Definición del alcance

Suppose a time history, as given in …gure 5.28, of the wave elevation during a su¢cient long but arbitrary period:

¿ = N ¢ ¢t

The instantaneous wave elevation is supposed to has a Gaussian distribution and zero mean.

The amplitudes ³_an can be obtained by a Fourier analysis of the signal. However, for each little time shift of the time history one will …nd a new series of amplitudes ³_an. Luckily, a mean square value of ³_an can be found: ³²_an.

When ³(t) is an irregular signal without prevailing frequencies, the average values ³²_an close to !n will not change much as a function of the frequency; ³²_a is a continuous function.

The variance ¾²_³ of this signal equals:

The wave amplitude ³_an can be expressed in a wave spectrum S³(!n), which expression is de…ned by:

where ¢! is a constant di¤erence between two successive frequencies. Multiplied with ½g, this expression is the energy per unit area of the waves (see equation 5.79) in the frequency interval ¢! see …gure 5.31.

Figure 5.31: De…nition of Spectral Density

Letting ¢! ! 0, the de…nition of the wave energy spectrum S^³(!) becomes:

¯¯¯

¯S^³(!_n) ¢ d! = 1 2³²_an

¯¯¯

¯ (5.110)

and the variance ¾²_³ of the water surface elevation is simply equal to the area under the

spectrum: ¯

Figure 5.32 gives a graphical interpretation of the meaning of a wave spectrum and how it relates to the waves. The irregular wave history, ³(t) in the time domain at the lower left hand part of the …gure can be expressed via Fourier series analysis as the sum of a large number of regular wave components, each with its own frequency, amplitude and phase in the frequency domain. These phases will appear to be rather random, by the way.

The value ¹₂³²_a(!)=¢! - associated with each wave component on the !-axis - is plotted vertically in the middle; this is the wave energy spectrum, S³(!). This spectrum, S³(!), can be described nicely in a formula; the phases cannot and are usually thrown away.

Figure 5.32: Wave Record Analysis

Spectrum Axis Transformation

When wave spectra are given as a function of frequency in Hertz (f = 1=T) instead of

! (in radians/second), they have to be transformed. The spectral value for the waves, S³(!), based on !, is not equal to the spectral value, S³(f), based on f. Because of the requirement that an equal amount of energy must be contained in the corresponding frequency intervals ¢! and ¢f, it follows that:

jS^³(!) ¢ d! = S^³(f) ¢ dfj or: S³(!) = S³(f )

d!

df

(5.112) The relation between the frequencies is:

! = 2¼ ¢ f or: d!

df = 2¼ (5.113)

0

Figure 5.33: Wave Spectra on Two Di¤erent Bases Then the wave spectrum on an !-basis is:

S_³(!) = S_³(f )

2¼ (5.114)

An example of a spectrum transformation is given in …gure 5.33, in which it is obvious that the ratio between the corresponding frequencies is 1=(2¼) while the ratio between the corresponding spectral values is 2¼. The areas of both spectra (signi…cant amplitudes) remain equal and the spectral moments provide equal average periods.

Wave Height and Period

Relationships with statistics can be found from computing the moments of the area under the spectrum with respect to the vertical axis at ! = 0.

If m denotes a moment, then mn³ denotes the n^th order moment given in this case by:

¯¯¯

This means that m0³ is the area under the spectral curve, m1³ is the …rst order moment (static moment) of this area and m2³ is the second order moment (moment of inertia) of this area.

As has already been indicated, m0³is an indication of the variance squared, ¾²_³, of the water surface elevation. Of course this m0³ can also be related to the various wave amplitudes and heights:

¯¯¾³ = RMS = pm_0³¯¯ (Root Mean Square of the water surface elevation)

¯¯¯³a1=3 = 2 ¢pm0³¯¯

¯ (signi…cant wave amplitude)

¯¯H1=3= 4 ¢pm0³

¯¯ (signi…cant wave height) (5.116)

Characteristic wave periods can be de…ned from the spectral moments:

m_1³ = !₁¢ m0³ with !1 is spectral centroid

¯ (mean zero-crossing wave period) (5.118) The mean zero-crossing period, T2, is sometimes indicated by Tz. One will often …nd the period associated with the peak of the spectrum, T_p, in the literature as well.

Rayleigh Distribution

Expressed in terms of m0x, the Rayleigh distribution of amplitudes xa is given by:

¯¯¯

¾¯¯¯¯ (Rayleigh distribution) (5.119) in which x is the variable being studied and m0x is the area under the spectral curve.

With this distribution, the probability that the amplitude, xa, exceeds a chosen threshold value, a, can be calculated using:

P fx^a> ag =

As an example for waves, the probability that the wave height, Hw, in a certain sea state exceeds the signi…cant wave height, H₁₌₃, is found by:

P©

Wave Record Length

An important problem in the conduct of irregular wave measurements is the required total duration of the measured time histories to obtain proper spectral shapes and statistical values. This duration is presented by the total number of wave cycles, N.

Figure 5.34: E¤ect of Wave Record Length

Figure 5.34 shows an example of the ‡uctuations in several characteristic values derived from calculated spectra based on records containing varying numbers of wave cycles. Using a wave record of a sea with a period T1 t 6 s, calculations were carried out for various record lengths, expressed in a varied number of N cycles. The ratios of the calculated characteristics H1=3, T1 and T2 for N cycles and those for a very large number of cycles are presented in the …gure. It can be seen that these ratios become more or less constant for N > 50. This value can be considered as a rough standard for the absolute minimum required number of cycles. It can be seen from the …gure that a much larger number is required to produce results approaching the base line, where this ratio becomes 1.0.

According to the 17^th International Towing Tank Conference in 1984, N = 50 should be taken as a lower limit. Larger values are to be preferred and it is more usual to take N = 100 as the usual standard. A number N = 200 or above is considered excellent practice by the I.T.T.C. In practice, a record length equal to 100 times the largest expected single wave period in the irregular waves is often used as a safe standard. This means a wave record length of about 15 to 20 minutes.

As well as in‡uencing the outcome of spectral analyses in the frequency domain, the number of wave cycles, N , also a¤ects the validity of statistical quantities calculated in the time domain such as mean values, probability densities and distributions of extreme values.