Spectrum model

𝑛𝑛=1

cos (𝜔𝜔_𝑛𝑛𝑡𝑡+𝜃𝜃) (Eq. 4.4) Where the coefficients are

𝜉𝜉= �𝐴𝐴𝑛𝑛2 +𝐵𝐵𝑛𝑛2 (Eq. 4.5)

And the phase angles are given by

𝑡𝑡𝑎𝑎𝑛𝑛𝜃𝜃 =−𝐵𝐵𝑛𝑛

𝐴𝐴𝑛𝑛

(Eq. 4.6) (Eq. 4.4) represents the irregular record by the sum of an infinite number of sine waves of amplitude (𝜉𝜉) and frequency (𝜔𝜔_𝑛𝑛). These frequencies correspond to (Eq. 4.2).

4.2 Spectrum model

Waves are usually generated from wind, called wind induced waves (illustrated in Figure 36).

The wave formation are dependent on water depth, wind duration, width of area affected, the fetch length and the wind speed. There are at least two physical processes involved, these being the friction between air and water and the local pressure fields associated with the wind blowing over the wave surface. A great deal of work has been done on the theory of wave generation by wind, but no completely satisfactory mechanism has yet been devised to explain the transfer of energy from wind to sea according to ref. [8].

Since the wind is fluctuating the waves will also be fluctuating, but it is common to assume that the sea state is stationary for a duration of 20 minutes to 3-6 hours. A stationary sea state can be characterised by a set of environmental parameters such as significant wave height Hs and spectral peak period Tp. Ref. [15].

Figure 36 How waves develop, ref. [10]

Short term stationary irregular sea states may be described by a wave spectrum. A wave spectrum is the power spectral density function of the vertical sea surface displacement. The wave spectrum is often defined in terms of Hs and Tp. When the time increase the sea spectrum changes to a fully developed sea the wave spectrum grow wider, increasing the standard deviation (ϭ).

There are several spectrum formulas that are used in the design of offshore structures. These formulas are derived from the observed properties of ocean waves. The most appropriate wave 60

spectrum depends on the geographical area with local bathymetry and the severity of the sea state. The Pierson-Moskowitz (PM) spectrum and JONSWAP (Joint North Sea Wave Project) spectrum are frequently applied for wind seas.

Swells are generally of long period and comparatively regular. Locally generated wave systems may therefore be contaminated by swells generated by distant storms. Moderate and low sea states in open seas areas are often composed of both wind-sea and swell. Any local environmental effects, e.g. the possibility of swell/waves at the lift-off site, should be identified and considered according to ref. [14]. A two peak spectrum may be used to account for both wind sea and swell. The Ochi-Hubble spectrum and the Torsethaugen spectrum are two-peak spectra which account for the swell waves according to ref. [15]

Different spectrum models for the same energy content distributes the energy differently across the frequency band. Thus the response of the structure will be different if a different spectrum models are used according to ref. [1]

4.2.1

Spectral moments

If the time history has a zero mean and the number of observations is very large. A time history represented by (Eq. 4.4) therefore has a variance (𝑚𝑚₀). The variance of the irregular wave time history is equal to the area under the wave energy spectrum.

𝑚𝑚𝑛𝑛 = � 𝜔𝜔^{∞ 𝑛𝑛}𝜉𝜉(𝜔𝜔)

0 𝑑𝑑𝜔𝜔𝑁𝑁 (Eq. 4.7)

Where 𝜔𝜔 is the angular wave frequency, and n =0,1,2.. according to ref. [21].

4.2.2

Significant wave height

The significant wave height is a commonly used measure of the height of the ocean waves. The significant wave height (Hs) is traditionally defined as the average of the highest one third of all the waves (H1/3) in the indicated time period from the trough to crest (Eq. 4.8). Nowadays it is usually defined as four times the square root of the standard deviation of the surface elevation (Eq. 4.9). The different definitions is given bellow:

Deterministic definition:

The significant wave height is an important parameter during planning of marine operations.

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4.2.3

Spectral peak period and crossing period

The spectral peak period TP is the wave period determined by the inverse of the frequency at which a wave spectrum has its maximum value. The zero-up-crossing period TZ is the average time interval between two successive up crossings of the mean sea level (illustrated in Figure 37).

𝑇𝑇_𝑧𝑧= 2𝜋𝜋�𝑚𝑚0

𝑚𝑚₂

(Eq. 4.10)

Figure 37 Typical wave record at a fixed point with definition of terms, ref. [8]

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4.2.4

JONSWAP

The JONSWAP spectrum extends Pierson-Moskowitz (PM) to include fetch limited seas. Both spectra describe wind sea conditions that often occur for the most severe seastates according to ref. [21] . The PM-spectrum was originally proposed for fully-developed sea according to ref.

[15]. The PM is given below in (Eq. 4.11).

The JONSWAP spectrum (Eq. 4.12) is formulates as a modification of the PM spectrum for a developing sea state in a fetch limited situation.

𝑆𝑆𝐽𝐽(𝜔𝜔) =𝐴𝐴𝛾𝛾𝑆𝑆𝑃𝑃𝑀𝑀(𝜔𝜔)𝛾𝛾^{exp�−0.5�}

𝜔𝜔−𝜔𝜔_𝑝𝑝

𝜎𝜎−𝜔𝜔𝑝𝑝�²� (Eq. 4.12)

Where 𝛾𝛾 is a non-dimensional peak shape parameter, 𝜎𝜎 is the spectral width parameter (Eq.

4.13) and 𝐴𝐴_𝛾𝛾 is a normalizing factor (Eq. 4.17). Average values for the JONSWAP experiment data are 𝛾𝛾= 3.3, 𝜎𝜎𝑎𝑎 = 0.07 and 𝜎𝜎𝑏𝑏= 0.09 according to ref. [1] and [42]. The shape parameter (𝛾𝛾) typically varies between 1 and 7, and for 𝛾𝛾 = 1 the JONSWAP spectrum reduces to the PM spectrum (illustrated in Figure 38) according to ref. [15]

𝜎𝜎= 𝜎𝜎_𝑎𝑎 𝑓𝑓𝑐𝑐𝑟𝑟 𝜔𝜔 ≤ 𝜔𝜔_𝑝𝑝 𝜎𝜎= 𝜎𝜎_𝑏𝑏 𝑓𝑓𝑐𝑐𝑟𝑟 𝜔𝜔 >𝜔𝜔_𝑝𝑝

(Eq. 4.13) 𝐴𝐴𝛾𝛾 = 1−0.287 ln(𝛾𝛾) (Eq. 4.14) If no particular values are given for the peak shape parameter 𝛾𝛾, the following value may be applied:

𝑇𝑇_𝑃𝑃≤3.6�𝐻𝐻_𝑠𝑠→ 𝛾𝛾 = 5 𝑇𝑇𝑃𝑃≥5�𝐻𝐻𝑠𝑠→ 𝛾𝛾= 1

�𝐻𝐻𝑠𝑠3.6 <𝑇𝑇𝑃𝑃< 5�𝐻𝐻𝑠𝑠→ 𝛾𝛾= exp (5.75−1.15𝑇𝑇𝑝𝑝/�𝐻𝐻𝑠𝑠)

(Eq. 4.15)

Figure 38 JONSWAP spectrum for Hs=4.0 m, TP=8.0s for shape parameter 1, 2 and 5, ref. [15]

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4.2.5

Wave spreading

In ideal conditions in the open ocean all the waves travel in the same direction. However changes in wind direction, the influence of coastlines and bottom topography and the presence of waves generated elsewhere ensure that the true long crested one directional wave systems is rarity. It is much more likely that the real waves in the ocean travel in many different directions, although an easily recognized “primary” direction more or less aligned with the local wind may be discernible according to ref. [10].

The presence of more than one wave system results in enhancement and cancelation of wave crest and troughs, and this phenomenon gives rise to the term short crested to describe the appearance of a wave system with a spread of wave directions (illustrated in Figure 39) according to ref. [10].

𝑆𝑆(𝜃𝜃) =𝑆𝑆(𝜔𝜔)𝐶𝐶(𝜃𝜃) (Eq. 4.16)

The wave energy spectrum derived from a record of surface elevations obtained at a particular point in the ocean will invariably contain contributions from several different wave directions.

The wave spreading have a profound influence on some vessel motions particularly roll.

� 𝐶𝐶(𝜃𝜃) = 1 (Eq. 4.17)

Figure 39 Illustrates the principle behind wave spreading, ref. [10] and [3]

A common directional function often used for wind sea is 𝐶𝐶(𝜃𝜃) = Γ(𝑠𝑠+ 1)

√𝜋𝜋Γ �1 2 +n

2�cos^𝑛𝑛(𝜃𝜃 − 𝜃𝜃𝑝𝑝) (Eq. 4.18) Where Γ is the Gamma function and �𝜃𝜃 − 𝜃𝜃_𝑝𝑝� ≤^𝜋𝜋₂.

Short crested sea with spreading n = 2 used in the directional function, should be applied for operations that are independent of vessel heading according to ref. [20].

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