6-Seakeeping - Ship Motions in Irregular Waves (2012)

the distance from the reference line to the crest of the wave ? is the apparent wave height, which is the vertical distance between a successive wave crest and wave trough Tz is the appa

SEAKEEPING –

SHIP MOTIONS IN IRREGULAR WAVES

Note: These notes are drawn from those issued by Dr Jonathan Duffy to students of JEE329 Seakeeping & Manoeuvring at

the Australian Maritime College Dr Duffy has used edited extracts from the main reference books for the subject and

which are listed at the end of the module

1 Irregular Waves

Ocean waves are not regular and are usually rather complex and extremely irregular The shape

of the ocean surface will constantly be varying as the wave velocity is a function of the wave length Therefore, the longer waves will be overtaking the shorter ones producing a continually varying surface There is a large body of measured ocean wave data, including wind speed, wind direction, wave height and wave length This data has been collected by oceanographic institutions and meteorological departments and various summaries have been produced for all oceans around the world For example, Global Wave Statistics Online provides worldwide coverage of wave climate in 104 sea areas, and an additional database providing a higher spatial resolution for the North European Continental Shelf The constantly varying ocean surface is not amenable to an exact mathematical definition Therefore statistical concepts are used to define an irregular seaway An example of an irregular wave is given in Figure 1

Figure 1 (a) Irregular seaway plotted to a base of time (b) Irregular seaway

plotted to a base of x (Bhattacharyya 1978)

H

Due to the random nature of irregular waves we need to establish a suitable method of describing the wave height and wave period Some of these parameters are explained below:

ζ is the instantaneous wave elevation from the reference line

ζ𝑎 is the apparent wave amplitude, i.e the distance from the reference line to the crest of the wave

𝐻 is the apparent wave height, which is the vertical distance between a successive wave crest and wave trough

Tz is the apparent zero-crossing period, which is the time between two upward or downward zero crossings

Tc is the apparent period, which is the time between two successive peaks (sometimes called the peak to peak period and denoted as Tp)

The mean zero-crossing period is the average of the apparent zero-crossing periods for multiple observations The mean apparent period is the average of the apparent periods for multiple observations The mean values of several readings are usually signified by a bar, e.g T z

The average wave height of an irregular seaway is the arithmetic mean of the heights of all the waves, for multiple observations When investigating ship motions we are often most interested

in the largest waves rather than the average, as these are the ones most likely to cause problems

A simple way of describing waves in this manner is to calculate the average height of the highest waves that make up 1/3rd of the total number of waves recorded This is called the significant

wave height and is denoted H 1/3 Sometimes the highest 1/10th waves are averaged and this is

denoted H 1/10

1.1 Histogram

A histogram can be used to describe the properties of an irregular seaway at a given time or at a given place An example histogram is shown in Figure 2 This histogram pertains to wave elevations for a collection of wave measurements Here the wave elevation has been split into bands of 1m For example, all records with a wave elevation between 0.5m and 1.5m are grouped together and the sum of records within this band are presented as a percentage of the entire population This is repeated for all other wave elevation bands Experience has shown that

a histogram of the wave elevations takes the shape of a Gaussian or normal distribution, as shown by the dotted line in Figure 2

Figure 2 Example of a wave elevation histogram (Bhattacharyya 1978)

A histogram may also be produced based on wave period An example is shown in Figure 3

Figure 3 Example of a wave period histogram

From wave measurements it has been found that a very irregular seaway will produce a low, relatively wide histogram, whereas a more regular seaway will produce a high, narrow histogram The location of the centre of gravity of the histogram in relation to the y axis gives the average value of the wave period (see Figure 3) It has been found from many ocean wave measurements that the theoretical Rayleigh curve fits the histograms for double amplitude (wave height) very well The Rayleigh distribution is given by Equation 1

𝑝 𝐻𝑖 =2𝐻𝑖

Where 𝑝 𝐻𝑖 is the probability density per metre or the percentage of times that any particular wave height 𝐻𝑖 will appear, with 0<p<1 If p=0 the wave height will never occur,whereas if p=1 the wave height will appear for every measurement Note that 𝐻 is the average wave height Using the Rayleigh curve the average of all wave heights squared is defined by:

𝐻 2 = 𝐻𝑖 2×𝑓 𝐻𝑖

Where

𝑓 𝐻𝑖 is the number of occurrences of 𝐻𝑖

2 Energy & Wave Spectrums

Another approach to describing an irregular seaway is to consider it as the sum of a number of regular sine wine waves of varying frequency and amplitude Therefore, regular wave theory can

be used to describe irregular ocean waves Figure 4 illustrates how an irregular wave can be described by considering it as a number of sine waves As can be seen, by superimposing 4 regular waves the result is a wave that is irregular both respect to wave height and period

Figure 4 Addition of sinusoidal waves to produce an irregular wave

(Bhattacharyya 1978)

2.1 Energy Spectrums

The total wave energy can be used to determine the severity of an irregular seaway This approach involves predicting the energy of the sinusoidal wave components that make up the irregular seaway Recall that the energy of a sinusoidal wave, per unit area of sea surface, is:

Therefore, the total energy per unit of area of sea surface for an irregular wave, with regular wave components of ζ𝑎1, ζ𝑎2, ζ𝑎3, ζ𝑎4, ,ζ𝑎𝑛, is given by:

𝐸𝑇 = 1 2 𝜌𝑔 ζ𝑎12 + ζ𝑎22 +ζ𝑎32 + ζ𝑎42 + ⋯ … + ζ𝑎𝑛2 (4) The ordinates of the energy spectrum, for each frequency bandwidth are given by:

1

Where 𝛿𝜔𝑤 is the bandwidth of wave frequency

The ordinates given by Equation 5 are plotted as a function of wave frequency to produce an energy spectrum For an irregular seaway made up of an infinite number of different regular waves the bandwidth approaches zero and the spectrum becomes a ‘continuous spectrum’

2.2 Wave Spectrums

Characteristics of irregular waves may also be presented using a wave spectrum The ordinates

of a wave spectrum are called the spectral density of wave energy and are denoted by the symbol

𝑆ζ(ωw) These ordinates are obtained by dividing the individual 1 2 ζ𝑎2 values at each frequency by the bandwidth (frequency interval on the x axis) Therefore the wave spectral ordinates for each frequency bandwidth are given by:

As can be seen, the wave spectrum has the same ordinates as the energy spectrum in the preceding section with 𝜌𝑔 divided out The wave spectrum is plotted such that the area under a part of the curve δωw wide is proportional to the energy per m2 of the sea surface for waves in that frequency band Thus the total area under the curve is proportional to the total energy per m2

of sea surface An example wave spectrum is shown in Figure 5

Figure 5 Definition of a wave spectrum (Lloyd 1998)

The area under the wave spectrum is defined as 𝑚0, which is given by:

Where

Sζ ωw is the spectral density (m2

s/rad)

ωwis the wave frequency (rad/s)

The total energy, per unit of sea surface area, can then be obtained from:

As a seaway is developed the wave energy spectrum and wave spectrum for a given point will change Imagine the situation when a wind starts blowing suddenly over a previously calm ocean In the early stages, for a constant wind speed, the waves are short Gradually the energy will be transformed into longer waves Eventually a fully developed sea is formed which is stable and does not change as the wind continues to blow Also, the maximum value of the energy spectrum shifts towards the lower frequency side as the seaway develops Figure 6 shows how the wave energy spectrum changes as the seaway develops

Figure 6 Energy build-up of partially and fully developed seas (Bhattacharyya 1978)

To define a seaway it is necessary to measure wave height and wave frequency over a suitably long period of time Although the wave pattern will never be repeated, the statistical characteristics of the seaway, for example the wave energy and wave spectrum, will remain the same The sinusoidal components that make up the irregular seaway are the same regardless of time and place and differ only in phase orientation from one record to another Therefore the energy of the wave system remains constant

Recall that a seaway record (histogram) for wave elevation is Gaussian The histogram for wave heights of an irregular seaway is considered to be close to a Rayleigh distribution However, the wave spectrum can be any functional form

The area under the wave spectrum can be used to estimate the significant wave height (double amplitude):

𝐻1

A wave spectrum can be used to determine the frequency at which the maximum energy is supplied, determine the content of energy at different frequency bands, determine the range of frequencies that are important for a given seaway and establish the existence of a low frequency swell The root mean square (RMS) value of the irregular wave height is 𝑚0 The significant wave amplitudes or other averages are multipliers of the RMS value From a known wave spectrum, based on wave amplitude, the wave amplitude and wave heights may be determined using the formulae given in Table 3

Average of 1/3rd highest waves 2.00 𝑚0 4.00 𝑚0

Average of 1/10th highest waves 2.55 𝑚0 5.09 𝑚0

𝑚𝑛is the nth moment of the area under the spectrum

Several wave characteristics can be obtained by taking moments about the vertical axis of the wave spectrum, 𝑚0, 𝑚1, 𝑚2, 𝑚3, 𝑚4 (similar to moments of a waterplane about one end) The average number of wave zero up-crossings per second, N0, can be obtained from moments of area under a wave spectrum, as follows:

N0= 1

2𝜋

𝑚 2

Also, the average zero crossing wave period, which is the average time interval between successive upward or downward crossings of the line representing the mean water level, can be obtained using:

The modal period, 𝑇0, corresponds to the frequency 𝜔0, which is the frequency that corresponds

to the peak of the spectrum Wave spectrums with different values of 𝜔0 are presented in Figure

7

Figure 7 Illustration of modal period (Lloyd 1998)

In general, the following factors influence the shape of a wave spectrum:

 Wind speed

 Wind duration

 Fetch (distance over which the wind blows)

 Location of other storm areas from which the swell may travel

The influence of wind on wave spectra is illustrated for an example case in Figure 8 As can be seen the peak of the wave spectrum shifts to the left and the spectrum becomes narrower at the higher wind speeds

Figure 8 Energy spectra of fully developed seas for various wind speeds

(Bhattacharyya 1978)

For swell dominated seas low frequency waves will be prominent, thus the peak of a wave spectrum for swell dominated seas will be at the lower frequency end of the spectrum and will be narrow However, wind generated seas give a wider band spectrum and the peak of the wave spectrum will be towards the high frequency end

Low frequency swell waves can propagate faster than the generating wind field and reach areas not influenced by this wind field or at least before the area is influenced directly by it This low frequency swell component will add to the locally generated wind sea and create double (or multiple) peak spectra Sea wave spectra can be quite complicated and be a result of several swell systems in addition to local generated waves Some spectral models have been developed

to give a realistic approach for double peak cases (Ochi and Hubble 1976; Torsethaugen 2004)

2.3 Idealised Wave Spectra

The wave spectrum derived from measurements at a particular place and time in the ocean will

be a unique result, which may not be representative of all conditions Although this may be a useful guide to likely wave conditions it has limitations for design purposes Therefore, when designing maritime structures it is customary to rely on families of idealised wave spectra Current practice is to use different formulae for open ocean and coastal conditions Some of the

idealised wave spectra are described below It should be noted that there are a number of additional wave spectrum formulations to what is presented here In the present notes only single peak spectrum formulations will be addressed

2.4 Open-Ocean Conditions

Bretschneider spectrum (ITTC two parameter spectrum)

The International Towing Tank Conference (ITTC) has adopted the Bretschneider spectrum for fully developed seas in open deep water This spectrum is often called the ITTC two parameter spectrum It is given by:

Pierson Moskowitz spectrum

The Pierson Moskowitz spectrum is applicable to fully developed seas and based on wave spectra measured on the North Atlantic Ocean The Pierson Moskowitz spectrum may be used to define a spectrum by a nominal wind speed at a height 19.5m above the sea surface It is given by:

JONSWAP spectral ordinates are given by:

𝑆𝐽ζ 𝜔𝑤 = 0.658 𝐶 𝑆𝐵ζ 𝜔𝑤 𝑚2/(𝑟𝑎𝑑/𝑠𝑒𝑐) (19) Where 𝑆𝐵ζ 𝜔𝑤 is the Bretschneider wave spectral density ordinate (Equation 17) The factor C

2 and

𝛾 =0.07 for 𝜔 <2𝜋

𝑇0and 𝛾 = 0.09 for 𝜔 >2𝜋

𝑇0Figure 9 shows a comparison between the JONSWAP and Bretschneider wave spectra for a significant wave height of 4 metres and a period of 10 seconds

Figure 9 Comparison between JONSWAP and Bretschneider wave spectra: significant wave

height = 4.0m and modal period 10s (Lloyd 1998)

2.6 Effect of Ship Velocity on Encounter Spectrum

Due to the ship velocity the encounter frequency of the waves will be different to the wave frequency Therefore it is necessary to change the wave frequency in the wave spectrum into the encounter frequency

This can achieved as follows As mentioned earlier in the notes the encounter frequency in regular waves is given by:

Equation 26 can be used to convert the wave frequency to the encounter frequency

2.7 Ship Response in an Irregular Seaway

The response of a ship in an irregular seaway may be described by using statistical measures similar to those used to describe irregular waves For example the response may be expressed as

an average, average one third highest (significant), average one tenth highest or average one hundredth highest

A histogram can be used to describe the properties of ship response in an irregular seaway at a given time or at a given place Recall that the theoretical Rayleigh curve can be used to fit the histograms for double amplitude wave height Similarly the theoretical Rayleigh distribution can

be used to approximate the histogram for double amplitude motion The theoretical Rayleigh approximation for double amplitude ship motion response is:

𝑝 𝑥𝑖 =2𝑥𝑖

𝑥 2 𝑒−𝑥𝑖2 𝑥 2

(27)