

The forces and moments on the caisson may be calculated in one of two ways:

1, By integrating the pressure resulting from the incoming wave and applying experimentally determined amplification factors to forces and moment.

2. By diffraction theory which takes into account the reflection of waves. This technique uses a large number of sinks

Fig. 1-64 Force calculation.

Fig. 1-64 Force calculation.

and sources distributed over the average wetted surface of the body.

The last method is preferred as the form of the structure could be more exactly taken into account than by the first method. It should, however, be mentioned that when using the diffraction theory, analytical solutions are not available for caisson forms other than cylindrical. Numerical solutions by means of computer are possible, and DnV has developed such a computer program for fixed and floating large bodies of any form 13. Some results are shown in Figure 1-66.

The maximum forces and moments on an offshore structure such as that shown in Figure 1-64 will occur when the wave position is as indicated in Figure 1-67. When drag forces are significant, the maximum horizontal force and

VMAX/ vVa'( H/a a)

Fig. 1-65 Velocity and acceleration distribution above base, overturning moment on the whole structure will occur at a wave position between the two shown. The exact wave position depends on the ratio of drag force to inertia force and the magnitude of the horizontal force on the caisson alone.

In order to minimize the total overturning moment and the horizontal force, the ratio of caisson height to caisson length should be chosen as small as possible. When this ratio is small enough, the caisson will have a stabilizing effect due to the moment contribution from the vertical wave force. This effect is illustrated in Figure 1-68.

• THEORETICAL (N.S.M.B. COMPUTER PROGRAM!
• experimental
• Dn V COMPUTER PROGRAM [ NV 459)

### top view; side view;

• THEORETICAL (N.S.M.B. COMPUTER PROGRAM!
• experimental
• Dn V COMPUTER PROGRAM [ NV 459)

top view; side view;

Wave Spectra

The wave spectrum most generally used by engineers in the last 10 years is the Pierson-Moskowitz spectrum proposed in 1964.14 A new spectrum which has not yet been extensively applied by engineers is the Jonswap spectrum. This spectrum is a result of the Joint North Sea Wave Observation Project-—a comprehensive international experiment undertaken in the North Sea off the Island of Sylt.15 The shape of the Jonswap spectrum is very different from that of the Pierson-Moskowitz

WAVE DIRECTION

 - -1- - — _ SWL --- --- i 1 1 i i

____ maximum vertical force maximum drag force

____ maximum vertical force maximum drag force maximum horizontal force on caisson — ■ maximum inertia force on tower.

### maximum overturning moment

1. 1-67 Wave profiles at maximum loads.
2. The major difference between these two spectra is that the jonswap spectTum is more peaked. When the original Pierson-Moskowitz spectrum is used for comparison, the Jonswap spectrum also contains more energy (Figure 1-69).

These two spectra cannot, however, represent the same sea state as the significant wave height for each is different due to the difference in the area beneath the spectra curves. Generally, it may be stated that when representing the same sea state, the Pierson-Moskowitz spectrum distributes the energy over a wider range of frequencies than does the Jonswap spectrum, provided that the areas beneath the spectra curves are equal (Figure 1-70),

The Jonswap spectrum and the original Pierson-Moskowitz wave spectrum with frequency f(hz) as a parameter may be expressed as:

E(f)=<Vg '(2ff) f exp.|- ^ (-■>-• a2 = 0.008

_ 0.07 for f «s fm 0.09 for f>fm f = Peak frequency

7 — peakedness parameter

7 =1 (Original Pierson-Moskowitz) j

exp.

See Figure 1-69

The modified Pierson-Moskowitz wave spectrum may be expressed as:

See Figure 1-70

Equation (35) is obtained from Equation (15) applying the following relationships:

id T

= 1.408 T (Valid only for Pierson-Moskowitz Spectrum) (37)

Which of these two spectra gives the best representation of reality depends obviously on the actual location. Analysis of

Fig, 1-69 Jonswap and original Pierson-Moskowitz wave spectra.

data recorded off the coast of Northern Norway do, however, strongly indicate that the Jonswap spectrum is the most representative for the North Sea and adjacent waters,15