ARCI-BEF
On the Truncation of Spectra
by
RE.D. Bishop
and
W.G, Price
Department of Mechanical Engineering University College London.
Lab. y. Scheepsbous1kAiIe
Technische Hogeschool
Summary
A rational method of truncating a Pierson and Moskowitz wave spectrum of wave elevation is explained. The fourth
order mouient of the spectrum,
(w)dw
o
is rendered finite and its value is given, once the cut-off frequency has been selected. The implícations of the theory as regards bandwidth parameter are examined and the particular features of the ITTC and ISSC formulations are discussed,
Introduction
The probabilistic theory of ship dynamics is heavily dependent on the possession of wave spectra. Theoretical wave spectra used in ship and offshore response calculations are almost invariably of the form suggested by Pierson andNoskowîtz
(1963) for fully developed waves
-A
= 5e
CA)
Here (w) is the mean square spectral density function of wave elevation (t) and c is the wave frequency. The constants are
A 8.1xl03g2 , B =
Q,74()4
where V is the wind speed measured 19.5m above the level of calm water, Other definitions of these constants exist such as in the ITTC and ISSC spectra but an exact form of the constants is not essential for the analysis to be discussed here.
The nth moment m of a spectrum is defined by
=
1w
(w)dw,n J
o
Interest centres on these moments because they are simply related to certain statistical properties of the wave random
process.
Thus the mean square values of elevation, velocity of elevation and acceleration of elevation areOEA in = i (w)dw o J t;; o J 2 .2 = w (w)dw = <.r (t)> o = 1
w4(w)dw
<2(t)> =where the angular brackets represent time averaging over the typical' realisation of the ergodic processes concerned. Again the 'spectral bandwidth parameter' is c where
2
2
ni2
c
=1
and there are many other parameters that may be cited. The theory underlying these results is given by Price and Bishop (1974).
It is an unfortunate and an awkward feature of Pierson
and Moskowitz spectra that ni4 is infinite and this presents the engineer with a real difficulty. For although it can of course be argued that some other representation should he employed, the fact remains that most of the spectra used in presenting wave data are of the Pierson and Moskowitz type.
On physical grounds, however, one would not expect such a spectrum to remain finite all the way from w=O to
w-4,
if onlybecause waves of extremely short wavelength are not propagated under the influence of gravity but, rather, by surface tension. If (w) is truncated at some values of w, then ni4 will remain finite; but
the questions then arise:
what truncation should be imposed? what is its effect on
Tickell (1977), for instance, notes that some cut-off frequency is usually used to overcome the difficulty but adds that "a rational choice of such a cut-off frequency i difficult to make".
The purpose of this note is to develop theory for a
particular form of truncation and to provide an answer to the second of these questions. Although the wave spectrum defined above is used in the discussion, ship or offshore structural response spectra
calculated using this type of spectrum may be treated in a similar,
though not identical, manner.
mm
o4
Selection of a Cut-Off Point
The maximum value of corresponds to
I
W = W = (O.8B)4 m
If the spectrum is truncated at some multiple n of the frequency
w at which this peak occurs, i.e. at nu , the moment
ni ni (flW 4 A -B/u m = i -e du 4 J u o
and, by the change of variable x-B/w4 this becomes
-x ( A
fe
Ai-ß
ni=-
i-
dx-E'--' 44x
411
4 B/(nw ) (nw ) ni mThe definite exponential integral form E1( ) is a
tabulated function, see Abramowitz and Stegun (1964), and it has a series expansion
mm
(-1) z
E1(z) = -y-1n(z)-i mn
The quantity y is the Euler constant, equal to 0,577216. In this particular context
ß
_5
4 4 (nu ) 4n
ni
and this is much less than unity if n2. The dominant contribution from E1(z) therefore comes from the first two terms of the
expansion and, as a reasonable approximation
E1(z) = - y - ln(l.25) + 4 in(n)
= - 0.8004 + 4 in(n) (n2).
Z
Table I, Approximate Variation of 4m4/A with Truncation Point
n
If the same truncation point is adopted for the calculations of the moments m and m2 (even though they are integrable over the entire frequency range
0w),
we find thatn&3 A m (w)d = -- exp(L4) o 4n and if n --A m o 4B
This is the result found for the original spectrum without
t run cat ion,
The second moment
n =_,/(2!) J = f 2 m2 (w)dw 4 B
[l_2 f{()}]
2nwhere the error function
Ix p 2 exp(- )dy1 erf(x) = i V'(2iî)o n 4m4 A 2 1,9722 3 3,5940 4 4.7448 5 5.6374 6 6.3666
witk the properties
erf(-x) = -erf(x) and erf() = 0.5,
Again, if n
--A
= ..v )
and this, too, is the result for the original spectrum.
with these results the bandwidth parameter c satisfies
the equation 2 2 m2 l_2erf{(2z))J2 £ = 1---
--1-mm
4 exp(-z) E1(z)where z = 514n4. The relationship between e and n is shown in figure 1, If m and m2 are evaluated over the entire frequency range 0w and only m4 is found from the truncated spectrum,
the bandwidth relationship reduces to
2 'it
e = i
E1(z)
which, to be greater than zero, requires that E1(z)>îr or
n2.68. This relationship, too, is shown in figure 1. Note that these previous relationships between e and n are independent of
the actual values of the constants A and B..
From wave data recorded at station 'India', Hoffman
(1974) formed a family of wave' spectra and placed them in ten
groups, each containing eight wave spectra. For this family,
Table II illustrates the variation of the measured bandwidth
parameter e for each group with the corresponding average mean period T1 and significant wave height h113. The values of n
which would give the measured values of c, i.e1 giving
appropriate truncation points, have been calculated using the. above more complicated expression for c2 These values indicate
how n should be selected for a range of significant wave heights and average wave periods. The increase of spectral bandwidth parameter with significant wave height indicates the presence of swells in the higher numbered groups.
Table II Measured data on a family of wave spectra at Station
'India' (Hoffman 1974)
Truncation of ITTC and ISSC Spectra
(i) ITTC Spectrum
For this spectrum the constants are defined as
A = 8.1x103g2 and B 3,l1/h,3 , 6, Group number Waveheights (m) h113(m) .
T1(s) c Approx. value n giving
suitable truncation Pt. 1 >0.91 0,68 6,97 0.5782 3.0 2 0,91- 1,83 1.59 7.33 0.5784 3.0 3 1.83- 2.74 2,30 8,46 0,6368 3,7 4 2,74- 3.66 3.24 8.31 0.6345 3.65 5 3.66- 4,88 4.28 9.03 0.6806
44
6 4.88- 6,40 .43 8,80 0,6803 4,4 7 6.40- 8,23 7.33 9.50 0.7075 5.1 8 8.23-10.36 8.82 9.93 0.7328 5.9 9 lO.36-12.80 11.35 11,21 0,7638 7.4 10 12,80> 14,65 11,61 0.7870 9.0After substitution of these constants the moments m o
and m2 of the truncated spectrums satisfy the relationship
5
h113 = 4.0 exp(---) /m 8n
and the average wave period
f
measured at the level of thecalm water surface is
T =
5)
2n 8n
for n2. When n -- so that the spectrum is defined over the frequency range (0,) then these results reduce to
h13 = 4,0 /in , h13 = 5,1 in2 o and h13 = 5,1m211-2erf{(
32
5 = 2.025x10g E1()
4n = 3.55 h 1/3(ii) ISSC Spectrum
The constants now are defined as
173 2
T1
B 691
T14
Now we find that for n2
_1
2
h13 = 4,0exp()Im,
T1 =l.086{1-2erf{}]
exp()
2n 8n 8n
= 43,25 h/3E(),T4
4n 2n }J -1 (n2)and when n =
K113 = 4,0 /m and T1 = l,086 o
Conclus ions
Naturally it is not possible for a purely mathematical argument to indicate where the cut-off frequency for a wave spectrum should be chosen. This theory does show, however, how the important moments m, m2 and m4 vary with the truncation selected, Further, the experimental data quotedin Table II indicates thevalue at which the truncation point should be chosen,
References
Abramowitz, M. and Stegun, I,A, (1964). "Handbook. of Mathematiáal.
Functions", National Bureau of Standards, Applied Mathematics
Series 55.
Hoffman, D, (1974). "Analysis of measured and calculated spectra". Symposium on the Dynamics of Marine Vehicles and Structures
in Waves (ed. R,E,D. Bishop and W,C,, Price). Mechanical
Engineering Publ. Ltd., London, p 8-18,
Pierson, W.J, and Moskowitz, L. (1963). "A Proposed Spectral
Form for Fully Developed Wind Seas based on the Similarity Theory of S,A, Kitaigorodsku", Tech. Rept, U.S. Naval Oceanographic
Office Contract No, 62306-1042.
'. Price, W.G. and Bishop, R,E,D. (1974). "Probabilistic Theory of Ship Dynamics". Chapman and Hall,
Tickell, R.G. (1977). "Continuous Random Wave Loading on
