NEDERLANDS SCHEEPSSTUDIEGENTRUM TNO
N E T H E R L A N D S S H I P R E S E A R C H C E N T R E T N OS H I P B U I L D I N G D E P A R T M E N T L E E G H W A T E R S T R A A T 5, D E L F T
SOME ASPEGTS OF SHIP MOTIONS I N
IRREGULAR BEAM AND FOLLOWING WAVES
( E N I G E A S P E K T E N V A N S C H E E P S B E W E G I N G E N I N O N R E G E L M A T I G E D W A R S - E N A G H T E R I N K O M E N D E G O L V E N )
b y
DR. IR. B . D E J O N G (Twente University of Technology)
De aanvangsmetacenterhoogte van een schip in vlalc water wordt nog steeds als een belangrijlce parameter beschouwd bij het be-oordelen van de veiligheid van een schip tegen Icapseizen. Ook aan de dynamische aspekten van de dwarsscheepse stabiliteit in golven wordt echter steeds meer aandacht besteed. Desondanks is het zeker niet algemeen gebruikelijk om in het ontwerp-stadium van een schip een prognose te maken van de slinger-bewegingen in golven. Voornamelijk wordt dit veroorzaakt door een gebrek aan kennis met betrekking tot het gedrag onder extreme omstandigheden.
Bij het bestuderen van de dynamische stabiliteit in golven, kunnen twee situaties als de meest signifikante worden be-schouwd, namelijk het varen in dwarsinkomende golven en het varen in achterinkomende golven.
Een bijzonder verschijnsel dat kan voorkomen bij liet slingeren in onregelmatige dwarsinkomende golven is het maken van kaaiers, d.w.z. het optreden van plotselinge grote slingerhoeken.
Bij varen in achterinkomende golven kan, vooral bij snelle schepen de onplezierige situatie optreden dat een eenmaal ont-stane slagzij gedurende enige tijd gehandhaafd blijft. Zelfs kan bij deze koers t.o.v. de golven, de optredende periodieke verande-ring in dwarsscheepse stabiliteit een spontane slingerbeweging veroorzaken.
Zoals in dit rapport wordt aangetoond, kunnen deze ver-schijnselen, die een gevolg zijn van de niet-lineaire eigenschappen van het systeem ,,schip in golven", worden afgeleid met mathe-matisch-fysische methoden en worden uitgedrukt in statistische grootheden.
Deze analytisclie benadering is oorspronkelijk gepubliceerd als het proefschrift ter verkrijging van de graad van doctor in de technische wetenschappen van de auteur.
Daar de genoemde verschijnselen van groot belang zijn bij het vaststellen van de veiligheid van een schip, werd het nuttig ge-oordeeld om dit proefschrift bij een ruimer publiek bekend te maken door er een enigszins verkorte versie van te publiceren.
H E T N E D E R L A N D S S C H E E P S S T U D I E G E N T R U M T N O
The initial metacentric height of a ship in still water is still considered an important parameter in determining the safety of a vessel with respect to capsizing. However, more and more attention is also being paid to the dynamical aspects of the transverse stability in waves. Even so, predicting the rolling motions of a ship in waves in the design stage is by no means a common practice. Mainly this is due to a lack of knowledge concerning the behaviour of the ship under extreme conditions. Considering the problem of dynamic stability in a seaway, two situations can be regarded as the most significant ones, namely sailing in beam waves and sailing in following seas.
A special phenomena that may occur during rolling in irregular beam seas is known as lurching, characterized by unexpected high amplitudes of roll.
When sailing in a following sea, especially with a fast ship the unpleasant situation may occur that if a hst is obtained, it remains for some time in this position. The periodic change of transverse stability at this course relative to the waves may even cause a spontaneous rolling motion.
As is shown in this report these phenomena, due to the non-linear properties of the system 'ship in a seaway', can be derived by mathematical-physical methods and expressed in statistical quantities.
This analytical approach originally appeared as the author's doctor's thesis.
As the phenomena mentioned are of great importance when considering the safety of a ship it was thought useful to make this thesis known to a larger public by publishing a somewhat concise version of it.
L i s t of symbols 6
Summary 7
1 Introduction 7
2 Preliminary results 9 2.1 The h a r m o n i c s o l u t i o n o f the D u f f i n g equation 9
2.2 R a n d o m phase m o d e l ; instantaneous a m p l i t u d e a n d f r e q u e n c y ;
quasi-stationary a p p r o x i m a t i o n m e t h o d 11
2.3 T h r e s h o l d crossing p r o b a b i l i t y 16
3 Some statistical characteristics with regard to the lurching motion in a
beam sea 19 3.1 T h e average n u m b e r per second o f lurches, experienced by a vessel
i n an irregular beam sea 19 3.2 T h e average n u m b e r o f lurches w h e n the r o l l i n g m o m e n t f u n c t i o n
has a very n a r r o w band w i d t h 22
4 The fraction of the time that a vessel is performing a lurching motion . 24
5 Probability density of the time intervals during which the vessel is
per-forming a lurching motion 30
6 Some notes on the transverse stability of ships 35 6.1 The d y n a m i c stability o f a ship i n an irregular f o l l o w i n g sea . . 35
6.2 R o l l i n g i n i r r e g u l a r l o n g i t u d i n a l waves 39
6.3 N u m e r i c a l example 41
References 45
K4>) R i g h t i n g a r m o f the stability m o m e n t Centre o f buoyancy Dr Residual d y n a m i c stability
Ar
D y n a m i c a l stability F R o l l i n g a m p l i t u d e G Centre o f gravity GM Metacentric height I R o l l i n g m o m e n t f u n c t i o n M, E x t e r n a l r o l l i n g m o m e n t M,, Stability m o m e n t R Envelope o f the r o l l i n g m o m e n t f u n c t i o n T Period o f encountery Specific gravity o f a fluid Surface wave a m p l i t u d e e Phase angle
V Frequency o f encounter co^ = 2nv <t> Heeling angle
C i r c u l a r frequency o f encounter V V o l u m e o f displaced fluid
SOME ASPECTS OF SHIP MOTIONS I N
IRREGULAR BEAM A N D FOLLOWING WAVES
by
D r . I r . B . D E J O N G
Summary
In the present report which is a concise edition of [1], three phenomena are considered which are essentially due to the nonlinear properties of the behaviour of a ship in a seaway, namely, lurching in an irregular beam sea, and loss of dynamical stability and spontaneous rolling in an irregular following sea.
Some statistical quantities are derived which characterize the sensivity of a vessel to these phenomena and, therefore, provide us with information with respect to the transverse stability of a vessel in an irregular sea.
We consider the power spectrum of the normal random sea to be relatively narrow, i.e., the amplitude and frequency of the waves vary slowly.
In virtue of these properties of the random sea, we apply the quasi-stationary approximation method to derive the desired statistical quantities.
It is assumed that the lurching motion occurs only for very high wave amplitudes. I t is shown that for this reason only the variation in amplitude of the sea is important for the derivation of the statistical quantities with respect to this phenomenon.
In our discussions with respect to the transverse stability of a ship in irregular longitudinal waves both the frequency and amplitude modulation of the irregular sea are talcen into account.
1 Introduction
A l r e a d y W i l l i a m F r o u d e recognized that the still water values o f the metacentric height a n d the d y n a m i c a l s t a b i l i t y are g i v i n g us o n l y sufficient i n f o r m a t i o n w i t h respect t o the transverse stability, w h e n the vessel is i n s t i l l water or i n a moderate seaway. U n d e r extreme c o n d i t i o n s , however, the p r o b l e m o f the stability o f a ship i n a seaway s h o u l d be considered f r o m the p o i n t o f view o f dynamics rather t h a n statics.
T h i s is clear f r o m the f a c t t h a t f o r a ship i n waves the r i g h t i n g arms o f the stability m o m e n t w i l l vary c o n d n u o u s l y . T h e r e f o r e , f o r a ship i n an irregular sea, the lengths o f these arms are r a n d o m f u n c d o n s , the stadsdcal properties o f these f u n c t i o n s being determined by the statistical characteristics o f the seaway a n d the geometry o f the ship. I n order t o be able t o p r e d i c t the safety o f a ship w i t h respect t o its transverse stability i n a n a n a l y t i c a l w a y , we s h o u l d have sufficient k n o w l e d g e a b o u t b o t h the behaviour o f a ship i n a n i r r e g u l a r sea a n d the external m o m e n t s w h i c h m i g h t w o r k o n a ship, e.g., due t o waves, w i n d , etc. H o w e v e r , since u p t o the present b o t h the behaviour o f a ship under extreme c o n d i t i o n s a n d sufficient k n o w l e d g e a b o u t t h e nature o f the sea o n various ship routes are still l a c k i n g , the present standards o f stability are t o a h i g h degree based o n a statistical i n v e s t i g a t i o n o f the experiences w h i c h have been encountered b y ships i n the past. T h i s i n v e s t i g a t i o n has been i n i t i a t e d b y J. R a h o l a ( " T h e J u d g i n g o f the Stability o f Ships a n d the D e t e r m i n a t i o n o f the M i n i m u m A m o u n t o f S t a b i l i t y " , H e l s i n k i , 1939). B y analysing the transverse stability o f b o t h a great n u m b e r o f casualties i n v o l v i n g capsizings and a great n u m b e r o f vessels w h i c h operated i n a satisfactory w a y he devised a h i s t o g r a m f o r the righting arms o f the stability m o m e n t s w h i c h yielded the curve o f the m i n i m u m a m o u n t o f static stability f o r safety at sea i n undamaged c o n d i t i o n .
T h i s m e t h o d has p r o v e d t o be rather successful.
B y analysing l o g - b o o k s c o n c e r n i n g dangerous situations o f ships at sea due t o phenomena where the stability o f the ship is i n v o l v e d , i t is concluded t h a t some special circumstances can be indicated w h i c h are i n p a r t i c u l a r responsible f o r large heeling angles. These are w h e n the ship is sailing i n beam waves or w h e n i t is i n f o l l o w i n g seas w i t h a speed w h i c h is a b o u t equal t o the celerity o f the d o m i n a n t waves. A l t h o u g h the r u n n i n g o f ships i n beam seas has numerous disadvantages a n d , therefore, this s i t u a t i o n w i l l be avoided w h e n resonance pheno-mena are causing excessive r o l l i n g m o t i o n s , there are occasions w h e n this is necessary, e.g., i n shallow restricted waters or w h e n the ship has lost p r o p u l s i o n p o w e r or r u d d e r c o n t r o l . T h i s s i t u a t i o n also occurs w h e n the c o n t r o l l a b i h t y o f a ship w h i c h is a t t e m p t i n g t o r u n downsea is overpowered b y the wave a c t i o n , w h e r e u p o n i t swings broadside, beam t o the sea. T h i s phenomena is k n o w n as b r o a c h i n g .
A quite hazardous manner o f roUing w h i c h m a y occur t o a ship i n beam seas is k n o w n as l u r c h i n g w h i c h m a y be described as a staggering m o t i o n , characterized b y sudden h i g h amplitudes a n d h i g h accelerations.
I n the present r e p o r t the d e r i v a t i o n o f the statistical characteristics w i t h respect t o the l u r c h i n g m o t i o n i n a beam sea w i l l be based o n the f o l l o w i n g m a t h e m a t i c a l m o d e l , (see [2])
i n w h i c h I{t) represents the r o l l i n g m o m e n t f u n c t i o n w h i c h is r a n d o m i n a n i r r e g u l a r sea. F u r t h e r , the representation o f the b u o y a n t f o r c e w h i c h is a n o d d n o n l i n e a r f u n c t i o n is i n o u r calculations restricted t o the first t w o terms o f its Taylor-series expansion.
A l t h o u g h the representation (1.1) is rather simple, i t w i l l be seen t h a t the m e t h o d o f c a l c u l a t i o n o f the statistical quantities o f the l u r c h i n g m o t i o n is easy t o extend t o m o r e c o m p l i c a t e d models.
I n case the i n p u t f u n c t i o n I{t) o f (1.1) is h a r m o n i c , we o b t a i n
T h i s e q u a t i o n is k n o w n as the D u f f i n g e q u a t i o n a n d has been t h o r o u g h l y investigated, (see e.g. [3, 4, 5, 6 ] ) . One o f the r e m a r k a b l e properties o f this e q u a t i o n is the occurrence o f discontinuities i n the i n p u t - o u t p u t r e l a t i o n . I n the a m p l i t u d e - f r e q u e n c y diagrams o f above oscillator this p r o p e r t y manifests itself as a j u m p i n the o u t p u t a m p l i t u d e w h e n the a m p l i t u d e a n d / o r the f r e q u e n c y o f the i n p u t passes t h r o u g h some c r i t i c a l value. I n systems w h i c h are governed b y (1.2) this p h e n o m e n o n is observed as a sudden increase or decrease i n a m p l i t u d e w h i c h corresponds w i t h the descriptions o f the l u r c h i n g m o t i o n i n log-books.
W i t h respect t o the d y n a m i c a l phenomena concerning the transverse stability o f a ship i n a f o U o w i n g sea, i t is r e p o r t e d t h a t ships exhibited sometimes signs o f a t e m p o r a r y loss o f transverse stability w h e n the ship speed was a b o u t the same as the wave celerity. These phenomena are i n p a r t i c u l a r observed f o r fast c o n t a i n e r ships w i t h a l o w p r i s m a t i c coefficient. I t happens that, i n case the ship has at some m o m e n t an angle o f i n c l i n a t i o n due t o an external m o m e n t , i t remains f o r a l o n g w h i l e i n t h a t p o s i t i o n a n d appears n o t t o be i n a h u r r y t o restore the u p r i g h t p o s i t i o n . O n l y , by m a k i n g a drastic course change, a q u i c k r e s t o r a t i o n o f the u p r i g h t p o s i t i o n can be effected.
I t w i l l be s h o w n i n chapter 6 t h a t these phenomena are essentially due t o a change i n the transverse s t a b i l i t y o f the ship. I t has been s h o w n by B . A r n d t and S. R o d e n [7] that i n a h a r m o n i c f o l l o w i n g wave the curve o f r i g h t i n g arms varies p e r i o d i c a l l y between t w o extreme positions w i t h a f r e q u e n c y equal t o the f r e q u e n c y o f encounter. These extreme positions are determined b y the geometry o f the ship, the wave l e n g t h a n d the wave height. F u r t h e r , f o r every wave height, the a m p l i t u d e o f o s c i l l a t i o n o f the stability curve has its largest value w h e n the wave l e n g t h is a b o u t equal t o the ship l e n g t h , a t t a i n i n g its lowest value w h e n the crest is i n the m i d s h i p r e g i o n and its highest value f o r a t r o u g h i n this region. F o r increasing or decreasing values o f the wave l e n g t h , the influence o f t h e waves o n the transverse s t a b i l i t y w i f l d i m i n i s h . I f n o w the lower fimit o f the curve o f r i g h t i n g arms is b e l o w the m i n i m u m prescribed stability curve, t h e n , d u r i n g each p e r i o d o f encounter, there is a time i n t e r v a l i n w h i c h the stability o f the ship does n o t f u l f i l the m i n i m u m requirements o f transverse stability. T h i s t i m e i n t e r v a l w i l l become longer w h e n the ship speed is v a r i e d i n such a w a y t h a t the f r e q u e n c y o f encounter has a smaller value.
I n c o n n e c t i o n w i t h this issue t w o papers o f G r i m [8, 9] s h o u l d be m e n t i o n e d . Starting f r o m t h e f a c t t h a t the largest change i n the transverse stability occurs w h e n the wave length is equal t o the ship l e n g t h , he investigates i n w h i c h measure the r a n d o m f o l l o w i n g sea o n the sides o f the ship can be a p p r o x i m a t e d b y a regular wave w i t h a wave length equal t o the ship l e n g t h [ 8 ] . T h e n the a m p l i t u d e o f this so-called effective wave is a r a n d o m t i m e f u n c t i o n . The p o w e r spectrum o f this f u n c t i o n is used f o r the d e t e r m i n a t i o n o f statistical quantities w i t h respect t o the a m p l i t u d e o f this wave w h i c h give an insight i n t o the changes o f the transverse s t a b i h t y i n t h a t p a r t i c u l a r r a n d o m sea. I n his other paper [9] G r i m p o i n t s o u t t h a t the periodic changing o f the transverse stability i n a regular l o n g i t u d i n a l wave can given rise t o a spontaneous r o l l i n g m o t i o n . T h e i n i t i a l metacenter height GM is assumed t o v a r y h a r m o n i c a l l y a b o u t some average value GMQ w i t h a m p l i t u d e AGM a n d f r e q u e n c y my equal t o the f r e q u e n c y o f encounter:
4> + 2k<p + a4> + b4>^ =I{1) (1.1)
^ + 2k(i) + a(j) + b(l)^ = 7? cos (cor+ 0) (1.2)
GM = GMg - AGM cos coj (1.3)
+ r V ( G M o - A G M cos coj)(l> = 0 (1.4)
T h i s equation is k n o w n as M a t h i e u ' s d i f f e r e n t i a l e q u a t i o n w h i c h can be w r i t t e n i n the f o l l o w i n g c a n o n i c a l f o r m
0 + ( f l - 2 ( / c o s 2 O 0 = 0 (1.5)
I n the literature this e q u a t i o n has been studied very extensively a n d the stable a n d unstable regions i n the a, q-plane o f the solutions o f this equation have been determined. O f course, these regions refer t o s t a t i o n a r y situations, where has a constant value.
I n this r e p o r t some statistical characteristics concerning above m e n t i o n e d nonlinear phenomena are derived by using methods w h i c h are essentially all based o n the F o u r i e r series m e t h o d o f Rice, (see Rice [10] or M i d d l e t o n [11]). W e f u r t h e r assume that the wave exciting m o m e n t has a relatively n a r r o w s p e c t r u m :
A v < v , „ (1.6)
where Av is the characteristic b a n d w i d t h and v,„ a representative m i d b a n d frequency, ( F i g . 1.1).
Vm V Fig. 1.1
T h e i n p u t o f e q u a t i o n (1.1) is then a quasi-harmonic signal w h i c h can be represented i n the f o r m :
7 ( 0 = i ? ( 0 cos (CO,,,/+ 0 ( 0 ) (1-7)
where R{t) and 0 ( 0 are r a n d o m f u n c t i o n s w h i c h v a r y slowly as compared w i t h cos (o,„t. F o r this reason we w i l l apply the quasi-stationary a p p r o x i m a t i o n m e t h o d f o r the d e t e r m i n a t i o n o f statistical quantities w i t h respect t o the j u m p p h e n o m e n a i n the o u t p u t . So, we assume t h a t the o u t p u t at t i m e / is o n l y determined b y t h e values w h i c h the instantaneous a m p l i t u d e R{t) a n d the instantaneous frequency ca,„ + 0(/) have at that very m o m e n t . T h e quasi-stationary m e t h o d w i l l also be a p p l i e d f o r calculations w i t h respect t o the unstable p h e n o m e n a o f a ship i n l o n g i t u d i n a l waves.
2 Preliminary results
2.1 The harmonic solution of the Duffing equation
T h e DuflRng e q u a t i o n (1.2) has been studied t h o r o u g h l y i n the literature [3, 4, 5, 6, 12]. I n a d d i t i o n t o a h a r m o n i c c o m p o n e n t F cos co/, the s o l u t i o n o f this e q u a t i o n contains also subharmonics, u l t r a h a r m o n i c s a n d u l t r a s u b -h a r m o n i c s , (see [ 3 ] ) . H o w e v e r , w -h e n Ic and \b\ are n o t t o o large t-he -h a r m o n i c s o l u t i o n w i l l be d o m i n a n t a n d f o r this reason we shall consider o n l y the h a r m o n i c c o m p o n e n t i n the s o l u t i o n .
S u b s t i t u t i n g t f ) = Fcos mt i n (1.2) and r e t a i n i n g o n l y the h a r m o n i c terms i n the resulting e q u a t i o n , we find by equating the coefficients o f cos mt a n d sin mt:
(a-m^+ibF^)F= R cose
i n w t i i c h the r o l l i n g a m p l i t u d e F and the phase angle Ö are u n k n o w n variables. Squaring a n d a d d i n g yields the w e l l - k n o w n amplitude-frequency r e l a t i o n
{{a-co^+%bF^Y + Am'-k^)F^ = (2.1.2)
The response curves, represented i n F i g . 2.1.1, are obtained by p l o t t i n g the relationship (2.1.2) f o r co > 0. The value f o r b is taken positive. I n this r e p o r t a l l results w i l l be derived f o r the case b>0. F o r è < 0 the calculations can be carried out along the same lines. I n t h a t case the curves i n F i g . 2.1.1 are leaned over t o the l e f t instead o f t o the r i g h t .
W e o b t a i n an expression f o r the lines F2(a>) and i^i(co) o n w h i c h the tangents o f the a m p l i t u d e - f r e q u e n c y curves are vertical by d i f f e r e n t i a t i n g (2.1.2) w i t h respect t o F a n d p u t t i n g dco/dF = 0. T h i s results i n the e q u a t i o n :
jyb^FU3bF\a~co^) + {a^ + (o^-2aco^ + 4kW) = 0 (2.1.3)
w i t h solutions:
Fl=^^(co^-a) + ^^{Uco^-ar-3kW}^
(2.1.4)
Since we consider o n l y the absolute values o f F we o b t a i n as f i n a l result:
l ^ i l =
\F2\ =
9b^ ' 9|5,
9/j
(2.1.5)
F r o m these expressions we see t h a t the amphtude-frequency curves have o n l y vertical tangents w h e n co satisfies the i n e q u a l i t y :
i ( c o ' - f l ) 2 - 3 A r ' c o ^ ^ 0 (2.1.6)
F o r the value (Ü = (ÜQ f o r w h i c h the equality sign is v a l i d we find:
Jo = [ a + 6/<" ± {36^-^ + 1 2 f c ' f l } * ] ^ è ^ 0 (2.1.7)
T h u s f o r some a r b i t r a r y frequency co = co,„ w h i c h satisfies the i n e q u a l i t y (2.1.6) the tangents o f the a m p l i t u d e -frequency curves are vertical w h e n the absolute value o f the o u t p u t a m p f i t u d e F satisfies the equations | F | =
l-^i(<Wm)i or \F\ = |F2(co„,)|, where the r i g h t - h a n d side o f these equations is given b y (2.1.5). T h e c o r r e s p o n d i n g values f o r the i n p u t a m p l i t u d e w h i c h we denote b y Ri,{co,„) a n d 7?j(co,„), (see F i g . 2.1.1), are f o u n d b y sub-s t i t u t i n g sub-succesub-ssub-sively | F | = |Fi(co,„)| a n d | F | = |F2(co„,)| i n e q u a t i o n (2.1.2) and sub-solving the resub-sulting t w o equationsub-s i n R. Replacing again cu,„ b y co we find:
Rj,{co) 8 / 2 N3 32 2J 2/ 2 X 64 g ^ ( c « ^ - a ) N ^ c « W - « ) - 3 ^ 8 . 2 x3 32 2 , 2 , 2 ^ 64 ^ ( c o ^ - « ) 3 + - c « W - ) + 8 j ^ { i ( c o ' - a ) ^ - 3 / c ^ c o ' } * { K c o ' - f l ) ' - 3 / ( ' c o ' } * (2.1.8) W e r e w r i t e e q u a t i o n (2.1.2) i n the f o r m :
w h i c h is an e q u a t i o n o f the t h i r d degree i n i n w h i c h the variable i? is a parameter. W e see f r o m F i g . 2.1.1. t h a t f o r any = CÜ,„ w h i c h satisfies (2.1.6) the e q u a t i o n (2.1.9) has f o r suitable values o f R three real r o o t s . We f u r t h e r see t h a t this e q u a t i o n has t w o equal roots F ^ = F i ( c o „ , ) f o r R = Rj,{o},„). Consequently, we m a y write (2.1.9) i n the f o r m :
^b'{F'-Flico,„)r{F'-Fl{co„:)} = 0
F r o m this e q u a t i o n the t h i r d r o o t FI((O,„) is easy to determine. Replacing again co,„ b y co, we o b t a i n :
(2.1.10)
1^31 = (2.1.11)
A l o n g the same lines, by s u b s t i t u t i n g co = co,,, a n d R = Riia>,„) i n (2.1.9), we o b t a i n an e q u a t i o n w h i c h has a d o u b l e r o o t F^ =Fl(a>,i,) and a t h i r d r o o t F^ =FI{O],„) w h i c h is given b y :
1^-41 =
(2.1.12)
T h e nonlinear restoring f o r c e acfy + bcj)^ o f the system (1.2) produces the w e l l - k n o w n j u m p and hysteresus pheno-mena. These properties are very well demonstrated by considering the f o l l o w i n g t w o cases:
We assume the a m p l i t u d e and f r e q u e n c y o f the i n p u t t o have the values R„ a n d co„„ respectively; so we start at p o i n t 1 o n the curve R = R„in F i g . 2.1.1. W h e n we increase R t h e n \F\ increases i n a c o n t i n u o u s w a y t i l l the p o i n t 2 is reached. F i g . 2.1.1 suggests t h a t a f u r t h e r increase o f R w i h cause a j u m p i n the o u t p u t a m p l i t u d e f r o m |Fi(co,„)| t o F3(ca,„)| after w h i c h | F | increases again i n a c o n t i n u o u s way. W h e n R is n o w decreased, we may expect a j u m p f r o m the p o i n t 4 t o the p o i n t 5. These j u m p phenomena have been verified experimentally.
|F| F3(ü)m)| Fjdü, F,(cuj| FJ (U , ''|F3(U)| ^ / 3 ' ^ jFj(u)| Fig. 2.1.1
A b o v e m e n t i o n e d phenomena occur also w h e n the a m p l i t u d e is k e p t constant, e.g., R = Rn and the f r e q u e n c y is decreased or increased. W e start f r o m the p o i n t P o n the curve R = Rnim,,,). A decrease o f f r e q u e n c y causes a j u m p f r o m p o i n t 2 t o p o i n t 3 and after that an increase of the f r e q u e n c y w i l l cause a j u m p f r o m p o i n t 6 t o p o i n t 7. Consequently, above m e n t i o n e d j u m p phenomena take always place f r o m a p o i n t o n the curve |Fi(co)| to a p o i n t o n the curve |F3(co)| or f r o m a p o i n t o n the curve |F2(co)| t o a p o i n t o n the curve |F4(co)|.
O b v i o u s l y , the values o f | F | a n d co between the curves |Fi(co)i and |F2(co)| never occur i n a stationary s o l u t i o n . These solutions are said t o be unstable (see [ 3 ] , chapter V I , section 6).
We notice t h a t the responsecurves s h o w i n g the v e r t i c a l j u m p s refer t o stationary situations. I n r e a l i t y , h o w -ever, there w i l l be a t r a n s i t i o n stage, d u r i n g w h i c h the a m p l i t u d e o f the o u t p u t changes at a finite rate w h e n the unstable region is passed.
2.2 Random phase model; instantaneous amplitude and frequency; quasi-stationary approximation metliod We assume t h a t the roUing m o m e n t f u n c t i o n / ( / ) i n e q u a d o n (1.1) w h i c h extends f r o m / = - o) t o r = co is a s t r i c t l y stationary, ergodic a n d Gaussian process. W i t h o u t r e s t r i c t i n g the generality o f the discussion we m a y assume t h a t the average value o f the process has zero value:
B E F O R E I N V E S T I G A T I N G V A R I O U S P R O P E R T I E S O F TLIE R A N D O M P R O C E S S / ( / ) , W E R E P R E S E N T IT I N A N A P P R O X I M A T E F O R M W L I I C H I S S U I T A B L E F O R ALL K I N D S O F A N A L Y T I C A L M A N I P U L A T I O N S . S E V E R A L R E P R E S E N T A T I O N S O F A G A U S S I A N P R O C E S S A R E P O S S I B L E [ 1 0 , 1 1 ] . I N O U R C A S E I T I S C O N V E N I E N T TO U S E T H E R A N D O M P H A S E M O D E L W H I C H I N T H E REAL C A S E H A S T H E F O R M : 1 ( 0 = E C „ C O S ( C O „ F - ( P „ ) ( 2 . 2 . 2 ) T H E A N G L E S (t>2, 4>n A R E I N D E P E N D E N T S T O C H A S T I C V A R I A B L E S W I T H A U N I F O R M D I S T R I B U T I O N O V E R T H E R A N G E ( 0 , 2 ? : ] . F U R T H E R C„ = [ 2M < V„ ) AV ] %
(O„ = 2nv„
A N D V„ = n Av ( 2 . 2 . 3 ) I N W H I C H N ' ( V ) R E P R E S E N T S T H E P O W E R S P E C T R U M O F T H E R O L L I N G M O M E N T F U N C T I O N W H I C H I S N O N - N E G A T I V E A N D C O N T I N U O U S . T H E C O R R E L A T I O N F U N C T I O N I S D E F I N E D B Y rP(x) = EI{t)I{t + ,)= L I M - L J- / ( 0 / ( F + T) D / C O M B I N I N G T H I S W I T H ( 2 . 2 . 2 ) A N D ( 2 . 2 . 3 ) , W E O B T A I N : 00 I/ ^ (T) = J \ V ( V ) C O S 2711'! D V 0 F O U R I E R ' S I N T E G R A L T H E O R E M Y I E L D S T H E I N V E R S E R E L A T I O N : CO ) I ' ( V ) = 4 J I/ ^ (T) C O S 27CVT D T ( 2 . 2 . 4 ) ( 2 . 2 . 5 ) ( 2 . 2 . 6 ) B Y U S I N G A B O V E E X P R E S S I O N S F O R I ( t ) , ^{T) A N D ) I ' ( V ) , W E M A Y E X P R E S S T H E V A R I A N C E O F T H E S I G N A L OR T H E TOTAL A V E R A G E P O W E R I N T E R M S O F / ( / ) A N D N ' ( V ) : El\t) = L I M ^ J l\t)At = m = J « < V ) D V ( 2 . 2 . 7 ) B Y R E P R E S E N T I N G I ( t ) B Y T H E S E R I E S ( 2 . 2 . 2 ) T H E C O N T I N U O U S S P E C T R A L D E N S I T Y )T'(V) I S R E P L A C E D B Y A D I S C R E T E S P E C T R U M W H I C H H A S O N L Y V A L U E S F O R V j , V j , I',„ ( S E E F I G . 2 . 2 . 1 ) : H ' ( V ) = i I C , ^ 5 ( V - V „ ) 11 = 1 ( 2 . 2 . 8 ) T L I I S S P E C T R U M B E L O N G S T O A S I G N A L W I T H A FINITE L E N G T H T = 1 / A V . T H E A V E R A G E P O W E R O F T H E H A R M O N I C O S C I L L A T I O N W I T H F R E Q U E N C Y V„ I S E Q U A L TO I I ' ( V „ ) A V . T H I S I S J U S T T H E A R E A O F T H E R E C T A N G L E W H I C H H A S ITS B A S E O N T H E V - A X I S w(v)i V„-i4V V„»i4V Fig. 2.2.1between v „ - ^ A v and v„ + ^Av as indicated i n F i g . 2.2.1. By a p p l y i n g the central l i m i t t h e o r e m i t can be shown t h a t the representation (2.2.2) converges t o a n o r m a l d i s t r i b u t e d process w i t h average value zero and variance
w h e n the discrete spectrum (2.2.8) approximates the corresponding c o n t i n u o u s one. T h i s p r o o f is, f o r example, given by Rice i n [10]. H e shows t h a t the d i s t r i b u t i o n o f the sum o f the series (2.2.2) approaches the n o r m a l l a w w h e n 7V-> CO and A v - > 0 such t h a t A'Av = F, where F i s any positive constant.
T h e m o d e l (2.2.2) w i l l be used to derive several results r e l a t i n g t o the process I{t). However, one o f the m a n i p u l a -tions i n these deriva-tions w i l l always be the l i m i t i n g o p e r a t i o n 7V-> oo and Av 0 so t h a t the final results always relate t o the o r i g i n a l n o r m a l process w i t h a continuous spectrum. A n a l o g o u s t o Rice's m e t h o d we choose a f r e q u e n c y co,,, w h i c h is a representative m i d b a n d frequency and w r i t e (2.2.2) i n the f o r m :
N
1(0 = E c„ cos {mJ - m,„t - <p„ + o i j ) = I , cos ro,,,/ - ƒ, sin ro,,,/ (2.2.9) n = l where N 4 = Z c „ c o s ( r o „ / - r o „ , / - ( p „ ) (2.2.10) N / , = Z c „ s i n ( c o „ / - r o , „ / - ( p „ ) 11=1
I t is obvious t h a t i n the fimit N^co a n d A v ^ O the processes 4 ( / ) a n d / ^ O have also a n o r m a l d i s t r i b u t i o n a n d a c o n t i n u o u s power spectrum and are also ergodic.
W e assume the power spectrum t o be relatively n a r r o w . T h e n the values o f c„ w i l l decrease very fast t o z:ero w h e n |/7 —/77| increases i n value. T h u s I ^ f ) and / , ( / ) are very slowly v a r y i n g f u n c t i o n s as c o m p a r e d w i t h cos ro,,,/
W e d e f i n e : R{t)=ll',+in^ (2.2.11) a n d 0(/) = arctg (2.2.12) T h e n (2.2.2) can be w r i t t e n i n the f o r m : / ( / ) = ^ ( / ) c o s ( r o „ , / - f 0 ( / ) ) (2.2.13)
I n v i r t u e o f the relations (2.2.11) a n d (2.2.12) the f u n c t i o n s R{t) and 0(/) are also s l o w l y v a r y i n g f u n c t i o n s as c o m p a r e d w i t h cos ro,,,/ when the spectrum is relatively n a r r o w . T h e r e f o r e , we m a y i n t e r p r e t R(t) as the envelope o f the signal / ( / ) . F r o m the physical p o i n t o f view this is also clear by considering an oscillogram of a Gaussian process w i t h a relatively n a r r o w spectrum, ( F i g . 2.2.2).
Fig. 2.2.2
I t can be s h o w n that 0 ( / ) is u n i f o r m l y d i s t r i b u t e d over the range (0,27T] w h i l e the envelope R{t) has the R a y l e i g h d i s t r i b u t i o n
R
represents here the t i m e derivative o f the phase f u n c t i o n 9{t).
T h e choice o f the m i d b a n d f r e q u e n c y m,„ is rather a r b i t r a r y and i t is n o t i m m e d i a t e l y evident whether or n o t a d i f f e r e n t choice o f this f r e q u e n c y leads to the same R{t) a n d c»,(?). A l s o i t is n o t clear w h a t becomes o f o u r physical concept o f envelope w h e n the process has a spectrum w h i c h is n o t relatively n a r r o w . I n order t o e l i m i n a t e this arbitrariness we shall give a m o r e direct f o r m u l a t i o n o f R{t) a n d cu,(/). W e i n t r o d u c e the c o m p l e x -valued f u n c t i o n
z ( 0 = £ c / f " " ' - " " * (2.2.15)
w h i c h is the complex f o r m o f the r a n d o m phase m o d e l (2.2.2) f o r the process / ( / ) .
W e n o w define the envelope R{t) a n d the instantaneous f r e q u e n c y C Ü,(/) o f the process / ( / ) , respectively, as the absolute value and the t i m e - d e r i v a t i v e o f the argument o f the c o m p l e x f u n c t i o n z(t).
T h u s representing z ( / ) b y z(0 = \z(t)\e''^'\ (2.2.16) we o b t a i n R{t) =
|z(0|
aj,.(0 = iA(0
(2-2.17) B y w r i t i n g z{t) i n the f o r m : z(0 = e"^'"' Z c„e 11 = 1 i{(w„-io,„)t-tpi,) (2.2.18)i t can be s h o w n t h a t the d e f i n i t i o n s (2.2.17) give the same result as the earher given d e f i n i t i o n s due t o Rice, f o r any choice o f the m i d b a n d f r e q u e n c y .
T h e d e f i n i t i o n s (2.2.17) are a special case o f the d e f i n i t i o n s f o r the envelope a n d instantaneous f r e q u e n c y as given b y D u g u n d j i [13, 14, 15] f o r a n a r b i t r a r y signal / ( / ) . H e started f r o m a c o m p l e x v a l u e d f u n c t i o n
z{t) = l{t) + il{0 (2.2.19)
w h i c h he called the pre-envelope o f the w a v e f o r m I{t). T h e f u n c t i o n I{t) represents the H i l b e r t t r a n s f o r m [16] o f the f u n c t i o n / ( / ) given by the p r i n c i p a l value o f the i n t e g r a l :
1(0 = 1 T l ^ d ^ (2.2.20)
T h e envelope and instantaneous f r e q u e n c y are o b t a i n e d b y s u b s t i t u t i n g expression (2.2.19) i n (2.2.17). So i n the general case:
R{t) = l l \ t ) + I \ t ) f rm^
ro,(0 = ^ a r c t g ^ ^ ^ ^ ^
T h e stochastic processes R{1) and cu;(/) are d e f i n e d o n the process I{t). A c c o r d i n g t o the elementary t h e o r y o f stochastic processes we m a y p r o v e t h a t the processes R{t) a n d co,(0 are also strictly s t a t i o n a r y a n d ergodic, since / ( / ) has also these properties.
W e note t h a t the f u n c t i o n
is the pre-envelope o f
00
f ( 0 = Z c„ cos ( ( « „ ( - ( ? „ ) ,
11= 1
since sin (o}„t-(p„) is the H i l b e r t t r a n s f o r m o f cos {mj — cp,).
W e saw i n the previotis section t h a t the stationary r e l a t i o n between the i n p u t and the o u t p u t variables display j u m p s f o r those values o f the a m p l i t u d e and frequency o f the i n p u t w h i c h satisfy the equations R = Rj{a)) a n d R = Rii(o]), where the r i g h t - h a n d sides o f these equations are given by (2.1.8). Consequently, we may expect t h a t under certain conditions the o u t p u t o f the system (1.1) w i t h i n p u t ƒ(/) = R{t) cos (co,,,/-I-0(0) w h i c h is b o t h a m p l i t u d e and frequency m o d u l a t e d w i l l also e x h i b i t discontinuous phenomena i n its a m p l i t u d e . W e n o w m a k e the decisive step by stating that j u m p s w i l l o n l y occur f o r those values o f the instantaneous a m p l i t u d e R{t) a n d f r e q u e n c y co,(/) = co,„-l-ö w h i c h satisfy the equations R = i?/(co;) and R = i?//(co,). W e come t o such a statement by assuming t h a t at t i m e / = /Q, w h e n the i n p u t f u n c t i o n has the instantaneous frequency cu, = co,„ + Ö(/o) a n d instantaneous a m p l i t u d e R{to), the h a r m o n i c o u t p u t f u n c t i o n is determined as i f the system is excited by the signal
R(to) cos {(co„, + Óit,))t + q>] (2.2.22)
a n d may therefore be w r i t t e n as F(/o) cos [{co„,-|-ö(/o)}/], where the a m p l i t u d e F is calculated b y s u b s t i t u t i n g R = R{to) a n d co = co,„ + Ö(/o) i n (2.1.2).
T h u s , i n this so-called quasi-stationary m e t h o d the o u t p u t at t i m e / = /Q is assumed t o depend o n l y o n the i n p u t at t i m e / = /Q and not on the t i m e h i s t o r y o f the i n p u t signal as is the case i n reality. F r o m an i n t u i t i v e p o i n t o f view i t is easy to understand that this a p p r o x i m a t i o n w i l l be m o r e accurate w h e n Rit) and 0 ( / ) are m o r e s l o w l y v a r y i n g f u n c t i o n s as c o m p a r e d w i t h cos co,,,/.
W e w i U derive n o w the rate o f a m p l i t u d e a n d f r e q u e n c y m o d u l a t i o n o f a Gaussian process w i t h a relatively n a r r o w b a n d w i d t h when the a m p l i t u d e values are very h i g h .
Consider the density p(R, R, 6) as derived i n appendix A , given b y
p(R,R,e) = ^ c x p
^ {b^R' + b^{R^ + R'Ö') - 2b,R'9} (2.2.23)T h e exponent o f this density, w h e n considered as a f u n c t i o n o f 0, has a m a x i m u m value w h e n 6 = bJbQ. Replacing i n (2.2.23) the variable 0 by 0', given b y ;
i' = 9 - ^ (2.2.24) we o b t a i n
p(R,R,9') = -£^cxp (2.2.25)
F r o m this expression i t is easy t o see t h a t f o r a Gaussian process w i t h a given p o w e r s p e c t r u m :
R = 0([) and 0 = O ( R - ' ) w h e n R - > oo (2.2.26)
P u t t i n g i'„, = 0 i n the expressions ( A . 3 ) f o r the coefficients b„, the instantaneous frequency co, is given b y co, = 0. T h e n (2.2.26) shows that, i n sense o f the p r o b a b i l i t y theory, according as the instantaneous a m p f i t u d e o f the signal has a larger value, the value o f the instantaneous frequency w i l l d i f f e r less f r o m the value
00
In J vn'(v) dv
= ° (2.2.27) ^ ° J v v ( v ) d v
The j o i n t density o f the instantaneous a m p l i t u d e a n d frequency are derived f r o m (2.2.23) by p u t t i n g CÜ,„ = 0 and i n t e g r a t i n g w i t h respect t o R over the range ( - c o , oo). This yields:
p(R, CO) = exp (~—fb2-2biOJ+ èoco')) (2.2.28) (IboBj^ \ IB' ) where B' = b,b2-bl (2.2.29) a n d &„ = (27i)"|M'(v)v"dv (2.2.30) 0
2.3 Threshold crossing probability
L e t / ( c ü ) be some c o n t i n u o u s a n d d i f f e r e n t i a b l e f u n c t i o n o f the instantaneous frequences co. I f we take the m i d -b a n d frequency co„, equal t o zero, we have co = 0 w h i c h is a f u n c t i o n o f t i m e . W e assume t h a t the r a n d o m envelope R{t) o f the Gaussian process intersects the g r a p h o f the f u n c t i o n f { t ) i n some p o i n t t + x in the t i m e i n t e r v a l (t, t + dt), where the f u n c t i o n s ƒ (r) a n d R(t) satisfy the r e l a t i o n :
^>^I (2.3.1)
dt dt ^ ^
W e f u r t h e r assume t h a t the l e n g t h dt o f the i n t e r v a l is so small t h a t i n this i n t e r v a l the f u n c t i o n s f { t ) and R(t) can be considered as hnear w i t h p r o b a b i l i t y almost one. I n the p o i n t o f intersection we have:
f(t+r) =m+h = R(t + r) = R{t) + RT (2.3.2) where 0 < T < d / Solving f o r T , we find: ^ ^ m - R i t ) ^2.3.3) R - f
w h i c h yields, since we assumed T t o be i n the i n t e r v a l (t, t + dt):
f + f d t - R d t <R<f (2.3.4) Since • ^ = d ö ^ ' we o b t a i n f r o m (2.3.1): R>%,& (2.3.5)
de
w h i l e (2.3.4) can be w r i t t e n i n the f o r m : / + ^ 0 d r - « d f < R < / (2.3.6)Assume that the simultaneous p r o b a b i l i t y d i s t r i b u t i o n o f the variables R, R, 9 a n d 0 is represented b y p(R, R, 9, 0).
T h e n the p r o b a b i l i t y that the envelope R(t) o f the Gaussian process
7(0
= ^ ( / ) cos0(0
intersects the f u n c t i o n f { t ) i n the i n t e r v a l (/, t + dt) o n the c o n d i t i o n that (2.3.1) is v a l i d , w h i l e at the same t i m e the instantaneous frequency o f / ( / ) has a value i n the i n t e r v a l (0, 0 + d0), is given b y :
00 00 ƒ
d0 J d0 J dk ]dR piR,R,Ó,ë) (2.3.7)
de do
T h e i n t e g r a t i o n intervals f o r R and R result f r o m the inequalities (2.3.6) and (2.3.5), respectively. Since
%Bdt-Rdt d6
is very small i n c o m p a r i s o n w i t h / , we p u t i n the i n t e g r a n d R = ƒ. I n t e g r a t i n g (2.3.7) w i t h respect t o R, we o b t a i n : 0Ö m
AS ;
d / d 0 J d0 J d R ( i ? - ^ 0 piR=fiÓ),9,R,Ö) (2.3.8)
W h e n we assume t h a t the envelope R{t) intersects the f u n c t i o n f ( t ) o n the c o n d i t i o n that instead o f (2.3.1) the i n e q u a h t y :
is satisfied, then i t can be s h o w n t h a t the i n e q u a l i t y (2.3.6) has to be replaced b y :
f < R < f + ^ ^ ö d t - R d t (2.3.10)
C o n d i t i o n (2.3.9) m a y be w r i t t e n i n the f o r m :
R<%0 (2.3.11)
The expression f o r the p r o b a b i l i t y w h i c h is similar t o (2.3.8) except f o r the c o n d i t i o n (2.3.1) w h i c h is n o w replaced by (2.3.9) can be derived i n the same way and is given b y :
-dtdÖ ƒ d ö ' j dR(^R-^^Ö^p{R=m,9,R,9) (2.3.12)
-co — GO
I n case the f u n c t i o n f ( t ) = c is a constant, then b y p e r f o r m i n g an a d d i t i o n a l i n t e g r a t i o n w i t h respect t o 0 over the range ( - c o , co) expression (2.3.8) can be reduced t o :
CO
dt^ Rp(R = c,R)dR (2.3.13)
0
w h i c h is Rice's result f o r the p r o b a b i l i t y t h a t the envelope R{t) passes i n the t i m e i n t e r v a l (/, / + dO t h r o u g h the constant level ƒ = c w i t h a positive slope. I n a similar w a y we o b t a i n f r o m (2.3.12) f o r the p r o b a b i h t y t h a t R(t) passes i n {t, t + dt) t h r o u g h the level ƒ = c w i t h a negative slope:
0
- d r ƒ Rp(R = c,R)dR
— 00
The integrals o f the expressions (2.3.13) a n d (2.3.14) are said t o be the intensities o f the stream o f u p w a r d a n d d o w n w a r d crossings, respectively. O n account o f the stationary properties o f the process R{t), i t is easy t o see t h a t these quantities are i n o u r case independent o f t i m e and are equal t o the corresponding average n u m b e r o f crossings per second. F o r a more f u n d a m e n t a l discussion o f this issue the reader is referred t o [17, Section 10.5] or [ 1 , Section 2.4].
D e n o t i n g the average n u m b e r per second o f u p w a r d a n d d o w n w a r d crossings by A^,, a n d N^, respectively, we o b t a i n : CO N„= ^ Rp(R = c,R)dR (2.3.15) Af^ = - J Rp{R = c,R)dR - CO
C o m b i n i n g the arguments w h i c h yielded the f o r m u l a s (2.3.8) a n d (2.3.12) we can derive an expression f o r the j o i n t p r o b a b i l i t y t h a t R{t) passes t h r o u g h the f u n c t i o n ƒ ( / ) i n the t i m e i n t e r v a l ( / j , + d / J and after t h a t t h r o u g h the f u n c t i o n g(t) i n the i n t e r v a l ( / j , / i + d / i ) o n the c o n d i t i o n t h a t i n the first intersection p o i n t the slopes o f R(t) a n d f ( t ) satisfy the i n e q u a f i t y d i ? / d / > d / / d / a n d the instantaneous frequency has a value i n the i n t e r v a l ( 0 i , ö i + d 0 i ) , w h i l e i n the second intersection p o i n t R(t) a n d g{t) satisfy dR/dt < dg/dt and the instantaneous frequency has a value i n ( Ö j , Öj + d Ö j ) . T h e f o U o w i n g result is o b t a i n e d :
00 OO m déz ( df \ / da
-dt,dt2dó,dÓ2 J dë, J dk, J d02 I dkJk,~-^9A(k2--^u2
d é / '•piR, =KÓ,),9„k„ü„R2 = g{92)A,R2,Ö2) (2.3.16)
where k , , 9,, 9,, R2, A j , 02, 02) represents the j o i n t density o f the variables R,, Rj^, 9,, 9, a n d R2, 02, 02 at t i m e / = t, and t = t2, respectively. W h e n the f u n c t i o n s ƒ a n d g are constants:
ƒ = Cl and g = C2 (2.3.17)
t h e n , after p e r f o r m i n g the integrations w i t h respect t o the variables 9,, 9,, 02 and 02 over the same range (—CO, co), we o b t a i n :
CO 0
-dt,dt2jdk,
j
dR2-RlR2P(Rl = C l , R l , K 2 = C2,R2) (2.3.18)0 - c o
w h i c h represents the p r o b a b i U t y t h a t R{t) passes i n (t,, tf^ + dt,) t h r o u g h the level ƒ = Ci w i t h a positive slope a n d t h r o u g h the level g = C2 w i t h a negative slope i n ( / j , 0 + d O ) .
A c c o r d i n g t o our discussion above, we find f o r the p r o b a b i l i t y t h a t the signal I(t) passes t h r o u g h zero i n the i n t e r v a l (t, t + dt):
dt Ï | / | p ( / = 0 , / ) d / (2.3.19)
- CO
where ; ; ( / , / ) represents the j o i n t density f o r the signal variables I(t) a n d / ( / ) = d / / d / . F r o m (2.2.2) we derive t h a t I(t) can be represented by
Using the central l i m i t theorem, we can prove that p is n o r m a l i n t w o dimensions w h e n the discrete spectrums o f (2.2.2) and (2.3.20) are replaced by the corresponding continuous ones:
1 ,-/2/2/iii-/2/2;i22 i n w h i c h : 27c(/<ll/i22)' CO / ' l l = ƒ w(v)dv 0 00 ^22 = 47c^ J v^iv(v)dv 0 A'12 = 0
Substituting (2.3.21) i n (2.3.19) and p e r f o r m i n g the i n t e g r a t i o n , we find
A«22
(2.3.21)
(2.3.22)
ch
n (2.3.23)
I n w h i c h the coefficient o f dt represents the intensity o f the stream o f zeros o f the signal I{t). T h e n the average n u m b e r o f zeros per u n i t t i m e w h i c h we denote b y P is given by
1^22
M i l
(2.3.24)
3 Some statistical characteristics with regard to the lurching motion in a beam sea
T h e discussions i n this chapter are based o n the m a t h e m a t i c a l m o d e l (1.1) f o r the roHing m o t i o n i n a beam sea. H o w e v e r , i n a r e m a r k at the end o f this chapter i t w i l l be s h o w n t h a t the argument i n this chapter is also applicable f o r more complicated models.
3.1 The average number per second of hirches, experienced by a vessel in an irregular beam sea
I t is r e m i n d e d t h a t a l u r c h corresponds w i t h a j u m p i n the o u t p u t a m p l i t u d e o f the system (1.2) w h i c h m a y occur w h e n the i n p u t a m p f i t u d e or f r e q u e n c y is varied. I t has been s h o w n i n 2.1 t h a t such a j u m p occurs either f o r the a m p l i t u d e value R = Rjj{aJ, w h e n f o r some constant frequency co = CÜ^ > CÜQ the a m p l i t u d e R o f the r o l l i n g m o m e n t f u n c t i o n R cos (co/ + ö) is increased f r o m some R < -/?j/(coJ u n t i l some R > Rjj(co^), or f o r the frequency co = co„ w h e n f o r some constant a m p l i t u d e R = R,> Rij(mo), where R, = i?„(co,), the f r e q u e n c y is decreased-from some a)> a>, u n t i l some co < co,. F o r this reason we m a y expect t h a t a simuhaneous v a r i a t i o n o f the frequency a n d the a m p l i t u d e o f the i n p u t signal can cause j u m p s i n the o u t p u t a m p l i t u d e . W e shall consider this m o r e closely.
W e represent at every t i m e instant t the i n p u t R{t) cos 6(t) as a p o i n t i n F i g . 3.1.1 w i t h coordinates (R, 9). T h u s the behaviour o f the i n p u t as a f u n c t i o n o f t i m e can be studied b y considering the movements o f the p o i n t (R, 9) i n the (R, cö)-plane. Consequently, according t o the quasi-stationary a p p r o x i m a t i o n m e t h o d w h i c h
we described i n 2.2, tlie i n p u t signal causes an u p w a r d j u m p i n the o u t p u t when the instantaneous a m p l i t u d e R(t) a n d f r e q u e n c y cu = 9Q) vary i n such a w a y t h a t the p o i n t (R, Ó) passes t h r o u g h the curve R = Ruico) f r o m region A i n t o region B .
By s u b s t i t u t i n g i n (2.3.8) f o r the f u n c t i o n / ( ö ) the expression f o r Ru(co) given by (2.1.8), where cu is replaced by 9, we o b t a i n the p r o b a b i l i t y t h a t i n the t i m e i n t e r v a l (t, t + dt) a j u m p occurs i n the o u t p u t a m p l i t u d e , w h i l e the absolute value o f the instantaneous frequency has a value i n the i n t e r v a l (Ö, 9 + d9):
CO OJ / dR \
dtdÓ J d ë ƒ d R { R - = ^ 9 ) { p i R = R„(9),9,R,9) + piR = Rn{9),-9,R,9)} (3.1.1) - c o dJJr, ,, \ d9 /
The first t e r m i n expression (3.1.1) refers t o the c o n t r i b u t i o n o f the positive values o f t h e instantaneous f r e q u e n c y , w h i l e the second t e r m takes its negative values i n t o account.
I n d e r i v i n g f o r m u l a (3.1.1) i t is assumed t h a t every passage t h r o u g h the level R^ f r o m region A i n t o r e g i o n B is attended by an u p w a r d j u m p .
This is true w i t h p r o b a b i l i t y almost one w h e n we assume t h a t the p r o b a b i l i t y o f R(t) h a v i n g values larger t h a n the level R^ is very smafi. This means t h a t any u p w a r d j u m p due t o a passing o f R{t) f r o m r e g i o n A i n t o region B is almost i m m e d i a t e l y f o U o w e d b y a d o w n w a r d j u m p due t o a passing o f R{t) f r o m r e g i o n A i n t o region C. Consequently, the p r o b a b i l i t y o f occurrence o f an a d d i t i o n a l passing o f R{t) f r o m r e g i o n A i n t o region B between an u p w a r d j u m p a n d the successive d o w n w a r d j u m p is almost zero.
F r o m F i g . 3.1.1 i t is seen t h a t the occurrence o f u p w a r d j u m p s i n the o u t p u t a m p l i t u d e is o n l y possible w h e n 0 has values i n the i n t e r v a l (CUQ, co). By i n t e g r a t i n g (3.1.1) w i t h respect to 0 over above m e n t i o n e d range o f values we o b t a i n the p r o b a b i l i t y that an u p w a r d j u m p occurs i n the o u t p u t a m p l i t u d e i n the t i m e i n t e r v a l
(t, t + dt) w i t h o u t regard t o the f r e q u e n c y o f the signal.
CO CO CO / r\R \
d f J d Ö J d Ö J d R i R - ^ 9 ) { p i R = Rn(9),9,R,9) + p(R = Rij0),~9,R,9)] (3.1.2)
too - d R f i „ \ d0 / dé
A n a l o g o u s t o the argument used i n 2.3, the average n u m b e r o f lurches per second is easy t o derive f r o m (3.1.2):
CO CO CC / d R \
N = J d Ö J d 0 J d R [ R - ^ ^ 9 \ { p { R = Rn{9\9,R,9) + p{R = Ru{9),-9X9)} (3.1.3)
dé '
The statistical q u a n t i t y has m u c h p r a c t i c a l interest. T h e r e f o r e , we shafi express i t i n terms o f the spectrum a n d system parameters and reduce i t t o a simpler f o r m .
W e substitute expression ( A . 1 3 ) (see A p p e n d i x A ) f o r the p r o b a b i l i t y density p(R, R, 9, 0) i n f o r m u l a (3.1.3). N e x t , we i n t r o d u c e the change o f variables:
^ = ^ 0 + ^ (3.1.4) d0 T h e n we o b t a i n : CO CO CO . 7 V = | d 0 J d9 j dR{I,iR,9,9) + I f R , - 9 , 9 ) } (3.1.5) RUMI^+MIJÓ^ + IM.^M^ié where: hil9,Ö) = - ^ R R ' n C x p IB"- M , + RUM,,+2M,2B) + M22[R' + [ ^ j 9' + 2 R ^ e + R - 2 M 2 3 R n ^ Ö ' + R.M + l ó f ö ^ + k f ] + M 3 3 (Rjj9' + 4 R J 9 ' ^ + d9 \ d0 / / \ d 0 + 4Rj9R + 4 9 ' 9 ' r ^ + 49'R' + W'ÖR ' d0
7
d0 (3.1.6)The integrations w i t h respect t o R and 0 are n o w easy t o p e r f o r m . T h i s yields: N=]dÓ{IM + h{-Ó)} (3.1.7) i n w h i c h : Rl ( ^ Y - 2 M , 3 [ R J ^ + 20 ( ^ Y l + M 3 3 ^ ^ „ + 20 ^ ^ ^ - ^ \ d0 / J d0 y d0 V d0 y " V dö
^
^
- | m . % - M . 3 r ^ . . 4 0 % ^
+
M 3 3 ( R J . 2 Ó ' ' - ^ ] ] \ d0 V d0 y v d0 + ^ M , 2 - 4 M „ Ö + 4 M , 3 0 ^ M M 2 2 ( ' ^ ^ Y - 2 M 2 3 f R u ^ + + M 3 3 (R,, + 2 0 ^ ' ' exp d0y " V " dö v
d0y y ^^v " dö
2B'I M , (3.1.8)Since the r e l a t i o n (2.1.8) between Rjj and co = 0 is very c o m p l i c a t e d , the i n t e g r a t i o n w i t h respect t o Ö has t o be p e r f o r m e d n u m e r i c a l l y .
Since we assume t h a t the p r o b a b i l i t y o f occurrence o f values o f R higher t h a n the levels Rj and Rji are very small, we m a y expect t h a t m o s t o f the t i m e intervals between an u p w a r d j u m p and a successive d o w n w a r d j u m p are very small. ( I n chapter 5 i t is s h o w n , f o r a special case, t h a t this is indeed true i f the o r i g i n a l signal R{t) cos
6{t) w h i c h is b o t h a m p l i t u d e a n d f r e q u e n c y m o d u l a t e d , is a p p r o x i m a t e d by the pure a m p h t u d e m o d u l a t e d signal R(t) cos CO,,/, where co„, is a suitable chosen m i d b a n d frequency.)
W e f u r t h e r assumed t h a t the p o w e r spectrum o f the r o l l i n g m o m e n t f u n c t i o n is relatively n a r r o w . Consequent-ly, d u r i n g such a t i m e i n t e r v a l , the instantaneous values o f the a m p l i t u d e a n d f r e q u e n c y o f the r o l l i n g m o m e n t f u n c t i o n r e m a i n a p p r o x i m a t e l y the same.
Assume t h a t at t i m e t an u p w a r d j u m p occurs i n the o u t p u t a m p l i t u d e . T h e n i t is seen f r o m F i g . 2.1.1 that d u r i n g such an i n t e r v a l between an u p w a r d j u m p and a successive d o w n w a r d j u m p the a d d i t i o n a l h e i g h t o f the r o l l i n g a m p h t u d e due t o the occurrence o f a l u r c h has values between |F3(0)| - | F i ( Ö ) | and |F2(0)| - 1 ^ 4 ( 0 ) 1 where Ö represents the instantaneous f r e q u e n c y at t i m e t.
T h e height S(6) o f the i n i t i a l j u m p i n the a m p l i t u d e o f the r o l l i n g angle, w h e n the p o i n t {R, 0) i n F i g . 3.1.1 passes f r o m region A i n t o r e g i o n B , is given b y :
5(0) = | F 3 ( 0 ) | - | F i ( 0 ) | (3.1.9) where | F i | and I F 3 I are given by the f o r m u l a s (2.1.5) and (2.1.11). T h i s q u a n t i t y can be conceived as a measure
f o r t h e seriousness o f the l u r c h .
Since S(Ó) is a m o n o t o n i e increasing f u n c t i o n , an expression f o r the average n u m b e r per second N j o f lurches w h i c h cause an a d d i t i o n a l i n i t i a l increase i n the r o l l i n g a m p h t u d e exceeding some value S j = S(cOj) is analogous t o (3.1.3) expressed by
OD CO CO / H ƒ? \
iV^.= J d 0 J ' d 0 J dR[R f ë ) { p i R = R„(Ó),9,R,Ö) + p(R = R„(Ó),-Ó,R,Ö)} (3.1.10)
t o j -OD d J i j j ^ \ d0 / ds
A p p l y i n g the same p r o c e d u r e w h i c h reduced f o r m u l a (3.1.3) t o (3.1.7) we o b t a i n f o r (3.1.10)
N j = ] d 9 { I M + l2i-9)} (3.1.11) mj
where the f u n c t i o n / j i n the i n t e g r a n d is given by (3.1.8).
By p l o t t i n g S j against N j we can j u d g e the seriousness o f the lurches, p e r f o r m e d by the vessel i n an i r r e g u l a r b e a m sea. Since S{0) is a m o n o t o n i e increasing f u n c t i o n , N f S j ) w i l l be a decreasing one.
I t is obvious t h a t f o r S j = 0 the q u a n t i t y N j is equal t o w h i c h is the average t o t a l n u m b e r o f j u m p s per second i n the a m p l i t u d e o f r o l l i n g angle as given by f o r m u l a (3.1.7).
has a power spectrum w i t h the shape:
n<v) = — ^ e - ^ " - * ' " ' » ' / ^ " ' (3.2.1)
as/in
i n w h i c h a is the characteristic b a n d w i d t h w h i c h is assumed t o be very n a r r o w . The q u a n t i t y v,„ = OJJITZ is the symmetry frequency and w i l l be dehned i n this section as the m i d b a n d frequency. F o r the coefficients b„, as defined by ( A . 3 ) , we o b t a i n :
(3.2.2)
(3.2.3)
b^,, = 2 ^ V " - V " F ( « + i ) , « = 0 , 1 , 2 , . . . b2„i=0 , n = l , 2 ,
-S u b s t i t u t i n g these values i n ( A . 7 ) we o b t a i n f o r the cofactors w h i c h appear i n (3.1.8): M i l = 6(27r(7)''; M 1 3 = 2 ( 2 7 r ( j ) ' ° ; 1^22 = 4(27r(T)'°
M 3 3 = 2(271(7)' ; M i 2 = M 2 3 = 0
T h e determinant o f the m o m e n t m a t r i x is calculated f r o m ( A . 8 ) and ( A . 9 ) :
M = B'= 4(271(7)" (3.2.4)
W e substitute (3.2.3) and (3.2.4) i n (3.1.7). T h i s yields:
N = - ] dÓR,J.(^]\2(2naf + 49') + 4 R j j ^ 9 + RÜ^e-'''''''''^^^^^^ (3.2.5)
27t<oo-<um l \ dó J d ö J
i n w h i c h the lower fimh COQ i n (3.1.7) has to be replaced b y O J Q - C Ü, , , , since the instantaneous f r e q u e n c y is here defined by c« = co„, — Ö.
W e consider first the case:
27iv„, = a,„ > a>o (3-2.6)
W e notice that f o r the range CMQ — «,„ < Ö < 00:
Rjm>Rhicoo-coJ>0 (3.2.7) P u t t i n g : a n d g 0 ) ^ l R ^ ^ \ ( ^ ) \ 2 ( 2 n a r + 4 9 ' ) + 4 R „ ^ Ó + R^,ye-'^"''' In I \ d9 J dO R', / , ( ö ) = - i ^ Ö ^ (3.2.9) 871 t h e n (3.2.5) is reduced to J g{9)e"'-'''^'d9 (3-2.10)
The f u n c t i o n /7(Ö) has its m a x i m u m value w h e n 0 = 0. W e w r i t e (3.2.10) as the s u m o f t w o integrals:
Each o f t h e integrals o n the r i g h t - h a n d side o f this expression can be a p p r o x i m a t e d by Laplace's m e t h o d , [18] w h i c h yields f o r b o t h integrals the same result, n a m e l y :
^ l ^ ^ e - ^ ' - ^ ' X - . + Oia') as , 7 - ^ 0 (3.2.12)
Consequently, we find f o r the average number o f lurches per second
N = af'2nRue'''"''% = o + 0{a^) where co,„ > CÜQ and ( T 0 (3.2.13)
I n a similar manner we a p p r o x i m a t e expression (3.2.5) w i t h Laplace's m e t h o d , f o r co,,, > CDQ and cr-^O. T h e f o l l o w i n g resuh is o b t a i n e d :
N = ^ Y 0 N 4 R , , ^ 0 + R ? , | ^ - ^ " ^ / ^ - ^ " ^ ' ' ^ ' ' ^ " ^ ' ' V ' + O(<T'^))|, = ,„„-„,,. (3.2.14) R j I V d ö / d0 J
where co,„ < CÜQ and cr 0
Corollary
A c c o r d i n g to section 2.3, the average n u m b e r o f times per second the envelope intersects the line R = Rjiiu),„) w i t h a positive slope is given b y :
]RpiR = R,M„),R)dR (3.2.15)
0
I n this expression p(R, R) represents the j o i n t p r o b a b i l i t y density o f the envelope variables R and R w h i c h is derived f r o m (2.2.25) by i n t e g r a t i n g w h h respect t o Ö' over ( - c o , co).
T h e n we find:
piR,R) = ^ ^-(b2R^+boR^)/2B (3.2.16) ^2nboB
S u b s t i t u t i n g this i n (3.2.15) a n d p u t t i n g 5 = fej = (27r(7)^ i n accordance w i t h (3.2.2), we find:
N = a^2^R,,e-''"'% = o (3.2.17)
C o m p a r i n g this result w i t h (3.2.13) we find that, i n case the spectrum o f the b o t h a m p l i t u d e a n d f r e q u e n c y m o d u l a t e d r o l l i n g m o m e n t f u n c t i o n i?(0 cos (CÜ,„/-t-0(0) is very n a r r o w , the n u m b e r o f u p w a r d j u m p s per second i n the o u t p u t is i n the first order a p p r o x i m a t i o n equal t o the n u m b e r o f j u m p s per second w h e n the i n p u t is the pure a m p l i t u d e m o d u l a t e d signal i?(0 cos co,,;. Consequently, f o r a very n a r r o w spectrum the con-t r i b u con-t i o n o f con-the frequency m o d u l a con-t i o n con-t o con-the j u m p s is negligible w i con-t h respeccon-t con-to con-the one o f con-the a m p l i con-t u d e m o d u l a t i o n w h i c h is also clear f o r physical reasons.
Remark
A s already has been p o i n t e d out earlier i n this r e p o r t , our calculations i n this chapter are based o n the rather simple m a t h e m a t i c a l m o d e l (1.1) f o r the r o l f i n g m o t i o n o f a vessel i n beam waves. I n reality, however, a m o r e c o m p l i c a t e d m o d e l is needed i n order to achieve m o r e satisfactory results. I t can be wondered n o w whether or n o t the m e t h o d o f c a l c u l a t i o n as expounded i n this chapter can be used f o r m o r e compUcated m a t h e m a t i c a l models. I n order t o answer this question we notice t h a t our m e t h o d is completely based o n the f u n c t i o n s R,{(o) a n d Ri,(co), as given by (2.1.8) w h i c h give f o r any frequency co > CÜQ the t w o values o f the a m p l i t u d e o f the r o l l i n g m o m e n t f u n c t i o n f o r w h i c h the corresponding response curves have a tangent i n the frequency p o i n t co, (see F i g . 2.1.1). Consequently, w h e n we are able to determine the f u n c t i o n s Ri a n d Rn as defined here, f o r a m o r e c o m p l i c a t e d m o d e l o f the roUing m o t i o n o f the vessel, the m e t h o d expounded i n this chapter can be a p p l i e d . Even a description o f R, a n d R„ o n basis o f an experimental d e t e r m i n a t i o n o f the response curves f o r the r o l l i n g m o t i o n is sufiicient.
4 The fraction of the time that a vessel is performing a lurching motion
We assume that the coefficient b i n the nonlinear t e r m o f e q u a t i o n (1.1) is positive and t h a t the s p e c t r u m o f the i n p u t w h i c h is assumed t o be symmetric is such that
ffl,„>o)o (4.1)
where m,„ is the s y m m e t r y f r e q u e n c y o f the spectrum and OJQ is defined by (2.1.7). The case co,„ < COQ w i l l n o t be considered, as i n t h a t case w h e n the characteristic band w i d t h a is small, the lurches o f the vessel are m a i n l y due to the frequencies i n the n e i g h b o u r h o o d o f cug a n d i n this r e g i o n the lurches have n o serious character.
We f u r t h e r assume t h a t the parameters o f the m a t h e m a t i c a l m o d e l (1.1) and o f the p o w e r s p e c t r u m o f the r o l l i n g m o m e n t f u n c t i o n I{t) are such t h a t the lurches o n l y occur at very h i g h levels o f the a m p l i t u d e o f the r o l l i n g m o m e n t f u n c t i o n . T h i s is f o r example the case w h e n the levels Rr(a>,„) a n d RI{CO,„~(T) are v e r y h i g h where a is the characteristic b a n d w i d t h o f the spectrum o f the signal.
I n m a t h e m a t i c a l t e r m i n o l o g y this means t h a t we consider the a s y m p t o t i c case
Ri^oo (4.2)
We showed i n o u r discussions at the end o f section 2.2 t h a t f o r very h i g h ampfitudes the f r e q u e n c y m o d u l a t i o n is o f a higher order t h a n the a m p l i t u d e m o d u l a t i o n , (see (2.2.26)) and t h a t t h a t instantaneous f r e q u e n c y w i l l d i f f e r less f r o m the value
2n I r t v ( v ) d v
~ . (4.3)
J M'(i') dv
0
For a symmetric power spectrum (4.3) is j u s t equal t o the s y m m e t r y frequency o f the spectrum. F o r this reason we replace i n the calculations i n this chapter the r a n d o m r o l l i n g m o m e n t f u n c t i o n R(t) cos 9(t) w h i c h is b o t h a m p l i t u d e a n d f r e q u e n c y m o d u l a t e d by the r a n d o m a m p l i t u d e m o d u l a t e d h a r m o n i c signal R{t) cos co,,,/.
A p p l y i n g again the quasi-stationary a p p r o x i m a t i o n m e t h o d , as e x p o u n d e d i n section 2.2, we state t h a t j u m p s i n the r o f i i n g a m p l i t u d e can o n l y occur w h e n the envelope passes t h r o u g h the levels Rji{(0,„) a n d Ri(m,„) w i t h a positive a n d negative slope, respectively. H e n c e f o r t h , we w i l l denote Ri{a>,„) and Rjj{co„,), b r i e f l y , b y R, a n d Rj,.
Similar t o our discussions i n section 3.1 we have again, i n v i r t u e o f (4.2), t h a t any u p w a r d passing o f Rit) t h r o u g h the level Rjj a n d any d o w n w a r d passing t h r o u g h Rj w i l l cause w i t h p r o b a b i h t y almost equal t o one, respectively, a sudden increase and decrease i n the r o l l i n g a m p l i t u d e . W e observe t h a t the r o l l i n g m o m e n t excites always a r o l l i n g m o t i o n w i t h a m p f i t u d e values above the unstable r e g i o n w h e n the c o r r e s p o n d i n g envelope satisfies the i n e q u a l i t y R > Rjj, w h i l e f o r R<Ri the c o r r e s p o n d i n g o u t p u t is below this r e g i o n . W e k n o w that f o r the values o f R{t) between Rj a n d R^ the r o l l i n g a m p f i t u d e has o n l y values above the unstable region w h e n R{t) has values o n a section between a n u p w a r d passing o f R(t) t h r o u g h the level R = Ru a n d the f o l l o w i n g d o w n w a r d passing t h r o u g h R = Ri. T h e r e f o r e , i n order t o be able t o decide whether or n o t some envelope element between the levels Rj a n d Rjj yields a r o l l i n g a m p l i t u d e above the unstable r e g i o n , we have to consider the t i m e h i s t o r y o f the i n p u t envelope.
The envelope R{t) o f the i n p u t signal passes t h r o u g h the level Rj alternately w i t h positive a n d negative slopes. T h e sudden increases a n d decreases i n the r o l l i n g angle occur i n the p o i n t s , denoted i n F i g . 4.1 by 2 a n d 4
R(t)I
Fig. 4.1
respectively, where R{t) passes w i t h an u p w a r d slope t h r o u g h the level Ru or w i t h a d o w n w a r d slope t h r o u g h RJ. So, f o r R{t) between the levels Rj and R^, the o u t p u t has values above the unstable r e g i o n w h e n R{t) has values i n the sections between the successive points 3 and 4. F o r this reason we w i l l consider the statistical properties o f the excursions o f R{t) above the level Rj w h i c h exceed i n a d d i t i o n the level R^. H e n c e f o r t h , we shall indicate w i t h the expression " e x c u r s i o n " always an excursion o f R(t) above the level Rj w h i c h begins at a p o i n t 1 and ends at the f o l l o w i n g p o i n t 4.
W e consider an i n t e r v a l o f l e n g t h F o f a realization o f the process / ( / ) . L e t Tg^^ be the l e n g t h o f t i m e t h a t the envelope has i n the i n t e r v a l T values above the level Rj and 7^^^ the l e n g t h o f t i m e t h a t R{t) belongs i n this i n t e r v a l t o excursions w h i c h exceed the level R,j. W e f u r t h e r denote the t o t a l n u m b e r o f excursions i n the i n t e r v a l T b y TV and the n u m b e r o f these excursions w h i c h have a w i d t h between T , _ I and T, a n d a peak height R,„ between Rj-1 and Rj by TV,; where T, = / AT and Rj = R,+jAR,„ w h i l e AT and AR,„ are real positive constants a n d i,j = 1,2,... T h e n the expression
J:JAR,„>R,,~Rr i= 1
j=i 1=1
(4.4)
is a n a p p r o x i m a t i o n f o r TRJJT^^ w h i c h is the f r a c t i o n o f the t i m e t h a t i n the i n t e r v a l Tg^^ the f u n c t i o n R{t) belongs t o excursions w h i c h exceed the level Rji. B y d i v i d i n g b o t h the n u m e r a t o r and d e n o m i n a t o r b y the t o t a l n u m b e r TV o f excursions i n T, we o b t a i n :
I I N i f j N
j:jAR,„>Ri,-Rii=l
J=l i = l
(4.5)
T h e accuracy o f the a p p r o x i m a t i o n is increased by decreasing the values o f the i n t e r v a l lengths AR„, and AT. F r o m (4.5) i t is observed t h a t the o n l y quantities o f a n excursion w h i c h are i m p o r t a n t f o r our discussions are its w i d t h a n d the height o f its peak. F o r m a l l y , a j o i n t d i s t r i b u t i o n o f these quantities can be o b t a i n e d f r o m a j o i n t d i s t r i b u t i o n o f the envelope variables at the successive intersection p o i n t s 1 and 4 a n d at the peak o f the excursion. H o w e v e r , i t is observed t h a t the poshions o f the peaks o f the excursions v a r y . I t appears t h a t the a n a l y t i c a l w o r k w h i c h has t o be c a r r i e d o u t t o determine i n this w a y the j o i n t density o f these variables is so c o m p l e x t h a t i t seems recommendable t o i n t r o d u c e a s i m p l i f i c a t i o n o f the excursion m o d e l . I t is assumed t h a t the p o s i t i o n o f the peak o f an excursion can be determined i n a sufficiently accurate w a y f r o m the properties o f the envelope at the intersection p o i n t s 1 and 4 o f this excursion w h i c h is achieved by a p p r o x i m a t i n g the excursion b y a p a r a b o l a w h i c h has at the p o i n t s o f intersection the same slope as the excursion a n d also the same w i d t h , (see F i g . 4.2).
T h e r e l a t i o n between the slopes A, a n d R2 at the intersection p o i n t s , the w i d t h T a n d the o r d i n a t e R,,, o f the peak is given b y : 2(R„-R,)iR,-R2) -R1R2 R(t) R, {111, t, t, + i t Fig. 4.2
