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Note on the intensity of encountered waves
ARTICLE IN PRESS
Ocean Engineering 34 (2007) 1561–1568 www.elsevier.com/locate/oceaneng
Note on the intensity of encountered waves Igor Rychlika, Sofia Aberga,, Ross Leadbetterb a
b
Centre for Mathematical Sciences, Lund University, Box 118, SE-22100 Lund, Sweden Department of Statistics and Operations Research, 210 Smith Building, University of North Carolina, Chapel Hill, NC 27599, USA Received 12 June 2006; accepted 6 December 2006 Available online 12 February 2007
Abstract An apparent wave is a part of the sea record observed between two successive upcrossings of the still water level. Integral formulas are given for intensities of encountered waves that overtake a ship sailing in directional sea with constant velocity. The formulas can be evaluated exactly in the case when the directional spectrum is known and the sea is assumed to be Gaussian, i.e. is a sum of noninteracting sinusoidal waves. r 2007 Elsevier Ltd. All rights reserved. Keywords: Encountered wave intensity; Rice’s formula; Wave velocities
1. Introduction and definitions In this note we present a general formula that can be used to compute the intensity of encountered waves that overtake a ship. Such waves can cause instability for a ship and hence it is of interest to also study their properties such as amplitude and wavelength. Current and planned future investigations are directed towards these properties of encountered waves and their effect on ship stability. This note answers the complementary question, basic for application of all such studies, of how many waves overtake a ship, on average, during a specified period of time. The formulae are simple, see (15) and (8)–(9) but to our knowledge not given in the literature. A short plan for this note is given next. First the intensity of waves and its relation to wave velocity is discussed for a deterministic sea model. It is also indicated that a similar formula may be expected for the random case. Then the Gaussian directional sea model is introduced, spectral moments defined and Rice’s formula for upcrossings of a fixed level by a stationary Gaussian process given. In the following subsection intensities of encountered waves are defined and some simple results given. Next the velocity with which the center of an Corresponding author. Tel.:+46 46 222 79 74; fax: +46 46 222 46 23.
E-mail address: [email protected] (S. Aberg). 0029-8018/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2006.12.004
apparent wave is moving is defined. The velocity will be used in the formulation of the main result for the intensity of encountered waves which overtake the ship. In the next subsection some examples will be given. Finally the appendix contains a proof of the formula and a short discussion of its validity for the nonGaussian sea model. 1.1. Encounter frequency for a deterministic wave Before turning to encountered wave intensities in a stochastic sea model we start by considering a simpler case, namely a single deterministic ocean wave. Let W det ðx; tÞ ¼ A cosðot kxÞ be a deterministic ocean wave having amplitude A, angular frequency o and wave number k. This is a wave that travels in the direction of the positive x-axis with (phase) velocity given by V ¼ L=T, where the wavelength L is defined by 2p=k and the period T by 2p=o. Now suppose that a ship is travelling along the x-axis at constant velocity v. Then the sea elevation Y det ðtÞ, say, at the center of gravity of the ship can be written as Y det ðtÞ ¼ W det ðvt; tÞ ¼ A cosðot kvtÞ ¼ A cosððo kvÞtÞ. The frequency o ¯ ¼ ðo kvÞ is often referred to as the encounter frequency since dividing it by 2p gives the number of waves encountered by the ship per time unit. Defining
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the intensity of encountered waves by mdet ¼ o=2p we may ¯ write mdet ¼
o kv 1 1 1 ¼ v ¼ ðV vÞ. 2p T L L
If the wave velocity V is larger than the ship velocity v the waves overtake the ship from behind. The intensity of such overtaking waves, here denoted by mþ det , can be expressed by introducing the positive part xþ ¼ maxðx; 0Þ, viz. mþ det ¼
1 ðV vÞþ . L
(1)
Note that in this case either all waves are faster than the ship or none are. However, for a random sea this is not the case. Yet the expression for the intensity of waves overtaking the ship from behind is very similar. In fact we will show that the intensity mþ of overtaking waves in a random sea is given by mþ ¼
1 E½ðV vÞþ , L
where in this case L is the expected wavelength and V a random wave velocity, first studied by Longuet-Higgins (1957). By E½X is meant the expectation of the random variable X. This result is derived and exemplified in the following sections.
able sample paths sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 VarðY_ ð0ÞÞ u2 exp . mY ðuÞ ¼ 2p VarðY ð0ÞÞ 2 VarðY ð0ÞÞ
(3)
2. Wave intensities A sea surface along the x- and t-axes, W ðxÞ; W ðtÞ, respectively, can be seen as a sequence of apparent waves, defined as parts of the sea between two consecutive zero upcrossings of the still water level, here assumed to be zero. The zero downcrossing lying between the upcrossings is taken as the center of a wave. Consider now a vessel sailing along the x-axis in the positive direction with constant speed v, having center of gravity at time zero at x ¼ 0. If one disregards the interaction between the ship and the waves, the sea surface Y ðtÞ seen at the center of gravity of the ship is given by Y ðtÞ ¼ W ðvt; tÞ. Let ti be the times when the center of gravity is passing through the still water level, i.e. Y ðti Þ ¼ 0. Now the ship encounters centers of waves at times ti if the slope of the wave is negative W x ðvti ; ti Þo0. At such ti the encountered sea Y may have a zero upcrossing Y_ ðti Þ40 if the wave is overtaking the ship or a zero downcrossing Y_ ðti Þo0 if the ship is passing the wave. The intensity of encountered waves overtaking the ship is defined as number of ti pt such that W x ðvti ; ti Þo0 and Y_ ðti Þ40 . t
mþ ¼ lim
t!1
(4) 1.2. Gaussian sea and Rice’s formula Let W ðx; y; tÞ be the sea elevation at location ðx; yÞ and time t. In order to simplify notation we shall write W ðxÞ ¼ W ðx; 0; 0Þ, W ðtÞ ¼ W ð0; 0; tÞ and W ðx; tÞ ¼ W ðx; 0; tÞ. Further, partial derivatives of W ðx; tÞ on x; t will be denoted by W x ðx; tÞ, W t ðx; tÞ, respectively. The sea surface elevation W ðx; y; tÞ will be modeled by means of a zero-mean, stationary Gaussian field with a directional spectrum having density Sðo; aÞ, o40, a 2 ½0; 2p, see Podgo´rski et al. (2000a) for a discussion and details of the model. With k ¼ ðo2 =gÞ cosðaÞ, define the following spectral moments: Z
1
Z
lij ¼ 0
2p
ki oj Sðo; aÞ da do.
(2)
0
We shall assume that the sea is ergodic, which is true e.g. if W possesses a spectral density function, in order to assure that the last limit exists and the intensity mþ is well defined. Similarly, the intensity of encountered waves that the ship is passing is defined by m ¼ lim
t!1
number of ti pt such that W x ðvti ; ti Þo0 and Y_ ðti Þo0 . t
(5) In this note we shall give an explicit formula to compute the intensities mþ ; m for a sea modeled by means of a Gaussian field with directional spectrum. The intensity of encountered waves is defined as m ¼ mþ þ m . Since for a zero-mean Gaussian field W ðx; y; tÞ has the same distribution as W ðx; y; tÞ we have that number of ti pt such that W x ðvti ; ti Þo0 and Y_ ðti Þo0 t number of ti pt such that W x ðvti ; ti Þ40 and Y_ ðti Þ40 lim , t!1 t
m ¼ lim
t!1
Some of the most commonly used variances and covariances of the process and its derivatives can be expressed in terms of the spectral moments. For example VarðW ð0ÞÞ ¼ l00 , VarðW t ð0; 0ÞÞ ¼ l02 , VarðW x ð0; 0ÞÞ ¼ l20 and CovðW x ð0; 0Þ; W t ð0; 0ÞÞ ¼ l11 , where VarðX Þ denotes the variance of the random variable X and CovðX ; Y Þ the covariance between two random variables X and Y. We will also need Rice’s formula for the intensity mY ðuÞ of upcrossings (downcrossings) of a level u by a stationary zero-mean Gaussian process Y ðtÞ, say, having differenti-
¼
and hence m is equal to m ¼ m þ þ m number of ti pt such that Y_ ðti Þ40 ¼ mY . ð6Þ t!1 t (Note that in general it may happen that mY amþ þ m .) Since for a zero-mean Gaussian sea the encountered sea ¼ lim
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level elevation Y ðtÞ is a zero-mean Gaussian process the intensity mY ¼ mY ð0Þ can be easily computed by means of Rice’s formula (3), viz. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 v2 l20 þ l02 þ 2vl11 mY ¼ . (7) 2p l00 On the other hand the intensities mþ , m cannot be computed using Rice’s formula (3). In the appendix we shall present the so-called generalized Rice’s formula which is applicable to their evaluation. In fact we shall prove that for a Gaussian sea the intensities mþ , m can be computed when the spectral moments l00 , l20 , l02 , l11 are available, viz. sffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 l20 l11 þ v (8) m ¼ v2 þ 2vl11 =l20 þ l02 =l20 4p l00 l20 and 1 m ¼ 4p
sffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l20 l11 2 þv . v þ 2vl11 =l20 þ l02 =l20 þ l00 l20 (9)
2.1. Example 1 þ
0.12 0.1 0.08 intensity [s-1]
ship and the other half is overtaken by the ship. At the same time the total intensity of encountering waves, mY , takes its smallest value. It turns out that this velocity is the mean velocity of the waves for this specific spectrum. Velocity of waves and mean wave velocity are both topics of the next section. 3. Expected wavelength and distribution of wave velocity Recall that the intensity (1) of overtaking waves for a deterministic sea wave contains the constant wavelength and velocity of this wave. With a random sea model these quantities can no longer be defined in the same way since the sea surface is no longer a single harmonic wave but a superposition of sine-waves with different frequencies having random amplitudes and phases. However, it turns out that the constant wavelength can be replaced by the expected wavelength and the constant velocity by a random velocity defined below. Let L be the average wavelength and let n ¼ 1=L be its inverse, i.e. the number of waves per length unit. To define n, let xi 40 be positions of centers of waves observed in W ðxÞ, i.e. along the x-axis at time zero. The intensity of waves in space n is then defined by number of xi ox . (10) x Using Rice’s formula for the process W ðxÞ we obtain that for the Gaussian sea sffiffiffiffiffiffi 1 l20 n¼ . (11) 2p l00 n ¼ lim
In Fig. 1 the intensities mY , m and m are shown as a function of ship velocity for a following unidirectional Gaussian sea having a JONSWAP type spectrum with parameters defined in Example 2. For low ship velocity most of the encountering waves are overtaking the ship. However, as the ship velocity increases this intensity, mþ , decreases, whereas the intensity of waves that the ship is passing, m , has the opposite behavior. For a certain velocity, slightly larger than 8 m/s, the intensities mþ and m are equal, i.e. half of the encountering waves overtake the
x!1
Again let xi 40 be positions of wave centers observed in W ðxÞ. With time the centers change their positions, due to variability of the sea surface in time. The center of the ith wave moves with velocity V i (at time zero) which can be easily evaluated by means of V i ¼ W t ðxi ; 0Þ=W x ðxi ; 0Þ. The velocities of random waves were first studied by Longuet-Higgins (1957), see also Baxevani et al. (2003) and the references therein. Here the variability of the velocities V i will be described by means of the empirical distribution defined as follows: F V ðvÞ ¼ PðV pvÞ number of xi ox such that V i pv . ¼ lim x!1 number of xi ox
0.06 0.04
For the Gaussian sea 0
0.02 0
1563
1
1B v v¯ C PðV pvÞ ¼ @1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; 2 ðv v¯ Þ2 þ s2 =l20 0
2
4
6
8
10
12
14
ð12Þ
v¯ ¼
l11 , l20
16
(13)
ship velocity [m/s] 2
Fig. 1. The intensities mY (solid), mþ (dashed) and m (dash-dotted) as a function of ship velocity for a following unidirectional Gaussian sea having a JONSWAP spectrum. The velocity at which two of the curves cross each other is the mean wave velocity for this specific spectrum.
l211 =l20
is a see Podgo´rski et al. (2000b), where s ¼ l02 measure of the bandwidth of the sea spectrum. Note that v¯ P is the average velocity, i.e. v¯ ¼ E½V ¼ limn!1 1n ni¼1 V i . Using the mean wave velocity the encountered intensity of
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waves can be written in an alternative way qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 mY ¼ n ðv v¯ Þ2 þ s2 =l20 ; s2 ¼ l02 11 . l20
where ( (14)
4. The main result Previously it was noted that mY can be computed by direct use of Rice’s formula (3), whereas this is not true for the intensities mþ and m . However, mþ and m are important quantities in studies involving effect of the ocean on ships, e.g. ship stability, and thus it is of great interest to obtain general expressions for how these can be computed. As a main result this note provides such expressions, proved in the appendix, and next the result is stated for the Gaussian sea. Proposition 1. Let W ðx; tÞ be a stationary zero-mean Gaussian process such that the vector of random variables ðY ð0Þ; Y_ ð0Þ; W x ð0; 0ÞÞ has a nonsingular distribution. Assume further that W ðx; tÞ has continuously differentiable sample paths. Then the intensities mþ and m are given by mþ ¼ n E½ðV vÞþ ;
m ¼ n E½ðV vÞ ,
s¼
sa ; sb ;
ooop ; oXop :
This spectrum is fully characterized by the set of parameters ðhs ; tp ; g; sa ; sb Þ, where hs is the significant wave height and tp is the peak period, related to the peak frequency op by tp ¼ 2p=op . The parameter g is a factor determining the concentration of the spectrum around the peak frequency that depends on hs and tp . The parameters sa and sb , are spectral width parameters. In this example hs ¼ 7 m, tp ¼ 11 s, g ¼ 3:3, sa ¼ 0:07 and sb ¼ 0:09. Moreover the cut-off angular frequency oc is set to 3 rad/s. We will use formulae (7)–(9) with the spectral moments computed as follows; let Z 1 li ¼ oi SðoÞ do, 0
and then by (2), l00 ¼ l0 , l20 ¼ l4 cos2 ðaÞ=g2 , l11 ¼ l3 cosðaÞ=g and l02 ¼ l2 .
(15)
where z ¼ z if z40 and zero otherwise.
5.1. Example 2
This result is valid even for nonGaussian seas, see the appendix for discussion. However, the intensity n and E½ðV vÞþ are then given in form of complicated integrals which need to be computed using suitable approximations.
In Fig. 2 we present the mean wave velocity as a function of the angle a and the intensity of encountered waves m ¼ mY as a function of the angle a and for ship velocities v ¼ 8; 12; 16 m=s. For low values of a the waves are, on average, traveling in the opposite direction to the vessel (negative velocity). Consequently, as is clearly seen in the right picture in Fig. 2, the ship will encounter many waves and the faster the vessel travels the more waves it will encounter. As the angle comes close to p=2 interesting things happen, e.g. the velocity in the left picture becomes very big (negative or positive). The explanation is that at this angle the wave direction is almost vertical to the ship, causing the spatial derivative in the x-direction, W x ðxi ; 0Þ, to be close to zero. Since the velocity is defined by V i ¼ W t ðxi ; 0Þ=W x ðxi ; 0Þ, a small spatial derivative gives rise to a large velocity. Another interesting phenomenon is that for a ¼ p=2 the intensity mY is the same for all ship velocities. This is due to the fact that for a long crested sea coming in from the direction p=2, the sea elevation is the same everywhere on the x-axis. Thus the sea surface elevation at the center of gravity of the ship will be the same regardless of the velocity of the vessel and measuring the elevation from a buoy fixed somewhere on the x-axis or at the center of gravity of the vessel would give rise to the same signal. Consequently, for a ¼ p=2, the intensity of waves in the encountered process Y ðtÞ is constant and equal to the intensity of waves in the process pffiffiffiffiffiffiffiffiffiffiffiffiffiffiW ffi ðtÞ ¼ W ð0; 0; tÞ. This intensity is given by ð2pÞ1 l02 =l00 . Finally, for a ¼ p the ship is sailing in a following sea with mean wave velocity close to 8 m/s, see left picture in
þ
4.1. Proof of (8) and (9) For a Gaussian sea, we have shown that mþ þ m ¼ mY and since E½ðV vÞþ E½ðV vÞ ¼ E½V v ¼ v¯ v one can compute the intensities by means of the following formulas: mþ ¼ 12ðmY þ nð¯v vÞÞ;
m ¼ 12ðmY nð¯v vÞÞ,
(16)
where mY is given by (14), the intensity of waves in space n by (11) and the average wave velocity v¯ by (13). Obviously, the intensities mþ , m can be computed when the spectral moments l00 , l20 , l02 , l11 are available, and are given by (8) and (9). In the following example we study the intensities mþ and m and compare mþ for longcrested and directional Gaussian seas with JONSWAP spectral density. 5. Examples Let SðoÞ be a JONSWAP frequency spectrum, i.e. a parametric spectrum of the form 5 h2s op 5 5=4ðop =oÞ4 expðð1o=op Þ2 =2sÞ SðoÞ ¼ e g , 16 op o oooc ,
ARTICLE IN PRESS I. Rychlik et al. / Ocean Engineering 34 (2007) 1561–1568
50
0.4
40
0.35
30 0.3
20
0.25
10 0
μY
mean wave velocity [m/s]
1565
-10
0.2 0.15
-20 0.1
-30
0.05
-40 -50
0
1
2
3
4
5
0
6
0
1
2
3
4
5
6
α [rad]
α [rad]
Fig. 2. Left—mean wave velocity v¯ as a function of angle a. Right—the intensity mY ða; vÞ of encountered waves as a function of the angle a for different velocities v ¼ 8 (solid), 12 (dashed), 16 (dotted) m/s.
0.4
0.12
0.35
0.1
0.3 0.25
0.06
μ-
μ+
0.08
0.2 0.15
0.04
0.1 0.02 0
0.05 0
1
2
3
4
5
6
α [rad]
0
0
1
2
3
4
5
6
α [rad]
Fig. 3. Left—the intensity mþ ða; vÞ of encountered waves that overtake the ship as a function of the angle a for different velocities v ¼ 8; 10; . . . ; 18 m=s. For a fixed a, mþ ða; vÞ is a decreasing function of ship velocity. Right—the intensity m ða; vÞ of encountered waves that the ship is passing as a function of the angle a for different velocities v ¼ 8; 10; . . . ; 18 m=s. For a fixed a, m ða; vÞ is an increasing function of ship velocity.
Fig. 2. If the velocity of the ship is of this magnitude the vessel travels along with the waves and it encounters few waves. As its velocity increases the intensity of encountered waves also increases. This explains the appearance of the right picture in Fig. 2 around a ¼ p. 5.2. Example 3 In Fig. 3 the intensities mþ and m of encountered waves that overtake the ship and that the ship is passing, respectively, are plotted as a function of the angle a and for velocities v ¼ 8; 10; . . . ; 18 m=s. For a fixed value of a, mþ is a decreasing function of ship velocity. This is easily explained since if a is fixed the intensity n in (15) is also fixed and the factor E½ðV vÞþ is a decreasing function of v. Similarly m is increasing with ship velocity. To get
some intuition of the qualitative appearance of these curves one can study the mean wave velocity in Fig. 2. The mean wave velocity is a location parameter of the velocity distribution given in (13). For small a the velocity distribution is centered around negative velocities and it is likely that the factor E½ðV vÞþ in the expression for mþ is small and thereby also the intensity mþ . However, as the angle a passes p=2 the mean wave velocity suddenly becomes positive and decreasing. This explains the discontinuity of mþ and it also makes it reasonable to believe that E½ðV vÞþ and mþ will decrease with a. Of course the shape of the velocity distribution as well as the interaction with the intensity n in (15) must be taken into consideration for a full explanation of the behavior of the intensity curve. Similar arguments can be presented for the shape of the m -curve.
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Directional Spectrum Level curves at: 2 4 6 8 10 12
90
0.8
120
60
0.6
150
0.12 30
0.4
0.1 0.2
0.08 0 μ+
180
0.06 0.04
330
210
0.02 240
300 270
0
0
1
2
3
4
5
6
α [rad]
Fig. 4. Left—the directional spectrum as function of a. Right—the intensity mþ ða; vÞ of encountered waves that overtake the ship as a function of the angle a for different velocities v ¼ 8; 10; . . . ; 18 m=s. For a fixed a, mþ ða; vÞ is a decreasing function of ship velocity v.
5.3. Example 4 Next we create a directional spectrum using a spreading function Gðo; yÞ of the cos 2s-type with parameter s ¼ 10. Here we should emphasize that this particular choice of parameter is for illustration purposes only. In Fig. 4 the directional spectrum as a function of a is shown together with the intensity mþ of encountered waves that overtake the ship. Comparing this intensity to the one obtained in the longcrested case shown in Fig. 3 shows that the intensity in the directional case essentially is a smoothed version of the intensity in the longcrested case. This is not surprising since the directional sea is a superposition of waves not only with different frequencies but also with slightly different directions.
Acknowledgments The second author was funded by the Swedish Foundation for Strategic Research, project Spatial statistics and image analysis for environment and medicine, A3 02:125.
Appendix A A.1. The generalized Rice’s formula Let ZðtÞ; Y ðtÞ be smooth processes, Y being continuously differentiable and Z continuous. Let N T ðzjuÞ be the number of t 2 ½0; T such that Y ðtÞ ¼ u, Y_ ðtÞ40 and
ZðtÞpz. Then, for almost all u, E½N T ðzjuÞ Z T E½jY_ ðtÞj1fY_ ðtÞ40;ZðtÞpzg jY ðtÞ ¼ u f Y ðtÞ ðuÞ dt, ¼
ð17Þ
0
where 1fAg is the indicator function equal to one if the condition A in brackets holds and zero otherwise. The formula (17) is called the generalized Rice’s formula. If the processes are jointly stationary, then define E½N T ðzjuÞ , (18) T the intensity of solutions to the equation Y ðtÞ ¼ u satisfying the conditions Y_ ðtÞ40 and ZðtÞpz. In addition, if the processes are jointly ergodic then the intensity can be obtained using
mðzjuÞ ¼ E½N 1 ðzjuÞ ¼
mðzjuÞ ¼ lim
T!1
N T ðzjuÞ . T
Under the joint stationarity assumption the intensity is given by, for almost all levels u, mðzjuÞ ¼ E½jY_ ð0Þj1fY_ ð0Þ40;Zð0Þpzg jY ð0Þ ¼ u f Y ð0Þ ðuÞ.
(19)
As we can see the generalized Rice’s formula is valid under minimal assumptions of smoothness of the processes involved and existence of the probability density f Y ð0Þ , see Leadbetter and Spaniolo (2004) for a detailed presentation of Rice’s formula and also Rice (1944, 1945) and Rychlik (2000). However (19) is valid only for almost all levels u and it may fail for a particular level u of interest, e.g. u ¼ 0. In order to prove the formula for a fixed level more restrictive assumptions are needed. For this it is sufficient
ARTICLE IN PRESS I. Rychlik et al. / Ocean Engineering 34 (2007) 1561–1568
to show that both sides of the equality in (19) are continuous functions of u. Continuity of the right-hand side is usually easier to check. For example, if the joint density of ðY ð0Þ; Y_ ð0Þ; Zð0ÞÞ exists and is sufficiently smooth then
A.2. Proof of Proposition 1
E½jY_ ð0Þj1fY_ ð0Þ40;Zð0Þpzg jY ð0Þ ¼ u f Y ð0Þ ðuÞ Z 1Z z ¼ jyjf Y ð0Þ;Y_ ð0Þ;Zð0Þ ðu; y; xÞ dx dy
Since
0
ð20Þ
0
m¯ þ ¼ E½jY_ ð0Þj1fY_ ð0Þ40;W x ð0;0Þo0g jW ð0Þ ¼ 0 f W ð0Þ ð0Þ.
jY_ ð0Þj1fY_ ð0Þ40;W x ð0;0Þo0g ¼ Y_ ð0Þþ 1fW x ð0;0Þo0g ¼ ðvW x ð0; 0Þ þ W t ð0; 0ÞÞþ 1fW x ð0;0Þo0g ¼ ðW x ð0; 0Þðv V ð0ÞÞÞþ 1fW x ð0;0Þo0g ¼ W x ð0; 0Þ ðV ð0Þ vÞþ ,
where V ð0Þ ¼ W t ð0; 0Þ=W x ð0; 0Þ, one has, again using (20), that m¯ þ ¼ E½W x ð0; 0Þ ðV ð0Þ vÞþ jW ð0; 0Þ ¼ 0 f W ð0;0Þ ð0Þ Z 1Z 0 ðx vÞjzjf W ð0;0Þ;W x ð0;0Þ;V ð0Þ ð0; z; xÞ dz dx. ¼ v
1
However, in many practical problems when nonGaussian models are employed the weaker version of Rice’s formula (19) is sufficient, see discussion in Rychlik (2004). This is also the case for the intensity of encountered waves. The problem that the intensity m may not be defined for every u is solved by introduction of the ‘‘averaged’’ intensity m¯ defined next. Suppose that the joint density of ðY ð0Þ; Y_ ð0Þ; Zð0ÞÞ exists and that right-hand side of Eq. (19) is computed using that (20) is a continuous function of u. Then the averaged crossing intensity mðzjuÞ is defined as ¯ follows: Z 1 uþ mðzjuÞ ¼ lim mðzjxÞ dx ¯ !0 2 u Z 1Z z ¼ jyjf Y ð0Þ;Y_ ð0Þ;Zð0Þ ðu; y; xÞ dx dy ð22Þ 0
1
with mðzjxÞ defined by (18). The average intensity will be used to prove that the formula (15) is also valid for nonGaussian sea models. Let mþ ðuÞ be the intensity of t such that Y ðtÞ ¼ u, W x ðvt; tÞo0 and Y_ ðtÞ40, defined by (18), then the intensity m¯ þ , defined in (22), is given by Z 1 þ þ m¯ þ ¼ lim m ðuÞ du. (23) !0 2 Obviously, m¯ þ ¼ mþ ð0Þ ¼ mþ if mþ ðuÞ is continuous, e.g. for a Gaussian sea. We also regard continuity of mþ ðuÞ as a property that any reasonable model for sea elevation should satisfy. Consequently, the intensity of encountered waves overtaking a ship can be defined by Z 1Z 0 zf Y ð0Þ;Y_ ð0Þ;Zð0Þ ð0; z; xÞ dx dz, (24) m¯ þ ¼ 0
Consider Eq. (24) and use (20) to write
1
and calculus can be used to proof the continuity right-hand side of Eq. (19). To prove that the left side in (19) is a continuous function of u is usually much harder. However, in the most important case when Y ðtÞ is Gaussian and ZðtÞ is a smooth function of Gaussian processes then, under mild regularity assumptions, see Leadbetter et al. (1983), the joint density of Y ð0Þ; Y_ ð0Þ; Zð0Þ exists and, for every u, Z 1Z z mðzjuÞ ¼ jyjf Y ð0Þ;Y_ ð0Þ;Zð0Þ ðu; y; xÞ dx dy. (21)
1
R1 R0 where Y ðtÞ ¼ W ðvt; tÞ, ZðtÞ ¼ W x ðvt; tÞ and 0 1 zf Y ð0Þ;Y_ ð0Þ;Zð0Þ ðu; z; xÞ dx dz is a continuous function of u, which usually is easy to verify.
1567
1
Now R0 f V 0 ðxÞ ¼
jzjf W ð0;0Þ;W x ð0;0Þ;V ð0Þ ð0; z; xÞ dz R0 1 jzjf W ð0;0Þ;W x ð0;0Þ ð0; zÞ dz
1
is the probability density of some random variable V 0 and R0 with nð0Þ ¼ 1 jzjf W ð0;0Þ;W x ð0;0Þ ð0; zÞ dz we have that m¯ þ ¼ nð0ÞE½ðV 0 vÞþ .
(25)
Now in the Gaussian case m¯ þ ¼ mþ , n ¼ nð0Þ and V 0 has the same density as V and hence (15) holds, concluding the proof in the Gaussian case. In the more general case we need to give physical interpretation to the density f V 0 ðxÞ and the intensity nð0Þ. This will be done next. Let V ðxÞ ¼ W t ðx; 0Þ=W x ðx; 0Þ and let xi 40 be solutions to W ðx; 0Þ ¼ u such that W x ðx; 0Þo0. Now V i ¼ V ðxi Þ is the velocity of the ith downcrossing of level u by W ðx; 0Þ. The empirical distribution of these velocities is F V u ðvÞ ¼ PðV pvÞ number of xi ox such that V i pv , ¼ lim x!1 number of xi ox
ð26Þ
while nðuÞ is the intensity of xi . Now using the generalized Rice’s formula one can show, see Rychlik (2004), that for almost all u the density of V u is given by R0 f V u ðxÞ ¼
jzjf W ð0;0Þ;W x ð0;0Þ;V ð0Þ ðu; z; xÞ dz . R0 1 jzjf W ð0;0Þ;W x ð0;0Þ ðu; zÞ dz
1
If the right-hand side of the last equation is a continuous and bounded function of u then Z 1 þ f V¯ ðxÞ ¼ lim f V u ðxÞ du ¼ f V 0 ðxÞ. !0 2
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R0 Similarly, nðuÞ ¼ 1 jzjf W ð0;0Þ;W x ð0;0Þ ðu; zÞ dz for almost all u, so that n¯ ¼ nð0Þ and hence m¯ þ ¼ n¯ E½ðV¯ vÞþ .
References Baxevani, A., Podgo´rski, K., Rychlik, I., 2003. Velocities for moving random surfaces. Probabilistic Engineering Mechanics 18, 251–271. Leadbetter, M.R., Spaniolo, G.V., 2004. Reflections on Rice’s formulae for level crossings—history, extensions and use. Australian and New Zealand Journal of Statistics 46, 173–180. Leadbetter, M.R., Lindgren, G., Rootze´n, H., 1983. Extremes and Related Properties of Random Sequences and Processes. Springer, Berlin.
Longuet-Higgins, M.S., 1957. The statistical analysis of a random, moving surface. Philosophical Transactions of the Royal Society, Series A 249, 321–387. Podgo´rski, K., Rychlik, I., Machado, U.E.B., 2000a. Exact distributions for apparent waves in irregular seas. Ocean Engineering 27, 979–1016. Podgo´rski, K., Rychlik, I., Sjo¨, E., 2000b. Statistics for velocities of Gaussian waves. International Journal of Offshore and Polar Engineering 10, 91–98. Rice, S.O., 1944,1945. The mathematical analysis of random noise I and II. Bell System Technical Journal 23, 24. Rychlik, I., 2000. On some reliability applications of Rice’s formula for the intensity of level crossings. Extremes 3 (4), 331–348. Rychlik, I., 2004. Five lectures on reliability applications of Rice’s formula for the intensity of level crossings. Reliability-Based Design and Optimisation. IFTR, Warsaw, ISSN 1642-0578, pp. 241–323.
Ocean Engineering 34 (2007) 1561–1568 www.elsevier.com/locate/oceaneng
Note on the intensity of encountered waves Igor Rychlika, Sofia Aberga,, Ross Leadbetterb a
b
Centre for Mathematical Sciences, Lund University, Box 118, SE-22100 Lund, Sweden Department of Statistics and Operations Research, 210 Smith Building, University of North Carolina, Chapel Hill, NC 27599, USA Received 12 June 2006; accepted 6 December 2006 Available online 12 February 2007
Abstract An apparent wave is a part of the sea record observed between two successive upcrossings of the still water level. Integral formulas are given for intensities of encountered waves that overtake a ship sailing in directional sea with constant velocity. The formulas can be evaluated exactly in the case when the directional spectrum is known and the sea is assumed to be Gaussian, i.e. is a sum of noninteracting sinusoidal waves. r 2007 Elsevier Ltd. All rights reserved. Keywords: Encountered wave intensity; Rice’s formula; Wave velocities
1. Introduction and definitions In this note we present a general formula that can be used to compute the intensity of encountered waves that overtake a ship. Such waves can cause instability for a ship and hence it is of interest to also study their properties such as amplitude and wavelength. Current and planned future investigations are directed towards these properties of encountered waves and their effect on ship stability. This note answers the complementary question, basic for application of all such studies, of how many waves overtake a ship, on average, during a specified period of time. The formulae are simple, see (15) and (8)–(9) but to our knowledge not given in the literature. A short plan for this note is given next. First the intensity of waves and its relation to wave velocity is discussed for a deterministic sea model. It is also indicated that a similar formula may be expected for the random case. Then the Gaussian directional sea model is introduced, spectral moments defined and Rice’s formula for upcrossings of a fixed level by a stationary Gaussian process given. In the following subsection intensities of encountered waves are defined and some simple results given. Next the velocity with which the center of an Corresponding author. Tel.:+46 46 222 79 74; fax: +46 46 222 46 23.
E-mail address: [email protected] (S. Aberg). 0029-8018/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2006.12.004
apparent wave is moving is defined. The velocity will be used in the formulation of the main result for the intensity of encountered waves which overtake the ship. In the next subsection some examples will be given. Finally the appendix contains a proof of the formula and a short discussion of its validity for the nonGaussian sea model. 1.1. Encounter frequency for a deterministic wave Before turning to encountered wave intensities in a stochastic sea model we start by considering a simpler case, namely a single deterministic ocean wave. Let W det ðx; tÞ ¼ A cosðot kxÞ be a deterministic ocean wave having amplitude A, angular frequency o and wave number k. This is a wave that travels in the direction of the positive x-axis with (phase) velocity given by V ¼ L=T, where the wavelength L is defined by 2p=k and the period T by 2p=o. Now suppose that a ship is travelling along the x-axis at constant velocity v. Then the sea elevation Y det ðtÞ, say, at the center of gravity of the ship can be written as Y det ðtÞ ¼ W det ðvt; tÞ ¼ A cosðot kvtÞ ¼ A cosððo kvÞtÞ. The frequency o ¯ ¼ ðo kvÞ is often referred to as the encounter frequency since dividing it by 2p gives the number of waves encountered by the ship per time unit. Defining
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the intensity of encountered waves by mdet ¼ o=2p we may ¯ write mdet ¼
o kv 1 1 1 ¼ v ¼ ðV vÞ. 2p T L L
If the wave velocity V is larger than the ship velocity v the waves overtake the ship from behind. The intensity of such overtaking waves, here denoted by mþ det , can be expressed by introducing the positive part xþ ¼ maxðx; 0Þ, viz. mþ det ¼
1 ðV vÞþ . L
(1)
Note that in this case either all waves are faster than the ship or none are. However, for a random sea this is not the case. Yet the expression for the intensity of waves overtaking the ship from behind is very similar. In fact we will show that the intensity mþ of overtaking waves in a random sea is given by mþ ¼
1 E½ðV vÞþ , L
where in this case L is the expected wavelength and V a random wave velocity, first studied by Longuet-Higgins (1957). By E½X is meant the expectation of the random variable X. This result is derived and exemplified in the following sections.
able sample paths sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 VarðY_ ð0ÞÞ u2 exp . mY ðuÞ ¼ 2p VarðY ð0ÞÞ 2 VarðY ð0ÞÞ
(3)
2. Wave intensities A sea surface along the x- and t-axes, W ðxÞ; W ðtÞ, respectively, can be seen as a sequence of apparent waves, defined as parts of the sea between two consecutive zero upcrossings of the still water level, here assumed to be zero. The zero downcrossing lying between the upcrossings is taken as the center of a wave. Consider now a vessel sailing along the x-axis in the positive direction with constant speed v, having center of gravity at time zero at x ¼ 0. If one disregards the interaction between the ship and the waves, the sea surface Y ðtÞ seen at the center of gravity of the ship is given by Y ðtÞ ¼ W ðvt; tÞ. Let ti be the times when the center of gravity is passing through the still water level, i.e. Y ðti Þ ¼ 0. Now the ship encounters centers of waves at times ti if the slope of the wave is negative W x ðvti ; ti Þo0. At such ti the encountered sea Y may have a zero upcrossing Y_ ðti Þ40 if the wave is overtaking the ship or a zero downcrossing Y_ ðti Þo0 if the ship is passing the wave. The intensity of encountered waves overtaking the ship is defined as number of ti pt such that W x ðvti ; ti Þo0 and Y_ ðti Þ40 . t
mþ ¼ lim
t!1
(4) 1.2. Gaussian sea and Rice’s formula Let W ðx; y; tÞ be the sea elevation at location ðx; yÞ and time t. In order to simplify notation we shall write W ðxÞ ¼ W ðx; 0; 0Þ, W ðtÞ ¼ W ð0; 0; tÞ and W ðx; tÞ ¼ W ðx; 0; tÞ. Further, partial derivatives of W ðx; tÞ on x; t will be denoted by W x ðx; tÞ, W t ðx; tÞ, respectively. The sea surface elevation W ðx; y; tÞ will be modeled by means of a zero-mean, stationary Gaussian field with a directional spectrum having density Sðo; aÞ, o40, a 2 ½0; 2p, see Podgo´rski et al. (2000a) for a discussion and details of the model. With k ¼ ðo2 =gÞ cosðaÞ, define the following spectral moments: Z
1
Z
lij ¼ 0
2p
ki oj Sðo; aÞ da do.
(2)
0
We shall assume that the sea is ergodic, which is true e.g. if W possesses a spectral density function, in order to assure that the last limit exists and the intensity mþ is well defined. Similarly, the intensity of encountered waves that the ship is passing is defined by m ¼ lim
t!1
number of ti pt such that W x ðvti ; ti Þo0 and Y_ ðti Þo0 . t
(5) In this note we shall give an explicit formula to compute the intensities mþ ; m for a sea modeled by means of a Gaussian field with directional spectrum. The intensity of encountered waves is defined as m ¼ mþ þ m . Since for a zero-mean Gaussian field W ðx; y; tÞ has the same distribution as W ðx; y; tÞ we have that number of ti pt such that W x ðvti ; ti Þo0 and Y_ ðti Þo0 t number of ti pt such that W x ðvti ; ti Þ40 and Y_ ðti Þ40 lim , t!1 t
m ¼ lim
t!1
Some of the most commonly used variances and covariances of the process and its derivatives can be expressed in terms of the spectral moments. For example VarðW ð0ÞÞ ¼ l00 , VarðW t ð0; 0ÞÞ ¼ l02 , VarðW x ð0; 0ÞÞ ¼ l20 and CovðW x ð0; 0Þ; W t ð0; 0ÞÞ ¼ l11 , where VarðX Þ denotes the variance of the random variable X and CovðX ; Y Þ the covariance between two random variables X and Y. We will also need Rice’s formula for the intensity mY ðuÞ of upcrossings (downcrossings) of a level u by a stationary zero-mean Gaussian process Y ðtÞ, say, having differenti-
¼
and hence m is equal to m ¼ m þ þ m number of ti pt such that Y_ ðti Þ40 ¼ mY . ð6Þ t!1 t (Note that in general it may happen that mY amþ þ m .) Since for a zero-mean Gaussian sea the encountered sea ¼ lim
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level elevation Y ðtÞ is a zero-mean Gaussian process the intensity mY ¼ mY ð0Þ can be easily computed by means of Rice’s formula (3), viz. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 v2 l20 þ l02 þ 2vl11 mY ¼ . (7) 2p l00 On the other hand the intensities mþ , m cannot be computed using Rice’s formula (3). In the appendix we shall present the so-called generalized Rice’s formula which is applicable to their evaluation. In fact we shall prove that for a Gaussian sea the intensities mþ , m can be computed when the spectral moments l00 , l20 , l02 , l11 are available, viz. sffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 l20 l11 þ v (8) m ¼ v2 þ 2vl11 =l20 þ l02 =l20 4p l00 l20 and 1 m ¼ 4p
sffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l20 l11 2 þv . v þ 2vl11 =l20 þ l02 =l20 þ l00 l20 (9)
2.1. Example 1 þ
0.12 0.1 0.08 intensity [s-1]
ship and the other half is overtaken by the ship. At the same time the total intensity of encountering waves, mY , takes its smallest value. It turns out that this velocity is the mean velocity of the waves for this specific spectrum. Velocity of waves and mean wave velocity are both topics of the next section. 3. Expected wavelength and distribution of wave velocity Recall that the intensity (1) of overtaking waves for a deterministic sea wave contains the constant wavelength and velocity of this wave. With a random sea model these quantities can no longer be defined in the same way since the sea surface is no longer a single harmonic wave but a superposition of sine-waves with different frequencies having random amplitudes and phases. However, it turns out that the constant wavelength can be replaced by the expected wavelength and the constant velocity by a random velocity defined below. Let L be the average wavelength and let n ¼ 1=L be its inverse, i.e. the number of waves per length unit. To define n, let xi 40 be positions of centers of waves observed in W ðxÞ, i.e. along the x-axis at time zero. The intensity of waves in space n is then defined by number of xi ox . (10) x Using Rice’s formula for the process W ðxÞ we obtain that for the Gaussian sea sffiffiffiffiffiffi 1 l20 n¼ . (11) 2p l00 n ¼ lim
In Fig. 1 the intensities mY , m and m are shown as a function of ship velocity for a following unidirectional Gaussian sea having a JONSWAP type spectrum with parameters defined in Example 2. For low ship velocity most of the encountering waves are overtaking the ship. However, as the ship velocity increases this intensity, mþ , decreases, whereas the intensity of waves that the ship is passing, m , has the opposite behavior. For a certain velocity, slightly larger than 8 m/s, the intensities mþ and m are equal, i.e. half of the encountering waves overtake the
x!1
Again let xi 40 be positions of wave centers observed in W ðxÞ. With time the centers change their positions, due to variability of the sea surface in time. The center of the ith wave moves with velocity V i (at time zero) which can be easily evaluated by means of V i ¼ W t ðxi ; 0Þ=W x ðxi ; 0Þ. The velocities of random waves were first studied by Longuet-Higgins (1957), see also Baxevani et al. (2003) and the references therein. Here the variability of the velocities V i will be described by means of the empirical distribution defined as follows: F V ðvÞ ¼ PðV pvÞ number of xi ox such that V i pv . ¼ lim x!1 number of xi ox
0.06 0.04
For the Gaussian sea 0
0.02 0
1563
1
1B v v¯ C PðV pvÞ ¼ @1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; 2 ðv v¯ Þ2 þ s2 =l20 0
2
4
6
8
10
12
14
ð12Þ
v¯ ¼
l11 , l20
16
(13)
ship velocity [m/s] 2
Fig. 1. The intensities mY (solid), mþ (dashed) and m (dash-dotted) as a function of ship velocity for a following unidirectional Gaussian sea having a JONSWAP spectrum. The velocity at which two of the curves cross each other is the mean wave velocity for this specific spectrum.
l211 =l20
is a see Podgo´rski et al. (2000b), where s ¼ l02 measure of the bandwidth of the sea spectrum. Note that v¯ P is the average velocity, i.e. v¯ ¼ E½V ¼ limn!1 1n ni¼1 V i . Using the mean wave velocity the encountered intensity of
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waves can be written in an alternative way qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 mY ¼ n ðv v¯ Þ2 þ s2 =l20 ; s2 ¼ l02 11 . l20
where ( (14)
4. The main result Previously it was noted that mY can be computed by direct use of Rice’s formula (3), whereas this is not true for the intensities mþ and m . However, mþ and m are important quantities in studies involving effect of the ocean on ships, e.g. ship stability, and thus it is of great interest to obtain general expressions for how these can be computed. As a main result this note provides such expressions, proved in the appendix, and next the result is stated for the Gaussian sea. Proposition 1. Let W ðx; tÞ be a stationary zero-mean Gaussian process such that the vector of random variables ðY ð0Þ; Y_ ð0Þ; W x ð0; 0ÞÞ has a nonsingular distribution. Assume further that W ðx; tÞ has continuously differentiable sample paths. Then the intensities mþ and m are given by mþ ¼ n E½ðV vÞþ ;
m ¼ n E½ðV vÞ ,
s¼
sa ; sb ;
ooop ; oXop :
This spectrum is fully characterized by the set of parameters ðhs ; tp ; g; sa ; sb Þ, where hs is the significant wave height and tp is the peak period, related to the peak frequency op by tp ¼ 2p=op . The parameter g is a factor determining the concentration of the spectrum around the peak frequency that depends on hs and tp . The parameters sa and sb , are spectral width parameters. In this example hs ¼ 7 m, tp ¼ 11 s, g ¼ 3:3, sa ¼ 0:07 and sb ¼ 0:09. Moreover the cut-off angular frequency oc is set to 3 rad/s. We will use formulae (7)–(9) with the spectral moments computed as follows; let Z 1 li ¼ oi SðoÞ do, 0
and then by (2), l00 ¼ l0 , l20 ¼ l4 cos2 ðaÞ=g2 , l11 ¼ l3 cosðaÞ=g and l02 ¼ l2 .
(15)
where z ¼ z if z40 and zero otherwise.
5.1. Example 2
This result is valid even for nonGaussian seas, see the appendix for discussion. However, the intensity n and E½ðV vÞþ are then given in form of complicated integrals which need to be computed using suitable approximations.
In Fig. 2 we present the mean wave velocity as a function of the angle a and the intensity of encountered waves m ¼ mY as a function of the angle a and for ship velocities v ¼ 8; 12; 16 m=s. For low values of a the waves are, on average, traveling in the opposite direction to the vessel (negative velocity). Consequently, as is clearly seen in the right picture in Fig. 2, the ship will encounter many waves and the faster the vessel travels the more waves it will encounter. As the angle comes close to p=2 interesting things happen, e.g. the velocity in the left picture becomes very big (negative or positive). The explanation is that at this angle the wave direction is almost vertical to the ship, causing the spatial derivative in the x-direction, W x ðxi ; 0Þ, to be close to zero. Since the velocity is defined by V i ¼ W t ðxi ; 0Þ=W x ðxi ; 0Þ, a small spatial derivative gives rise to a large velocity. Another interesting phenomenon is that for a ¼ p=2 the intensity mY is the same for all ship velocities. This is due to the fact that for a long crested sea coming in from the direction p=2, the sea elevation is the same everywhere on the x-axis. Thus the sea surface elevation at the center of gravity of the ship will be the same regardless of the velocity of the vessel and measuring the elevation from a buoy fixed somewhere on the x-axis or at the center of gravity of the vessel would give rise to the same signal. Consequently, for a ¼ p=2, the intensity of waves in the encountered process Y ðtÞ is constant and equal to the intensity of waves in the process pffiffiffiffiffiffiffiffiffiffiffiffiffiffiW ffi ðtÞ ¼ W ð0; 0; tÞ. This intensity is given by ð2pÞ1 l02 =l00 . Finally, for a ¼ p the ship is sailing in a following sea with mean wave velocity close to 8 m/s, see left picture in
þ
4.1. Proof of (8) and (9) For a Gaussian sea, we have shown that mþ þ m ¼ mY and since E½ðV vÞþ E½ðV vÞ ¼ E½V v ¼ v¯ v one can compute the intensities by means of the following formulas: mþ ¼ 12ðmY þ nð¯v vÞÞ;
m ¼ 12ðmY nð¯v vÞÞ,
(16)
where mY is given by (14), the intensity of waves in space n by (11) and the average wave velocity v¯ by (13). Obviously, the intensities mþ , m can be computed when the spectral moments l00 , l20 , l02 , l11 are available, and are given by (8) and (9). In the following example we study the intensities mþ and m and compare mþ for longcrested and directional Gaussian seas with JONSWAP spectral density. 5. Examples Let SðoÞ be a JONSWAP frequency spectrum, i.e. a parametric spectrum of the form 5 h2s op 5 5=4ðop =oÞ4 expðð1o=op Þ2 =2sÞ SðoÞ ¼ e g , 16 op o oooc ,
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50
0.4
40
0.35
30 0.3
20
0.25
10 0
μY
mean wave velocity [m/s]
1565
-10
0.2 0.15
-20 0.1
-30
0.05
-40 -50
0
1
2
3
4
5
0
6
0
1
2
3
4
5
6
α [rad]
α [rad]
Fig. 2. Left—mean wave velocity v¯ as a function of angle a. Right—the intensity mY ða; vÞ of encountered waves as a function of the angle a for different velocities v ¼ 8 (solid), 12 (dashed), 16 (dotted) m/s.
0.4
0.12
0.35
0.1
0.3 0.25
0.06
μ-
μ+
0.08
0.2 0.15
0.04
0.1 0.02 0
0.05 0
1
2
3
4
5
6
α [rad]
0
0
1
2
3
4
5
6
α [rad]
Fig. 3. Left—the intensity mþ ða; vÞ of encountered waves that overtake the ship as a function of the angle a for different velocities v ¼ 8; 10; . . . ; 18 m=s. For a fixed a, mþ ða; vÞ is a decreasing function of ship velocity. Right—the intensity m ða; vÞ of encountered waves that the ship is passing as a function of the angle a for different velocities v ¼ 8; 10; . . . ; 18 m=s. For a fixed a, m ða; vÞ is an increasing function of ship velocity.
Fig. 2. If the velocity of the ship is of this magnitude the vessel travels along with the waves and it encounters few waves. As its velocity increases the intensity of encountered waves also increases. This explains the appearance of the right picture in Fig. 2 around a ¼ p. 5.2. Example 3 In Fig. 3 the intensities mþ and m of encountered waves that overtake the ship and that the ship is passing, respectively, are plotted as a function of the angle a and for velocities v ¼ 8; 10; . . . ; 18 m=s. For a fixed value of a, mþ is a decreasing function of ship velocity. This is easily explained since if a is fixed the intensity n in (15) is also fixed and the factor E½ðV vÞþ is a decreasing function of v. Similarly m is increasing with ship velocity. To get
some intuition of the qualitative appearance of these curves one can study the mean wave velocity in Fig. 2. The mean wave velocity is a location parameter of the velocity distribution given in (13). For small a the velocity distribution is centered around negative velocities and it is likely that the factor E½ðV vÞþ in the expression for mþ is small and thereby also the intensity mþ . However, as the angle a passes p=2 the mean wave velocity suddenly becomes positive and decreasing. This explains the discontinuity of mþ and it also makes it reasonable to believe that E½ðV vÞþ and mþ will decrease with a. Of course the shape of the velocity distribution as well as the interaction with the intensity n in (15) must be taken into consideration for a full explanation of the behavior of the intensity curve. Similar arguments can be presented for the shape of the m -curve.
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Directional Spectrum Level curves at: 2 4 6 8 10 12
90
0.8
120
60
0.6
150
0.12 30
0.4
0.1 0.2
0.08 0 μ+
180
0.06 0.04
330
210
0.02 240
300 270
0
0
1
2
3
4
5
6
α [rad]
Fig. 4. Left—the directional spectrum as function of a. Right—the intensity mþ ða; vÞ of encountered waves that overtake the ship as a function of the angle a for different velocities v ¼ 8; 10; . . . ; 18 m=s. For a fixed a, mþ ða; vÞ is a decreasing function of ship velocity v.
5.3. Example 4 Next we create a directional spectrum using a spreading function Gðo; yÞ of the cos 2s-type with parameter s ¼ 10. Here we should emphasize that this particular choice of parameter is for illustration purposes only. In Fig. 4 the directional spectrum as a function of a is shown together with the intensity mþ of encountered waves that overtake the ship. Comparing this intensity to the one obtained in the longcrested case shown in Fig. 3 shows that the intensity in the directional case essentially is a smoothed version of the intensity in the longcrested case. This is not surprising since the directional sea is a superposition of waves not only with different frequencies but also with slightly different directions.
Acknowledgments The second author was funded by the Swedish Foundation for Strategic Research, project Spatial statistics and image analysis for environment and medicine, A3 02:125.
Appendix A A.1. The generalized Rice’s formula Let ZðtÞ; Y ðtÞ be smooth processes, Y being continuously differentiable and Z continuous. Let N T ðzjuÞ be the number of t 2 ½0; T such that Y ðtÞ ¼ u, Y_ ðtÞ40 and
ZðtÞpz. Then, for almost all u, E½N T ðzjuÞ Z T E½jY_ ðtÞj1fY_ ðtÞ40;ZðtÞpzg jY ðtÞ ¼ u f Y ðtÞ ðuÞ dt, ¼
ð17Þ
0
where 1fAg is the indicator function equal to one if the condition A in brackets holds and zero otherwise. The formula (17) is called the generalized Rice’s formula. If the processes are jointly stationary, then define E½N T ðzjuÞ , (18) T the intensity of solutions to the equation Y ðtÞ ¼ u satisfying the conditions Y_ ðtÞ40 and ZðtÞpz. In addition, if the processes are jointly ergodic then the intensity can be obtained using
mðzjuÞ ¼ E½N 1 ðzjuÞ ¼
mðzjuÞ ¼ lim
T!1
N T ðzjuÞ . T
Under the joint stationarity assumption the intensity is given by, for almost all levels u, mðzjuÞ ¼ E½jY_ ð0Þj1fY_ ð0Þ40;Zð0Þpzg jY ð0Þ ¼ u f Y ð0Þ ðuÞ.
(19)
As we can see the generalized Rice’s formula is valid under minimal assumptions of smoothness of the processes involved and existence of the probability density f Y ð0Þ , see Leadbetter and Spaniolo (2004) for a detailed presentation of Rice’s formula and also Rice (1944, 1945) and Rychlik (2000). However (19) is valid only for almost all levels u and it may fail for a particular level u of interest, e.g. u ¼ 0. In order to prove the formula for a fixed level more restrictive assumptions are needed. For this it is sufficient
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to show that both sides of the equality in (19) are continuous functions of u. Continuity of the right-hand side is usually easier to check. For example, if the joint density of ðY ð0Þ; Y_ ð0Þ; Zð0ÞÞ exists and is sufficiently smooth then
A.2. Proof of Proposition 1
E½jY_ ð0Þj1fY_ ð0Þ40;Zð0Þpzg jY ð0Þ ¼ u f Y ð0Þ ðuÞ Z 1Z z ¼ jyjf Y ð0Þ;Y_ ð0Þ;Zð0Þ ðu; y; xÞ dx dy
Since
0
ð20Þ
0
m¯ þ ¼ E½jY_ ð0Þj1fY_ ð0Þ40;W x ð0;0Þo0g jW ð0Þ ¼ 0 f W ð0Þ ð0Þ.
jY_ ð0Þj1fY_ ð0Þ40;W x ð0;0Þo0g ¼ Y_ ð0Þþ 1fW x ð0;0Þo0g ¼ ðvW x ð0; 0Þ þ W t ð0; 0ÞÞþ 1fW x ð0;0Þo0g ¼ ðW x ð0; 0Þðv V ð0ÞÞÞþ 1fW x ð0;0Þo0g ¼ W x ð0; 0Þ ðV ð0Þ vÞþ ,
where V ð0Þ ¼ W t ð0; 0Þ=W x ð0; 0Þ, one has, again using (20), that m¯ þ ¼ E½W x ð0; 0Þ ðV ð0Þ vÞþ jW ð0; 0Þ ¼ 0 f W ð0;0Þ ð0Þ Z 1Z 0 ðx vÞjzjf W ð0;0Þ;W x ð0;0Þ;V ð0Þ ð0; z; xÞ dz dx. ¼ v
1
However, in many practical problems when nonGaussian models are employed the weaker version of Rice’s formula (19) is sufficient, see discussion in Rychlik (2004). This is also the case for the intensity of encountered waves. The problem that the intensity m may not be defined for every u is solved by introduction of the ‘‘averaged’’ intensity m¯ defined next. Suppose that the joint density of ðY ð0Þ; Y_ ð0Þ; Zð0ÞÞ exists and that right-hand side of Eq. (19) is computed using that (20) is a continuous function of u. Then the averaged crossing intensity mðzjuÞ is defined as ¯ follows: Z 1 uþ mðzjuÞ ¼ lim mðzjxÞ dx ¯ !0 2 u Z 1Z z ¼ jyjf Y ð0Þ;Y_ ð0Þ;Zð0Þ ðu; y; xÞ dx dy ð22Þ 0
1
with mðzjxÞ defined by (18). The average intensity will be used to prove that the formula (15) is also valid for nonGaussian sea models. Let mþ ðuÞ be the intensity of t such that Y ðtÞ ¼ u, W x ðvt; tÞo0 and Y_ ðtÞ40, defined by (18), then the intensity m¯ þ , defined in (22), is given by Z 1 þ þ m¯ þ ¼ lim m ðuÞ du. (23) !0 2 Obviously, m¯ þ ¼ mþ ð0Þ ¼ mþ if mþ ðuÞ is continuous, e.g. for a Gaussian sea. We also regard continuity of mþ ðuÞ as a property that any reasonable model for sea elevation should satisfy. Consequently, the intensity of encountered waves overtaking a ship can be defined by Z 1Z 0 zf Y ð0Þ;Y_ ð0Þ;Zð0Þ ð0; z; xÞ dx dz, (24) m¯ þ ¼ 0
Consider Eq. (24) and use (20) to write
1
and calculus can be used to proof the continuity right-hand side of Eq. (19). To prove that the left side in (19) is a continuous function of u is usually much harder. However, in the most important case when Y ðtÞ is Gaussian and ZðtÞ is a smooth function of Gaussian processes then, under mild regularity assumptions, see Leadbetter et al. (1983), the joint density of Y ð0Þ; Y_ ð0Þ; Zð0Þ exists and, for every u, Z 1Z z mðzjuÞ ¼ jyjf Y ð0Þ;Y_ ð0Þ;Zð0Þ ðu; y; xÞ dx dy. (21)
1
R1 R0 where Y ðtÞ ¼ W ðvt; tÞ, ZðtÞ ¼ W x ðvt; tÞ and 0 1 zf Y ð0Þ;Y_ ð0Þ;Zð0Þ ðu; z; xÞ dx dz is a continuous function of u, which usually is easy to verify.
1567
1
Now R0 f V 0 ðxÞ ¼
jzjf W ð0;0Þ;W x ð0;0Þ;V ð0Þ ð0; z; xÞ dz R0 1 jzjf W ð0;0Þ;W x ð0;0Þ ð0; zÞ dz
1
is the probability density of some random variable V 0 and R0 with nð0Þ ¼ 1 jzjf W ð0;0Þ;W x ð0;0Þ ð0; zÞ dz we have that m¯ þ ¼ nð0ÞE½ðV 0 vÞþ .
(25)
Now in the Gaussian case m¯ þ ¼ mþ , n ¼ nð0Þ and V 0 has the same density as V and hence (15) holds, concluding the proof in the Gaussian case. In the more general case we need to give physical interpretation to the density f V 0 ðxÞ and the intensity nð0Þ. This will be done next. Let V ðxÞ ¼ W t ðx; 0Þ=W x ðx; 0Þ and let xi 40 be solutions to W ðx; 0Þ ¼ u such that W x ðx; 0Þo0. Now V i ¼ V ðxi Þ is the velocity of the ith downcrossing of level u by W ðx; 0Þ. The empirical distribution of these velocities is F V u ðvÞ ¼ PðV pvÞ number of xi ox such that V i pv , ¼ lim x!1 number of xi ox
ð26Þ
while nðuÞ is the intensity of xi . Now using the generalized Rice’s formula one can show, see Rychlik (2004), that for almost all u the density of V u is given by R0 f V u ðxÞ ¼
jzjf W ð0;0Þ;W x ð0;0Þ;V ð0Þ ðu; z; xÞ dz . R0 1 jzjf W ð0;0Þ;W x ð0;0Þ ðu; zÞ dz
1
If the right-hand side of the last equation is a continuous and bounded function of u then Z 1 þ f V¯ ðxÞ ¼ lim f V u ðxÞ du ¼ f V 0 ðxÞ. !0 2
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I. Rychlik et al. / Ocean Engineering 34 (2007) 1561–1568
R0 Similarly, nðuÞ ¼ 1 jzjf W ð0;0Þ;W x ð0;0Þ ðu; zÞ dz for almost all u, so that n¯ ¼ nð0Þ and hence m¯ þ ¼ n¯ E½ðV¯ vÞþ .
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Longuet-Higgins, M.S., 1957. The statistical analysis of a random, moving surface. Philosophical Transactions of the Royal Society, Series A 249, 321–387. Podgo´rski, K., Rychlik, I., Machado, U.E.B., 2000a. Exact distributions for apparent waves in irregular seas. Ocean Engineering 27, 979–1016. Podgo´rski, K., Rychlik, I., Sjo¨, E., 2000b. Statistics for velocities of Gaussian waves. International Journal of Offshore and Polar Engineering 10, 91–98. Rice, S.O., 1944,1945. The mathematical analysis of random noise I and II. Bell System Technical Journal 23, 24. Rychlik, I., 2000. On some reliability applications of Rice’s formula for the intensity of level crossings. Extremes 3 (4), 331–348. Rychlik, I., 2004. Five lectures on reliability applications of Rice’s formula for the intensity of level crossings. Reliability-Based Design and Optimisation. IFTR, Warsaw, ISSN 1642-0578, pp. 241–323.