Waves generated by a load moving on an ice sheet over water

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W aves G en era ted b y a Load M ovin g

on an Ice S h eet over W ater

by

W'iclijaiitc) S a ty o X ugroho

B.Sc.. B a n d u n g I n s t i t u t e o f Technology. 1984 M .M a th .. U niv ersity o f W aterlo o . 1988

D is se rta tio n S u b m i t t e d in P a r ti a l Fuilfillrnent o f th e R e q u ir e m e n ts for th e Degree of

D O C T O R O F P H I L O S O P H Y

in th e D e p a r tm e n t of M a th e m a ti c s a n d S ta tis tic s We a c c ep t th is d is s e r ta t io n as conform ing

to t h e re q u ire d s t a n d a r d

Dr. F. .\[ilina^;4H, S u p e r v is o r ( D e p a r t m e n t of M a th e m a tic s a n d S ta t is tic s I

Dr. H.M . Sri vast ava. D e p a r t m e n t a l M e m b e r ( D e p a r tm e n t of M a t h e m a t i c s a n d S tatis tic s)

Dr. P. van den Driessche. D e p a r t m e n t a l .Member ( D e p a r tm e n t of M a t h e m a t ics aru 1 /SThf istics )

Dr. I. B a r r o d a l e ^ ^ u t s i d e M e m b e r ( D e p a r tm e n t of C o m p u t e r Science

Dr. R..1. Hosking. E x t e r n a l E x a m in e r ( la m e s C o o k L'niversity

© W 'idijanto S. X u groho. 1997 L'niversity o f \*ictoria

All rig h ts reserved. D is s e r ta tio n m a y n o t be re p ro d u c e d in w hole o r in p a r t , by m im e o g ra p h o r o t h e r m e an s, w ith o u t th e perm ission o f t h e a u t h o r .

11

S upervisor: Dr. F. Milinazzo

A b stract

A load m oving o n a floating ice sheet produces a deflection of t h e ice slieet. In th i s D is se rta tio n , th re e pro b lem s a s so ciated w ith m a th e m a tic a l m o d e ls of the ic e w a t e r system a re ex am in ed .

. \ m a t h e m a t i c a l m odel involving a steadily m o v in g r e c ta n g u la r load on an ice s h e e t where th e s u p p o r t i n g fluid is of infinite d e p t h is analyzed. T h e solution is w r i t t e n as a F o u rie r integral a n d is e s tim a te d usin g a n a s y m p t o tic m e th o d . T h e re s id ts show t h a t th e a m p litu d e o f th e ice deflection is sim ilar to th e case w here t h e s u p p o r tin g fluid is of finite d e p t h . T h e only significant difference is th a t, in c o n t r a s t to th e case where the s u p p o r t i n g fluid is o f finite d e p t h w h ere a c[uiescent zo n e a p p e a rs b e h i n d th e load w hen its speed exceeds th e speed o f g ra v ity waves o n shallow w a te r, wave's a p p e a r b e h in d th e load for all s u p e rc ritic a l load speeds.

m a t h e m a tic a l m odel of a n ice p la te th a t ta k e s into a cc o u n t th e thickness o f t h e ice is dc'rived by as s u m in g t h a t th e vertical sh earin g forces vary linearly th r o u g h the ice p la te . T h e e q u a tio n s o b ta in e d a r e s im ila r to th o s e used to dc'- sc rib e a m a t h e m a t i c a l m odel using a th in p late a p p r o x im a tio n s u b je c te d to in­

p la n e forces. c o m p a ris o n of th e dispersion r e la tio n is carried o u t between th e

m a t h e m a t i c a l m o d e l of an ice p la te t h a t takes in to account th e p la te thickness, t h e m a th e m a tic a l m odel of an ice p la te using th e th i n p la te a p p r o x im a tio n , a n d th(' m a th e m a tic a l m odel of a n ice p la te using th e t h i n p la te a p p r o x im a ti o n s u b ­ j e c t e d to in -p lan e forces. T h e re s u lts show t h a t ta k in g th e ice thickness in to

c o n s id e ra tio n decre ase s th e m in im u m phase speed. However, th is effect is sm all. T h e m a jo r c o n t r i b u ti o n of th is D is se rtatio n is t h e d e t e r m in a tio n o f the large

Ill

tim e re s p o n s e o f th e deflection of an ice sheet ca u s e d by t h e s te a d y m o t io n o f an im p u ls iv e ly -s ta rte d p o in t load. T h e resu lts o b ta in e d a re new. T h e s o lu tio n of th e ice deflection is w ritte n as a Fourier integ ral a n d a s y m p t o t i c m e th o d s a r e used to e s ti m a t e t h e large tim e b e h av io u r of th e r a te o f c h a n g e o f the ice defle ction w ith re s p e c t to tim e. T h e large tim e b e h a v io u r o f t h e ice deflection itse lf is inferred from this e s tim a te . T h is is d one for th e full r a n g e o f load speeds a n d th e results a re verifted n u m e rically using th e Fast F o u rie r T ra n s fo rm . T h e re s u lts in this D is s e rta tio n show t h a t th e m in im u m of th e p h a s e s p e e d is th e o n ly c ritic a l speed, in t h e sense th a t no finite s te a d y - s ta te is a t t a i n a b l e . . \ t this s p eed th e ice deflection grow s lo g a rith m ic a lly w ith tim e. T h is is in c o n t r a s t w ith th e c ase o f a line load w h ere th e re are two critical speeds: th e m i n im u m o f th e phase s p e e d at which t h e ice deflection grows as the s q u a re -r o o t of tim e , a n d th e speed o f g r a v ity waves in s h a llo w w ater at which th e ice deflection grow s as th e c u b e-ro o t o f rim e. For a p o in t load, it is found t h a t th e tr a n s ie n t p a r t of th e ice deflection decays as th e c u b e - r o o t o f tim e w hen the load sp eed is t h e s p eed of gravity waves in shallow w a te r. T h e a s y m p t o tic e s tim a te s also show t h a t th e decay or th e g ro w th rate of th e tra n s ie n t c o m p o n e n t of th e ice deflection d o e s not d ep en d o n e i th e r the re la tiv e o r ie n ta tio n of th e o b servation p o in t a n d th e load o r on th e d is ta n c e between t h e load a n d th e observation point.

I V

E x am in ers:

Dr. F. S u p e rv is o r ( D e p a r tm e n t o f M a tiie m a tie s a n d S tatistic s)

Dr. H.M. S riv a s ta v a . D e p a r t m e n t a l M em b er ( D e p a r t m ent of M atln m ia tics a n d S ta tis tic s)

D r. P. van d e n Driessche. D e p a r tm e n ta l M e m b e r (D e p a r tm e n t of M a th e m a ti c s and^^^^tistics)

Dr. I. Barrodale>CTiitsWe M e m b e r ( D e p a r tm e n t o f C o m p u t e r Science

______________________________ Dr. R..I. Hoskitm. E x te rn a l E x a m i n e r (.James C o o k C niversitv)

C on ten ts

T itle p a g e

i

A b stra ct

ii

C o n ten ts

v

List o f F igu res

viii

A ck n o w led g em en ts

ix

D e d ic a tio n

x

C h ap ter 1

In tro d u ctio n

1

1.1 I n t r o d u c t i o n ... 1

C h ap ter 2

S te a d y S o lu tio n d u e t o th e M o tio n o f a R ecta n g u la r Load o n a n Ice

S h eet over W ater o f In fin ite D e p th

9

2.1 I n t r o d u c t i o n ... 9 2.2 T h e D ispersion R e l a t i o n ... 13 2.3 T h e S o lution . M e t h o d ... 17 2.3.1 T h e D efinition of A-... 18 2.3.2 Poles in t h e C om plex P l a n e ... 19 2.3.3 T h e E v a lu a tio n of th e I n t e g r a l ... 27 2.3.4 T h e S t e a d y S ta te S o l u t i o n ... 30 2.3.5 T h e F a r-h e ld Solution ... 32

C h ap ter 3

V ariation on th e M a th e m a tic a l M o d e l

38

3.1 Ice P la te w ith F in ite T h i c k n e s s ... 38

3.2 T h e D ispersion R e l a t i o n ... 49

C h ap ter 4

V I

4.1 I n t r o d u c t i o n ... -36

4.2 T h e B eh a v io u r of th e Ice Deflection for Large T i m e ... 39

4.2.1 C ase 1: Load s p e e d I ' = ( Q r l t n i n ... ... 63

4.2.2 C ase 2: Load s p e e d ( Q r l n i i n < ' < <^min Q i i n < I ■ < \ / ^ ... 69

4.2.3 C ase 3; Load S p e e d 1 = 70 4.2.4 C ase 4: Load S p e e d C — \ / g H ... 72

4.2.3 C ase 3: Load S p e e d V > \Jg H ... 74

4.2.6 C ase 6 ; Load S p e e d C appro ac h es \ / g H ... 73

4.3 T h e b e h a v io u r of th e ice deflection for large tim e ... 77

4.3.1 C ase 1: Load s p e e d 1 ' = (C'gr)ndn ... 80

4.3.2 C ase 2: Load s p e e d ( Q r ) n d n < ^ < <^min ^ m i n <

C <

80

4.3.3 C ase 3: Load S p e e d \ ' — 81 4.3.4 C ase 4: Load S p e e d 1 = \ / g H ... 82

4.3.3 C ase 3: Load S p e e d I > \ / g H ... 82

4.3.6 C ase 6 : Load S p e e d \ ' appro ac h es \ / g H ... S3 4.4 X un ieric al C o m p a ris o n s ... 84

C h a p ter 5

S u m m a ry

92

3.1 S u m m a r y ... 92

B ib lio g r a p h y

95

A p p e n d ix A

T h e fu n c tio n

D i { \ )

99

.A..0.1 C ase 1: Small values of A... 100

.A.0.2 C ase 2: Large values of |A|. A > 0 ... 100

v i l

List o f F igu res

2.1 T h e dispersion c u rv e for the case o f infinite d e p t h ... 15

2.2 T h e dispersion cu rv e for the case o f finite d e p t h ... 16

2.3 T h e branch cut in th e A,-p la n e ... 18

2.4 Zeroes of JF... 19

2.Ô T h e relation betw een th e zeroes in th e A’-p lan e a n d th e poles in th e A t- p l a n e ... 2 0 2.6 P o s itio n of th e zeroes in th e first a n d the fo u r th ((u ad ran t of th e A -plane... 23

2.7 P o sitio n of zeroes in th e A-plane a n d poles in t h e A,-p la n e as a fu n c tio n of A-j for F ' < 24 2.8 P o s itio n of zeroes in th e A-plane a n d poles in t h e A,-plane as a fu n c tio n of Am for F - = F ^ ... 2-3 2.9 P o s itio n of zeroes in th e A-plane a n d poles in th e Ai-plane as a fu n c tio n of Am for F ‘ > F,;^... 26

2.1Ü P oles in the A [ - p la n e ... 27

2.11 In te g ra tio n p a t h in th e u p p er h a lf p la n e ... 28

2.12 In te g ra tio n p a t h in th e lower h a lf p l a n e ... 29

2.13 T h e w avelength as a function of loa d s p e e d ... 36

2.14 T h e depression d e p t h as a fu n c tio n o f load s p e e d ... 37

3.1 T h e forces a c tin g on a b ending p l a t e ... 39

3.2 T h e stress c o m p o n e n ts ac tin g on a b e n d in g p l a t e ... 40

C o n te n t» ____________________________________________________________________ v i i i

3.4 T h e force c o m p o n e n ts a c t in g on a b e n d in g p l a t e ... 42

3.5 T h e in-p lan e forces a c tin g o n th e m id -p la n e o f a p la te ... 48

3.6 T h e d isp ersio n relation for th e case of in fin ite d e p t h - case I. . . . 52

3.7 T h e d isp ersio n relation for t h e case of in fin ite d e p t h - case 2. . . . 33

3.8 T h e d isp ersio n relation for t h e case of finite d e p t h - case 1... 34

3.9 T h e dispersio n relation for t h e case of finite d e p t h - case 2 ... 33

4.1 T h e roots of (.•*.-= Ü... 63

4.2 T h e ice deflection versus ti m e a t th e load s p e e d ... 8 6

4.3 T h e ice deflection versus t i m e a t the load s p e e d - a h e a d of

th e loa d... 87

4.4 T h e ice deflection versus ti m e a t th e load s p e e d - b e h in d th e

lo a d ... 8 8

4.3 T h e ice flefiection versus ti m e a t the load s p e e d (XV^min < ^ <

Q n i n ... 89

4.6 T h e ice deflection versus ti m e a t th e load s p e e d < 1 < \ / g H

- 23 ni b e h in d th e loa d... 90

4.7 T h e ice deflection versus ti m e a t th e load s p e e d < I < s / g T J

IX

A ck n ow led gem en ts

I th a n k G o d th e A lm ighty for giving me th e light d u rin g m y study. I would also like to t h a n k my s u p e rv is o r. Dr. F. .Milinazzo. for his e n c o u ra g e m e n t a n d countless discussions in th is research area. Several people have assisted in m a n y ways a n d th e ir generosities have helped d u rin g th e course o f th is research; m y th a n k s to Prof. R..J. H osking from th e D e p a r tm e n t of .M athem atics a n d S t a t i s ­ tics a t .James C ook U niversity for his discussions. Prof. H.M. S rivastava for his advice. Prof. P. van den D riessche for her valuable feedback. Prof. P.O. S affm an. T h e o d o r e von K a r m a n P rofessor of .Applied M a th e m a tic s a n d .Aeronautics, a n d Prof. D. Meiron from th e .Applied .Mathematics D e p a rtm e n t a t C altech for pro­ viding CR.AA tim e a t th e .National Energ}' Research Scientific C o m p u t in g C e n te r. Berkeley. C alifornia. My d e e p ap p re c ia tio n also goes to my fam ily an d my fath e r, w ith o u t their s u p p o r t I will n o t have been able to go this far.

D ed ication

This thesis is dedicated to my family. Rini. R iiiinta. and Arvin.

who have been waiting patiently for the com pletion o f this study.

C h a p ter 1

In tro d u ctio n

1.1

In tr o d u c tio n

For s o m e areas on e a r t h , ice is s o m e t h in g t h a t never e x ists in n a t u r e . B u t in o th e r a re a s , ice becom es p a r t of daily life t h a t needs to be u n d e rs to o d . W ork in g in a cold e n v iro n m e n t s u r ro u n d e d by ice can pose a c e r ta in level o f difficulty th a t is

u s u a lly associated w ith a harsh a n d d y n a m ic e n v iro n m e n t. G old [6 ] s ta t e s th a t

.\s early as th e mid-1800s. ice was imposing significant constraints on activities such as the construction and operation of hydro-electric systems and shipping on inland waterways and in coastal regions.

F l o a t i n g ice sh eets on lake w a te r in th e .\r c tic a n d in th e . \ n t a r c t i c regions are r o u ti n e ly used for t r a n s p o r t a t i o n . For exam ple, ice-covered lakes a n d rivers serve as s e a s o n a l runw ays a n d roadw ays a n d provide w in te r access to r e m o t e areas. It is now well known t h a t a load trav e llin g above a critic a l speed on a floating ice sh eet g e n era tes waves in the ice. w hereas th e r e s p o n s e is q u a s i- s ta t ic a t slower loa d s p e e d s (Squire et al. [26]). T h i s wave has a n a m p l i t u d e t h a t d e p e n d s on th e load s p e e d and th e ice-w ater p a r a m e te r s , such as th e ice rigidity a n d th e d e p th of th e u n d e rly in g w a te r. By u n d e r s ta n d i n g th e resp o n se o f these flo a tin g ice sheets to a m o v in g load, t h e o p e ra tio n a l safety of t r a n s p o r t a t i o n can b e assessed. In m o re te m p e r a t e c lim a te s , the e m p h a s is is on t h e effort req u ire d to break ice. C o n seq u en tly , it is i m p o r ta n t to d e te r m in e loads t h a t can be s u p p o r t e d by ice s h eets.

1.1 In tro d u ctio n

T h i s s tu d y falls in to th e m o re general a re a o f th e response o f contin u o u sly

s u p p o r t e d beam s a n d plates to m o v in g loads. F o r exam ple. T im o s h e n k o [33]

s t u d ie d t h e response o f a railro ad tr a c k to d y n a m ic loads a n d K e r r [9] an aly zed

an infin ite beam s u b je c te d to a m o v in g load a n d a c o n s ta n t a x ia l force. large

n u m b e r o f these stu d ie s are discu ssed in the surv ey by K e rr [10] a n d also in th e m o re recent book by Squire et al. [26].

In o r d e r to a p p ly a n a ly tic te c h n iq u e s to s t u d y th e response o f a floating ice s h e e t to a load m o v in g on its surface, th e ice sheet is usually tr e a t e d as an elastic (see. for exam ple, K e rr [11]. D avys et al. [.5], Schulkes et al. [24]. Milinazzo et

al. [19]) o r a viscoelastic (see H o sk in g et al. [8 ]) h om ogeneous p la te of infinite

h o r iz o n ta l extent. To m odel th e fiuid-ice in te ra c tio n , one a p p ro a c h is to take th e fluid s u p p o r tin g th e ice sheet to b e a W in k ler b ase w here it is a s s u m e d th a t th e re a c tio n pressure of th e fluid b ase on th e ice sh eet is p r o p o r ti o n a l to the local d eflection of the ice sheet. T h is a p p r o a c h is a c c u r a t e eno u g h to describe th e c h a ra c te ris tic s of a s u p p o r tin g fluid for a s ta tic p ro b le m (K e rr [10]. Livesley [17]).

F or d y n a m ic p roblem s, however, a m ore realistic m odel of t h e huid-ice in te r­ a c tio n is needed to ta k e into a c c o u n t th e in e rtia o f th e fluid base. In this case, th e p re s s u re th a t th e fluid exerts o n th e ice is o b t a i n e d by solving t h e e q u atio n s o f m o t io n of the fluid. For ex a m p le , th e fluid base c a n be co nsidered to consist of a n ideal, hom ogeneous, in com pressible fluid a n d t h e c o rre s p o n d in g m a th e m a tic a l m o d e l for the fluid-ice system c a n be derived. G re e n h ill [7] used th is la tte r a p ­ p ro ach . T h a t article, one of th e first th e o re tic al s tu d ie s of th e s u b je c t , includes a n a n a ly s is of th e response of a flo a tin g ice p la te to a w ater wave p ro p a g a tin g in o n e direction. G reenhill a s s u m e d t h e ice p la te to be of u n ifo rm thickness a n d th e w a te r base to be o f uniform d e p t h a n d gave th e d e riv a tio n o f th e dispersion r e l a t i o n ‘ for the waves in the s v ste m .

Fa. dispersion relation states the nature o f a dispersive process where, in this case, waves of different lengths, propagating at different speeds, disperse or separate. .A dispersive property

1.1 In tro d u ctio n

KheLsin [12j as s u m e d t h e fluid base to b e ideal, hom ogeneous, a n d in c o m p ress­ ible. and s tu d ie d th e resp o n se o f a n ice p l a t e to b o th a m o v in g point load a n d a moving line load. T h is p io n e erin g article e x a m in e d the s t e a d y ice deflection for various load speeds a n d n o te d t h a t th e defle ction depen d s on th e load s p eed . In th e article, it was shown t h a t for a line load th e re are two c ritic a l speeds, defined to be those speeds where t h e ice deflection is infinite. C ritic a l speeds were found only for a line load.

.\evel f'2 1] s tu d ie d th e s t e a d y deflection a n d stress of a floating ice sh eet d u e

to a moving load unifo rm ly d is tr i b u te d o ver a c irc u lar area. T h e ice p la te was assu m ed to be s u p p o r te d by w a te r of u n ifo rm d e p th . T h e ice deflection u n d e r th e center of th e load was o b ta in e d by s u p e r im p o s i n g th e deflections d u e to con­ c e n tra te d loads. Xevel [21] n o te d t h a t t h e r e does exist a c ritic a l speed a n d t h a t

Kheisin ([12]. [13]) h ad in c o rre c tly c o n clu d ed t h a t for a point load the ice deflec­ tion is b o u n d e d at all load speeds.

In trying to u n d e r s ta n d t h e origin of c r itic a l speeds. K heisin [14] m ade a n o t h e r m a jo r c o n trib u tio n to th e field by an a ly sin g th e associated in itia l value p ro b lem . In t h a t p ap er. K heisin a n a ly z e d the tim e - d e p e n d e n c e of t h e ice deflection for a line load a n d d e te rm in e d t h a t th e re is a s p e e d a t which th e ice deflection grows

w ith time. K err [1 1] identified th is critical s p e e d as the m in im u m phase speed*

of th e system of waves. F u rth e r m o re . K e r r [11] found t h a t this critical speed varies with the m a g n itu d e of in-plane forces w hich, for e x am p le, could be d u e to th e rm a l s tra in s in th e ice. T h e p a p e r show ed t h a t a force field in co m pre ssion reduces th e critical speed, w hile a force field u n d e r tension increases th e c ritic a l speed. However, for a given set o f ice p a r a m e te r s , these force fields must be large in o rd er to have any significant effect (Schulkes et al. [24]).

of a wave means that the wave speed depends on the wavelength, and possibly also on the direction of propagation (see Light hill [16]. Kundu [15]. also section 2.2 o f this Dissertation).

■The phase speed of a wave is the rate at which th e phase of the wave propagates (see Kundu [15]). The minimum phase speed is calculated from the dispersion relation.

1.1 In tro d u ctio n

D avys et al. [5] investig ated th e p ro p a g a tio n o f flexural-gravity waves in a floating ice sheet over w ater o f finite d ep th . T h e Fourier tr a n s f o r m was used to e x a m in e the h y b r id wave p a t t e r n s genera ted by a s tead ily m o v in g point source, wliere th e waves a re p r e d o m in a n tly flexural a h e a d an d p r e d o m i n a n tl y g r a v i t a ­ tio n a l behind t h e source. It was n o te d th a t a t th e critical sp eed no s te a d y wave p a t t e r n can exist, a n d th a t a shad o w zone' a p p e a r s b ehind th e source w hen its s p eed exceeds th e shallow w a te r wave speed. Schulkes et al. [24]. e x te n d in g th e

work of Davys et al. [ôj. confirm ed K err's [1 1] result th a t a com pressive stress in

th e p lane of th e ice p la te causes a slight decrease in th e phase sp e e d . In a d d itio n . Schulkes et al. showed th a t th e existence of a uniform flow in th e u n d e rly in g fluid changes th e o rie n ta tio n o f th e wave p a t t e r n , a n d t h a t in te r n a l waves a re g e n e ra te d if it is stratified.

In further work. Schulkes a n d Sneyd [25] an a ly z e d th e tim e -d e p e n d e n t re­ s p o n se of a floating ice sheet over w ater of finite d e p t h to a m o v in g line load, as previously done by Kheisin [14]. T h e Fourier tr a n s fo rm a n d t h e m e th o d of s te e p ­ est descent were used to o b ta in th e a s y m p to tic e x p an sio n of th e ice disp la cem e n t for large time. It was shown t h a t the ice deflection grows w ith ti m e at th e load s p eed Cjjjjjj. th e m in im u m p h a s e speed of the waves in th e sy ste m : and also a t th e load speed CVpj. th e speed o f grav ity waves on shallow w ater. T h e load s p e e d a t w hich the deflection of a floating ice sheet b ecom es infinite is referred to in th e li te r a t u r e as th e c ritica l speed or critical velocity. T his critic a l speed coincides

w ith in th e deep w ater lim it provided t h a t th e ice sheet acc eleration can

le g itim ate ly be o m i t t e d (see S q u ire et al. [26]). Recently. M ilinazzo et al. [19] considered the s te a d y response of a n ice sheet of infinite e x te n t to a m oving r e c t­ a n g u l a r load. M ilinazzo et al. a s s u m e d th a t a s te a d y solution e x ists a n d o b ta in e d b o th num erical a n d a s y m p to tic e s tim a te s of th e so lution. In p a r ti c u la r , the p a p e r

1.1 In tro d u ctio n

finite a t th e load s p e e d

S everal e x p e rim e n ta l s tu d ie s o f waves on ice h a v e also been c o n d u c te d . .\ large n u m b e r of th e se e x p e r im e n ta l s tu d ie s are d is c u s s e d in d e ta il by S quire ct

al. [26i. It is in te re s tin g to no te t h a t t h e th e o re tic a l p r e d ic tio n s are in re m a rk a b ly

g o o d a g re e m e n t w ith e x p e r im e n ta l resu lts. In W ils o n [36]. th e co u p lin g between m o v in g loads an d fiexural waves in flo a tin g ice s h e e ts was s tu d ie d e x p e rim e n ta lly a n d t h e results were c o m p a r e d w ith th e th e o re tic a l p re d ic tio n s o b ta in e d using th e a n a ly s is of G reen h ill [7]. T h e e x p e r im e n ts w ere c a r r i e d o u t for b o t h one an d two vehicles as m oving loads. It was n o te d for a s in g le vehicle t h a t th e m a x im u m d eflection occurs a t a c ritica l load sp e e d . For tw o vehicles, d e p e n d in g on th e d i s ta n c e betw een th e vehicles, th e m a x im u m d e fle c tio n differs from t h a t of a single vehicle an d th e wave p a t t e r n a p p e a r s as th e s u p e r p o s i t i o n of tw o s e p a r a te sets o f wave p a t te r n s c a u s e d by two single vehicles. B e lta o s [2] also o bserved the e x iste n c e o f a critical s p e e d an d a d e p e n d e n c e of t h e w ave a m p l i t u d e on th e speed of th e load.

T a k iz a w a ;3(). 31] c a rrie d o u t a n u m b e r of r e m a r k a b l e e x p e rim e n ts on ice- covered Lake S aro n ia on th e .Iapane.se island of H o k k a id o u sing a snow m obile. T h e a r tic le s noted th e existence o f a critical s p e e d below w hich th e response is c[uasi-static. an d a b o v e which tw o ice waves t r a i n s a p p e a r - one o f s h o rte r w a v e le n g th a h ea d o f th e m oving snow m obile, a n d t h e o t h e r o f la rg e r w avelength tr a i li n g b eh in d . W i t h increased sn o w m o b ile s p e e d t h e le ad in g wave s h o rten s, while t h e tra ilin g wave le n g th en s a n d even tu ally v an ish es. In s u b s e q u e n t work. T a k iz a w a [32] fu r th e r a n a ly z e d th e e x p e r im e n ta l r e s id t s a n d confirm ed th e earlier findings.

F u r t h e r e x p e rim e n ts were c a rrie d o u t by S q u ire et al. [27] on b o t h lake a n d sea ice. .\g a in . it was observed t h a t tw o different w aves a p p e a r w h en th e load sp eed exceeds th e c ritic a l speed. However, for th e s a m e load ty p e a n d t h e sam e load s p e e d , th e wave a m p l i t u d e for fresh w a te r lake ice was la rg e r t h a n for sea ice.

1.1 In tro d u c tio n

S q u ire et al. [27] pointed o u t t h a t th is is m ost likely d u e t o th e higher d a m p i n g o f sea ice c o m p a r e d to fresh w ater ice. T h e y also a s c e r ta in e d e x p e rim e n ta lly th e wave p a t t e r n s th e o re tic ally p red ic ted [cf. D avys et al. [5]).

P re v io u s th e o re tic al a n d e x p e rim e n ta l s tu d ie s have c o n c e n tr a te d not only on an a ly z in g or m e asu rin g th e ice deflection n e a r th e load region, in order to assess th e p o te n tia l to break ice. b u t also on a n a ly z in g or m e a s u rin g th e ice deflection in th e fa r field - for exam ple, to assess th e th e o re tic a l a n d e x p e r i m e n t a l a g re e m e n t (S q u ire et al. [26]). T h u s K heisin [14]. Xevel [21]. a n d T a k iz a w a [31] c o n c e n tr a te d m o re on th e ice deflection n e a r th e load region. Davys et al. [5] a n d Schulkes et

al. [24] used a s y m p to tic an aly sis to pred ic t th e wave p a t t e r n s a t some d is ta n c e

from th e load, whereas Schulkes a n d S n ey d [25] co n sidered b o t h the im m e d ia te v ic in ity a n d th e sp a tia l a s p e c ts in th e case o f a line load. M ilinazzo et al. |19l c a lc u la te d th e s te a d y ice deflection in th e e n tir e flow field a n d o b ta in e d a s y m p t o tic e s ti m a t e s for th e far field ice deflection.

S everal p a p e rs , for e x a m p le Kheisin [14] a n d Xevel [21]. assum ed t h a t th e

s u p p o r t i n g fluid is w ater o f u n lim ite d d e p t h : w hereas o th e rs . K e rr [1 1]. D avys et

al. [5]. Schulkes et al. [24]. a n d M ilinazzo et al. [19] a s s u m e d t h a t the fluid is of

finite d e p th . F u r th e r analysis is carried o u t in this D is s e r ta ti o n to e x am in e th e ice deflection w hen the fluid base is a s s u m e d to be of infinite d e p t h . T h e work of M ilinazzo et al. 119] is re-visited, b u t for a fluid of u n li m it e d d e p t h w ith a view to sim p lify in g t h e analysis. T h e results show t h a t th e c h a ra c te ris tic s of th e ice deflection a re very sim ilar for th e two cases (cf. Davys et al. [5]. Takizaw a [31]. M ilinazzo et al. [19]). However, it tu r n s o u t t h a t th e a n a ly s is is essentially th e s a m e as t h a t for th e case w here the fluid base is o f finite d e p t h . T h e infinite d e p t h a s s u m p tio n does not seem to affect th e c h a ra c te r is tic s of t h e waves, except t h a t as e x p e c te d th e long waves d is a p p e a r w h en t h e load s p e e d is g re a te r th a n for t h e case w here the fluid base is of finite d e p t h . For t h e case where th e fluid base is of infinite d e p th th e waves a p p e a r b e h in d th e lo a d a t all speeds g r e a t e r

1.1 In tro d u ctio n

t h a n t h e load speed .A.s discussed by Squire et al. [26]. th is d is tin c tio n

c o r re s p o n d s to th e respective in tersectio n s o f an o r d in a t e re p re s e n tin g tiie load s p e e d w ith the two d is p e rs io n curves.

In d eriving th e m a t h e m a t i c a l m ode l for the w ater-ice problem , it is usually a s s u m e d t h a t th e ice thickness is sm all so t h a t th e stress a t th e surface a n d a t the b o t t o m o f th e ice sheet are ta k en to be th e sam e. O ften , a fu rth e r s im p lific atio n is m a d e by o m ittin g t h e te r m t h a t c o n ta in s th e ice thickness. T h is is equivalent to a s s u m in g th a t th e w avelengths of t h e waves of in terest are m uch la rg e r th a n t h e ice thickness (see for ex am ple. D avys et al. [5] or Schulkes a n d S n e y d [24]). S t r a t h d e e et al. [28] a n a ly z e d th e response of a floating ice sheet to a m o v in g load, w here th e ice is a s s u m e d to be a n isotropic viscoelastic p la te w ith finite thickness. T h e y a r g u e d th a t th e d is tr i b u tio n o f stress and s tr a i n a ro u n d a c o n c e n tr a te d load receive significant c o n trib u tio n from waves w ith w avelength c o m p a r a b l e to p la te thickness a n d hence th e ex ac t d escrip tio n of th e thickness effects need to be in c lu d e d in rhe m a t h e m a t i c a l m odel. However, f u r th e r analysis by S q u ire et

al. [26] w here they c o m p a r e d the so lu tio n from S tr a th d e e et al. [28] for a s t a t i o n a r y

load w ith th e so lu tio n o b ta in e d from th e th in p la te th e o ry due to W y m a n [-37] (see Stpiire et al. [26] section 5.8 for details) showed t h a t th e results a r e in fact id e n tica l. In this D is se rta tio n , a m a th e m a tic a l m odel t h a t takes in to acc o u n t th e ice thickness is d e riv e d w here th e ice thickness is no longer c o n s id e re d sm all. T h e result shows t h a t th e effect of th e ice thickness is negligible, b u t th e re are s o m e sim ilarities b etw een th is m odel a n d th e model a n a ly z e d by K e rr [11] w here in -p la n e forces are ta k e n into account.

.A.S m e n tio n ed above, th e previous the o re tic al stu d ie s have e ith e r a s s u m e d a s t e a d y s t a t e exists or considered th e tim e -d e p e n d e n t response to a n im pulsively s t a r t e d o n e-d im en sio n al line load. For th e tw o-dim ensional case how ever, to d a t e no a t t e m p t has been m a d e to d e te rm in e th e tim e -d e p e n d e n t b e h a v io u r o f th e ice d eflection as a fu n c tio n of load speed {cf. Squire et al. [26]).

1.1 In tro d u c tio n

T h e m o s t i m p o r t a n t a s p e c t of this D is se rta tio n is th e investigation o f th e tim e- d e p e n d e n t b e h a v io u r of th e ice deflection d u e to a n im p u ls iv e ly -sta rte d s te a d ily m oving p o in t load on a n ice sheet. T h e load is a s s u m e d to move over a ho m o ­ geneous ice sheet of infinite horizontal e x te n t a n d th e fluid base is a s s u m e d to be w a te r o f finite d e p t h . T h e Fourier tra n s fo rm is used to solve th e e q u a tio n s o f m o tio n , a n d th e s in g u la ritie s of th e c o r re s p o n d in g in te g ra n d are a n a ly z e d as a fu n ctio n o f t h e load sp eed. T h e results show t h a t th e tim e -d e p e n d e n t p a r t of th e

ice deflection grows w ith tim e when th e load sp eed is However, a t th e load

s p eed th e tr a n s ie n t p a r t of the ice deflection decays w ith time. .Assuming

t h a t a s te a d y s o lu tio n is possible, for a r e c t a n g u la r load. .Milinazzo ct al. , 19]

found t h a t a s te a d y s o lu tio n exists a t th e load s p eed b u t th a t p a p e r did not

ad d re s s t h e a t t a i n a b i l i t y o f th e solution. T h e findings in th is D is se rta tio n suggest

t h a t at th e load s p eed th e ste a d y s t a t e so lu tio n s c o m p u te d in M ilinazzo ct

al. [19j a re large tim e s o lu tio n s of th e c o rre s p o n d in g in itia l value p ro b le m a n d

a re consecpiently physically a tta in a b le .

T h e following is t h e o rg a n iz a tio n of th is D is se rta tio n . C h a p t e r 2 an a ly z e s th e s te a d y s t a t e waves ca u s e d by a moving r e c t a n g u la r load on an ice sheet s u p p o r t e d iyv w a te r of infinite d e p t h . C h a p t e r 3 deals w ith th e m a th e m a tic a l m o d e lin g of th e ice-w ater s y s te m w hen t h e thickness of th e ice p la te is ta k e n into account. C h a p t e r 4 d e s crib e s th e m a t h e m a t i c a l model of th e ice-w ater s y ste m a n d analyzes in d e ta il th e tim e - d e p e n d e n t s o lu tio n of the ice deflection d u e to a point load. Finally, in C h a p t e r -3. a s u m m a r y is presented.

C h a p te r 2

S te a d y S o lu tio n d u e to th e M o tio n o f a R ecta n g u la r Load

on an Ice S h eet over W ater o f In fin ite D ep th

2.1

In tr o d u c tio n

In th is c h a p t e r , rhe s te a d y s ta t e ice deflection caused by a r e c ta n g u la r lo a d m oving on an ice sheet s u p p o r te d by a fluid base of infinite d e p t h is consid ere d . T h e m a t h e m a t i c a l model is th e sam e as t h a t an alyzed by M ilinazzo et al. [19]. except th a t in t h a t p a p e r th e d e p th of t h e fluid base is a s s u m e d to be finite. T h e y

found t h a t th e ir s te a d y solution is finite a t all loa d sp e e d s except In

th e ir a n a ly sis , th e so lu tio n is e x p ressed as a tw o -d im en sio n al Fourier in t e g r a l a n d th e poles o f th e in te g ra n d are used to o b ta in b o th a n u m e rical d e s c rip tio n and a s y m p t o t i c e s tim a te s of th e ice deflection. T h e m o tiv a tio n in m aking t h e infinite d e p t h a s s u m p ti o n in th is c h a p te r is to see if th e a n a ly s is of the ice deflection b ecom es s im p le r th a n w hen th e fluid base is of finite d e p t h . T h e an a ly sis is also c a rrie d o u t to confirm t h a t th e d eflection is finite for all load speeds o t h e r th a n

th e load specxl a n d to c o m p a r e th e results w ith th o se o b ta in e d for t h e case

w here th e fluid base is of finite d e p t h . In w h at follows, th e te rm the ca.se o f f i n it e

depth is u sed to refer to t h e pro b lem w h ere th e s u p p o r t i n g w a te r is of finite d e p th ,

a n d th e t e r m the ra.se o f infinite d ep t h is used to refer to th e problem w h e re the s u p p o r t i n g w a te r is of infinite d e p t h .

T h e m a t h e m a t i c a l m odel t h a t d e s c rib e s th e ice d is p la cem e n t is as follows. T h e J-. : p la n e is taken to coincide w ith th e b o tt o m o f th e ice sheet. T h e elastic, h o m o g e n e o u s, th in ice sheet of in fin ite e x te n t in th e x . z p la n e is c o n s id e re d to

2.1 In tro d u c tio n __________________________________________________________

10

have c o n s ta n t th ic k n e s s h and c o n s ta n t d en sity . T h e s u p p o r tin g w ater is

a s s u m e d to have c o n s ta n t density p. th e u n d is tu r b e d w ater surface is taken to he a t // = 0. a n d th e b o tto m is ta k e n to be a t y = — oc. T h e th e o ry of th e b e n d i n g of a t h i n p la te (T im oshenko a n d W oinow sky-K rieger [.35]. T im oshenko

a n d G e re [34]) is used to model th e floating ice p la te. If j } ( x . z . t ) represents

th e ice deflection a n d f i s . z ] rep resen ts th e dow n w a rd applied s tress d u e to th e u n ifo rm ly d i s t r i b u t e d rec ta n g u la r lo a d on th e ice s h e e t, th e n th e e q u a tio n of th e ice deflection (S z ila rd [29] section 4.2) in a fram e of reference m oving with th e load a t speed \ ' in th e x direction is given by

''* 1

*

for - 3C < -V. : < (2 .1)

w h ere X — x — 1 7 . rj a n d / have been redefined as functions of .V a n d :. a n d /( . V . z) is given by

X o te t h a t in eq. (2 .2 ). is a c o n s ta n t . T h e c o n s ta n t D > Ü is given in term s of

Y o u n g 's m o d u lu s E a n d Poisson s r a tio for ice n (0 < u < 1). by th e relation

T h e u p w a rd w a te r p ressure p is d e te r m in e d , by a s s u m in g th a t th e w a te r base is in com pressible a n d its flow is ir r o ta tio n a l. w hen th e flow can be describ e d by th e velocity p o te n tia l o ( t . X . ij. z) satisfy in g

d ' o d ~ o d ’o

—— -t- — — 7 -4- — 7 — 0 — o c < y < 0 . — 3C < . \ . c < (2.4)

a \ - o n - a

z-T h e c o n d itio n o f no n o rm a l flow a t th e b o t t o m is enforced by s e tt in g th e vertical

velocity to zero a t ij = —oc. T h e n o n -c a v ita tio n k in e m a tic c o n d itio n at the ice-

2.1 In tro d u c tio n __________________________________________________________ ^

is enforced by s e t t i n g t h e v ertical velocity of th e w ater to be equal to t h a t o f th e

ice-w ater interface, i.e. ^ ^ 4- \ ' -^)r]. Since the analysis is lim ited to waves

of sm all a m p lit u d e , th e B ernoulli equation a t y = 0 is given by

fjn — P + ( ^ 1 - r ^ ) o ( t . A . 0. c ) = 0 . (2 .Ô)

f) a t u \

T h e g r a v it a ti o n a l a c c e le ra tio n g in eq. (2.5) is taken to be in the n e g a tiv e g direction.

In o r d e r to d e t e r m i n e th e s te a d y solution of eqs. (2.1)-(2.5) using th e F ou rier tra n s fo rm , it is n e c e ss a ry to en s u re th a t th e solution satisfies th e correct r a d ia ­ tion c o n d itio n a t infinity. T h i s is achieved by using a te c h n iq u e in tro d u c e d by

Lighthill [16] w h e re t h e p re s s u re / is replaced by /*’ = <5 > 0. T h u s a pres­

sure t h a t was zero in t h e d i s t a n t past has grown to / a t f — 0 . c o rre s p o n d in g to a

r a d ia tio n c o n d itio n t h a t e lim in a te s incoming waves. T h e s te a d y solution is th e n o b ta in e d by le tti n g () —> O '.

T h e tim e d e p e n d e n c e of th e dependent variables c an be ta k en to be f c o n ­ sequently

(2.6)

o { t . X . g . z ) = A', y. ; ) . (2.7)

p ( t . X . z ) = / Y ( A ' . : ) . (2.8)

S u b s tit u tin g e<is. (2.6)-(2.8) in to eqs. (2.1)-(2.5) and tra n s fo rm in g th e re s u ltin g eciuations. in c lu d in g th e b o u n d a r y conditions, using Fourier tra n s fo rm s in t h e A

a n d : d ire c tio n s y ie ld ( c/. M ilinazzo al. [19]):

[ Dk^ + p,re h{6 - = - U . (2.9)

= 0 for - DC < y < 0 . (2.1 0 )

2.1 In tro d u ctio n __________________________________________________________ ^

(]H^ —'P'* -i- (d — i\ Ali ( Ai 1. Ai-). 0) = 0. (2.12) P

whore Ai Ls givon hv k~ = k\ + K.j and ' W (kx. Kt). <Ï>'’(k i. k -j.//). P '*(ki.k-j). a n d

n(A, |. Ai-j) have been in tro d u c e d to d en o te th e F ou rier tr a n s f o r m s o f i f . o ' , p ' a n d / . respectively. T h e F ourier tra n s f o r m of / ( .V . r) in eq. (2.2) is given by

a b

n = y d.V

—ti — 6

2 Po sin(«Ai[ ) sin(6Aij)

- A i 1 K -2

T he s o lu tio n of ecjs. (2 .I0 ) - ( 2 .I 1 ) is

M 3 )

4»" = (2.14)

S u b s tit u tin g etp (2.14) into eq. (2.1 2 ) gives

{ < ) - l V K x f ^

(J

P ' = - p (2.13)

w hich can be used to g e th e r w ith ecp (2.9) to o b ta in

_ n ( |A-|

H" = — ---:--- —--- (2.16)

PfJ VP-(A-, + t<)ï-(d\k\ + 1) - \ k \ ( l + a k - ^ ) ,

w here th e following te rm s have been in tro d u ced to simplify t h e n o ta tio n .

Pire , h - d(l r--, ' D

ft = ---. .5 = / A — . () = — . r = — . a =

f) a I g a pga^

A’l = a s i . k> = aK>. k = an. (2.17)

L'sing ec[. (2.13) in eq. (2.16). th e final form of the s t e a d y ice deflection c a n be seen to be

pg 0 F- ( k i + /d)-(.A|A| + 1) — |A'|( 1 + oA '

sin(Ai) s i n f U ’-j)

d k i d k ) . (2.18)

2.2

The Dispersion Rel ati on

13

N ote t h a t th e n o rm a liz a tio n has been d o n e using a. the le n g th of th e lo a d in the A direction, as th e length scale r a t h e r t h a n th e w ater d e p t h as is d o n e for the case of finite d e p t h (see M ilinazzo et al. [19]).

From ecp ( 2 .IS), th e poles of th e i n t e g r a n d are given by th e zeroes o f the

d e n o m in a to r F ' i k i — tô)-{. j\k\ -t- 1) — |Ar|( 1 -roA*^) = Ü. which is a p o ly n o m ia l in

k. T his is in c o n tra s t to th e case o f finite d e p t h (rf. M ilinazzo et al. [19]). where

t h e d e n o m in a to r is a n e q u a tio n involving tra n s c e n d e n ta l functions.

2.2

T h e D isp e r sio n R e la tio n

T h e difference betw een th e case o f finite d e p t h a n d th e case o f infinite d e p th c a n be u n d e rs to o d by c o m p a r in g th e d is p e r s io n relation for t h e two p ro b le m s . .-V dispersion rela tio n is th e re la tio n s h ip b e tw e e n th e frequency o f a pla n e wave an d

its w avelength a n d is derived by looking for travelling wave s o lu tio n s of eqs. (2 .1)-

(2 .Ô) of th e form ^ ' ' - ^ - « 1 . \ - k j o /(A*, z) = Ü. Such so lu tio n s exist provided

th e relation

is satisfied. T h e c o r re s p o n d in g re la tio n b e tw ee n the phase sp eed C of th e free

waves a n d th e w av en u m b e r k is given by

= ,2.201

V K / ^ ( 1 f fill |k| )

T h e corresp o n d in g re la tio n s h ip for w a te r o f finite dep th . H . is given in C h a p t e r 4

(see eq. (4.10). In t h a t e q u a tio n , ta k in g H zxz gives C ’ = ( D k ^ / p + g ) / k . which

is th e sam e as eq. (2 .2 0 ) w hen w a v e le n g th s o f interest a re m uch la rge r t h a n the

ice thickness. D avys et al. [ô] n o te t h a t for a s te a d y wave p a t te r n , th e c o m p o n e n t o f the load velocity n o rm a l to a n y wave c r e s t m ust equal th e crest p h a s e speed, i.e.

■I ) _{The D ispersion R elation 14}

where J is th e angle betw een th e w a v e n u m b e r vector a n d th e d ire c tio n o f travel of th e load, a n d cos i = ^ . Hence, o n ly waves co rre s p o n d in g to th e p oles in the integrand of eq. (2.18) will a p p e a r in th e s te a d y p a t t e r n of th e ice deflection.

T h e dispersion rela tio n curve for eq. (2.20) is shown in F igure 2.1. It is

p lo tted using som e ty p ical p a r a m e te r s given in Takizaw a [31]. X ote how t h e curve ap proaches inflnity as th e w av en u m b e r appro ac h es zero. By c o n tra st, in th e case of finite d e p t h (see F ig u re 2.2) th e c o rre s p o n d in g d isp ersio n curve in te rs e c ts th e vertical axis & = 0 a t C = \J<jH. T h e g ro u p s p e e d ', which is defined as Cgr = is also shown in these figures. F ro m th e dispersion curves, it is clear t h a t the m a jo r difference is for long waves, i.e. waves w ith a sm all w av enum ber. T his difference has been n o te d previously in th e fluid m echanics lite ra tu re , as well as in present context (e.g. by Davys et al. [ô]). In the next section, th e s o lu tio n for the case of infinite d e p t h is analyzed.

group speed is the speed at which the envelope of a wave group travels. T he wave components of the wave group propagate with the speed C (Kundu [15]).

2.2 T he D isp ersio n R ela tio n 15

Dispcraon rcladon for the case of infinité dcpüi

■0 B S

a

V I

mm

Figure 2.1: T h e d isp e r sio n curve for th e case o f in fin ite dep th. The plot

labeled

C

denotes the phase speed (eq. (2.20) and the plot labeled

Cgr

denotes

the group speed. The wavenumber is represented by the horizontal axis and the

speed is represented by the vertical axis. The parameters are

E

= 5x10*.

^ h -

0.175 m. ^ = 9.8 m/sec^.

p

= 1026 m/kg*, and

pice

= 850 m/kg*.

2.2 T h e D isp ersio n R ela tio n 1 6

Dispcrsioa relation for the case of finite dqrtfa

mm

Warenmnber

Figure 2.2: T h e d isp e r s io n curve for th e ca se o f fin ite d ep th . The plot

of the dispersion relation for the case of finite depth (eq. (4.10) is shown as the

plot of the phase speed C. The plot of the group speed is shown as the plot

labeled

Cgr-

The parameters are

E

= 5x10*.

v = ^. h =

0.175

m. H =

6.8 m.

2. 3_____ The S o l u t i o n Method__________________________________________________^

2.3

T h e S o lu tio n M e th o d

To ca lc u la te a/ , tlie d ouble in te g ra l given in eq. (2.18) is w r itt e n as a n ite r a te d

in te g ra l in a n d k-,. T h e in n e r integral in k\ is ev a lu a te d for a fixed k> using

c o n t o u r in teg ratio n . T h e o u te r in te g ra l in k> is th e n e s tim a te d u s in g th e m e th o d of s t a t i o n a r y pluise (Xayfeh [20]). T h is is in c o n t r a s t w ith M ilin azzo et al. [19] w here th e k) in te g ra l is also e v a l u a te d n um erically using an a d a p t i v e G a u ssian (quadrature schem e (see .Milinazzo et ai. [19] s ectio n 4).

T h e k[ integral of eq. (2.18) c a n be w ritten as a function o f k> in th e form

1 /-oc

w here JF'* is

JF'*(k [. k >) = [F"( k\ + /())“( 1 4- djA‘| ) — A ( 1 4- oA' ' )] — -. 12.23 )

1^ I

Recall th a t à in is used to d e te rm in e th e s o lu tio n th a t satisfies the ra d ia ­

tion conditio n a t infinity. T h e poles of eq. (2.22) a r e th e zeroes o f t h e e q u a tio n

!F^(k\.k>) = 0. T h e zeroes of eq. (2.23) co rresp o n d to th e poles o f t h e in te g ra n d

of ecj. (2.22) in th e Aj-plane. F or à = 0. the real poles are identified w ith th e waves g en era ted in th e system . W h e n these poles a re on the real axis, d eform ing th e c o n to u r above o r below th e m d e te rm in e s w h e th e r th e c o rre s p o n d in g waves are o u tg o in g waves or incom ing waves. By in tro d u c in g 6 > 0. th e se poles move off th e real axis, th e re b y m a k in g it possible to identify which poles a r e to be included w hen ev a lu a tin g t h e integral u s in g c o n to u r in te g ra tio n . In w h a t follows. !F will

be used to d e n o te .F"'* when r) = 0 .

F ro m c(j. (2.18) it can be seen t h a t it is n ecessary th a t A > 0 for real A,, k-,.

. \ n a p p r o p r ia te choice of b r a n c h c u t for the fu n c tio n A = ^ k ' \ - t - A.j is given in th e

2.3 The S o lu tio n M ethod

₁₈

2.3.1

T h e D e fin itio n o f

k.

T h e function k = \/k'i + k'i h as b r a n c h p o in ts at ki = ±ik-,. k , is a s s u m e d to he real a n d k-, > 0. F igure 2.3 illu s tr a t e s th e branch cu t. d e n o te d by d a s h e d lines along the v ertical axis, a n d th e tw o b ra n c h points.

A

1

-p la n e

i k>

t k

-Figure 2.3:

T h e branch c u t in th e

At

-plane.

T h e branch c u t is shown as

dashed lines on th e vertical a x is s t a r t i n g from the bran ch points ±ik>.

T h e angles a n d m e a su re d in th e co u n te r clockwise direction a n d shown in

F igure 2.3. satisfy

3 — - - 3

-(2.24)

3 - - % 3 “

■— < ^^1 < - : 7 < 0 > < —

a n d th e fu nction k is defined by th e expression

k =\ At - ik-2 | ‘ -| At + ik-2 2.23)

T h is gives a definition of A in t h e co m p lex plane for which A > 0 for all real At a n d A_>. X ote t h a t on the h o r iz o n ta l axis in th e A fp la n e . ^t = —#). a n d hence

A = i At — /A) I’ ‘ I At + ik> i‘ ‘

— I At — i k ) I ’ " I At - I - /Aj “ > 0 for At real.

T hus, from th e definition given in eq. (2.25). A is always real a n d n o n -n eg ativ e on the real At axis.

2.3 T he S o lu tio n M ethod 1 9

2 .3.2

P o le s in th e C o m p le x P lan e

-A.S previously n oted, to o b ta in th e poles of th e in te g ra n d o f eq. (2.22). it is neees- -sarv to d e te rm in e th e zeroes o f eq. (2.23). Since th e in teg ral given by eq. (2.22) is a fu nction of A j. th e loc ation of these zeroes, a n d hence th e location o f t h e poles, is also a function o f k>. To d e te rm in e the se poles. k\ is r e w r itt e n in t e r m s o f k i.e. A’l = ± \ J k - — k.j. a n d th e equation.

T = F -{k~ - A-.])(I + S k ) - A-(l 4 - oA-^) = 0 (2.26) is considered as a fu n c tio n o f k w ith fixed F~ a n d k \ . T h e d e p e n d e n c e o f th e

F igure 2.4:

Z eroes o f

T . For a fixed F a n d A-_) > 0. th e real zeroes are i l l u s t r a t e d

as th e intersection o f th e two curves in th e u p p e r right h a lf o f th e p la n e . indicates th e m in im u m of th e left hand side of eq. (2.27).

zeroes in th e A-plane on load speed (i.e. given by F'~) a n d k j can be seen by rew riting eq. (2.26) in t h e form

A- ( 1 + JA) ^ k'^'

(2.27)

where k > k-, > 0 . F ig u re 2.4 illustrates th e zeroes of eq. (2.26) for a fixed F

2.3 T he S o lu tio n M ethod 2 0

f'q. (2.26) t h a t o c c u r in the first or th e f o u r th q u a d r a n t of th e c o m p le x A-plane. T h i s a s s e r tio n can he d e m o n s t r a te d by c o n s id e r in g th e following four different cases.

A-plane

A’l-p la n e

p o les

I

zeroes

F ig u re 2.3:

T h e rela tio n b e tw e e n th e z e r o e s in th e A -plane and th e

p o le s in th e Aq-plane.

T h e illu s tra tio n s h o w s th e zeroes o f eq. (2.23) for

A‘i

—

± y A'- — A.]. ()

7^ Ü. k> 7^ 0. a n d th e c o r r e s p o n d in g poles o f t h e in te g ra n d of

eti.

.Assume th e zero Aq is in th e first q u a d r a n t o f th e Aq-plane: In th is q u a d r a n t

— ^7 S. Si ^ . a n d O'l ^ —d\. C o nsequently. () ^ ^ rr

a n d so 0 < arg(A ) < f . A" is in th e first q u a d r a n t .

.Assume th e zero Aq is in th e second q u a d r a n t o f the Aq-plane: In th is

q u a d r a n t — ^ S S ^ S ^2 S "• a n d G-y < —0[. In this case.

— M < (Gi -f- Gy) < 0 a n d —f < arg(A-) < 0. Hence. A- is in th e fo u rth

cpiadrant.

.\s s iim e th e zero Aq is in th e th ir d q u a d r a n t o f t h e Aq-plane: In th is q u a d r a n t

S ^2 S ^7". —~ ^ —fy. a n d Go ^ —Gi- Therefore. Ü ^ (f^i -t- Go) < ~

a n d 0 < arg(A) < f . A is in th e first q u a d r a n t .

.Assume th e zero Aq is in th e fo u rth q u a d r a n t o f th e A i-plane: In th is

2.3 The S o lu tio n M eth o d _________________________________________________ 2 ^

- - < (Oi -r (i>) < Q a n d — ^ < arg(A:) < 0 . k is in th e f o u rth q u a d ra n t.

Since only th e zeroes in th e first a n d t h e fo u r th q u a d r a n t o f th e A-plane are im p o r ta n t , o t h e r zeroes in th e left half of t h e A-plane can be ignored. F igure 2.3 illu strates th e zeroes of eq. (2.23) found in t h e right half of th e A-plane a n d th e co rre s p o n d in g poles in th e Aq-plane. T h ese zeroe s in th e A-plane are co m p u te d for

a fi.xed F. Aq # 0. a n d ^ 0 from eq. (2.23) by s u b s t itu t in g Aq — ± \ J k - — A.y T h e

zeroes in th e A-plane d e n o t e d by r-, a n d r., c o rre s p o n d to th e zeroes of eq. (2.23) w hen Aq = ^ s j k ' ~ — A-.j; w h ereas the zeroes d e n o t e d by /q a n d r , co rrespond to t h e zeroes of ecj. (2.23) w h e n Aq = - ^ k - — A\]. W h e n Aq = 0. th e re is also a root a t th e origin o f th e A-plane. T h is root c o r r e s p o n d s to Aq = 0 in th e Aq-plane.

In su m m a ry , it has been show n th a t t h e po le s o f th e in te g ra n d of eq. (2.2 2) can

o nly be th e result of zeroes of eq. (2.26) t h a t a re in the right h a lf of th e A-plane.

Let th e m i n im u m of th e left-hand side o f (2.27) be defined as If F~ < F,;,.

th e zeroes in th e A-plane a r e com plex for all values of Ay. L s in g M .\ P L E . these zeroes can be found from eq. (2.26) for a fixed F a n d a given range of Aq. T h e result in F ig u re 2.6 shows t h a t th e zeroes a p p e a r n e a r th e origin when Aq is sm all

a n d move aw ay from th e origin as Aq —)■ 3C. Since th e zeroes in th e A-plane are

com plex, th e poles are also com plex. T h e b e h a v i o u r of these poles as a function o f Aq is illu s tra te d in F ig u re 2.7.

W hen F - = F,;,. from F ig u re 2.4 it can b e seen t h a t eq. (2.27) has a positive

real root w hen Aq = 0. In this case, t h e ro o t is com plex w hen Aq > 0 . T h e

beh av io u r of th e zeroes a n d th e poles as a f u n c tio n o f Aq can be seen in Figures 2.6 a n d 2 .8.

However, w h en F~ > F ^ . eq. (2.27) possesses two real, positive zeroes for a ra n g e of Aq (see F ig u re 2.9). -A.s a result, t h e c o r re s p o n d in g poles of the in te g ra n d o f eq. (2.22) a re also real for t h a t range of Aq. F ig u re 2.4 shows t h e two real zeroes in th e positive h a lf of A-plane for a fixed A-j. F ig u re 2.6 shows th e location of th e

2.3 The S o lu tio n M ethod_________________________________________________ 2 2

zeroes as a fu n ctio n of k-z- T h e zeroes s t a r t on the real axis when k> is sm all, one near t h e origin a n d th e o th e r away from th e origin. .A.s ky incretises. these zeroes a p p r o a c h each o th e r along the real axis, coalesce a n d becom e co m p lex . T h e b e h a v io u r o f th e c o rre s p o n d in g poles in th e A,-plane as a function o f k-y is illustrated in F igure 2.9.

2.3 The S o lu tio n M eth o d 2 3

complex

k-plane

0

0 0

0

F ig u r e

2.6: P o s itio n o f th e zeroes in th e first a n d th e fou rth q u a d ra n t o f

th e A-plane.

T h e plot shows th e lo c a tio n of the zeroes as a fu n c tio n o f Aj. T h e c o rr e s p o n d in g load s p e e d is also show n on th e right h a n d side of t h e p lo t.

2.3 The S o lu tio n M ethod 2 4

(a)

(I)

k-plane

Ü

k-plane

V 0

/

‘‘2

/

I

M

F ig u r e 2.7: P o s i t i o n o f z e r o e s i n t h e Ar-plane a n d p o l e s i n t h e A i - p l a n e a s a f u n c t i o n o f A j. P lo t (a) shows th e zeroes (in th e first a n d fo u r th q u a d r a n t of A-plane) of eq. (2.26) as a fu n c tio n of k>- P lo t (b) shows th e c o rr e s p o n d in g poles as a function of Aj in th e Aq-plane.

•2.3 The S o lu tio n M ethod 2 5

(a)

(b!

k-plane

0

F ' = F

m

0

F ig u re 2.8;

P o s itio n o f ze r o e s in th e Ar-plane an d poles in th e Ai-plane as

a fu n ctio n o f Aj.

P lot (a) show s th e zeroes (in th e first and t h e fo u rth q u a d r a n t o f A-plane) of eq. (2.26) as a fu n c tio n of ky. P lo t (b) shows t h e c o rresp o n d in g poles as a fu nction of A_) in th e Aq-plane.

2.3 The S o lu tio n M ethod 2 6

(a)

k-plane

0

0

F igure 2.9;

P o sitio n o f zeroes in th e Ar-plane and p o le s in th e A'l-plane as

a fu n c tio n o f

k). P lo t (a) shows th e zeroes (in th e first a n d fourth q u a d r a n t of

A-plane) o f eq. (2.26) a s a function of A-.>. P lo t (b) shows t h e c o rre s p o n d in g poles as a fu n c tio n of k> in t h e Aq-plane.

2.3 The S o lu tio n M e th o d 2 7

2 .3 .3

T h e E v a lu a tio n o f th e Integral

In section 2.3.2. th e poles o f th e in te g ra n d of eq. (2.22) were a n a ly z e d . By a n a l y z ­

ing — 0 for s m all d > 0 . it is easy to see t h a t t h e real poles closest to th e o rig in

move into th e lower h a lf p la n e { k [ a n d —k [ ) a n d those fa rth e s t from th e o rig in

move into th e u p p e r h a lf p la n e (k[ a n d —k[ ). F or all c o n to u r in te g ra tio n c a n

A t-P la n e

- H ' _{^ e}

" - Â f A-r

F ig u re 2.10;

P o le s in t h e At-plane.

T he p lo t shows th e p o s itio n of the po le s

in the Ai-plane. T h ese a r e th e poles of th e in te g r a n d of th e in tegral given in eq. (2.2 2 ).

h e used to e v a lu a te th e in te g ra l in eq. (2 .2 2 ) by closing th e p a t h of in te g r a tio n

e ith e r in th e lower or u p p e r h a lf plane.

Since it is necessary t h a t th e integ ran d goes to zero a t infinity, w h e th e r t h e in te g ra tio n p a t h is closed in th e u p p e r or lower h a l f plane d e p e n d s on th e sig n

o f th e e x p o n e n ts in th e e x p o n e n tia l term s o f eq. (2 .2 2 ). In front of th e lo a d .

( ^ ± 1) < 0 . so th e p a t h o f in te g ra tio n in eq. (2 .2 2 ) must be ta k e n in the u p p e r

h a lf plane. T h e c o n to u r is ta k e n in such a way t h a t all the poles in th e u p p e r h a l f p la n e are included. U sing t h e closed curve X>. w hich consists o f t h e p a th s -q. - i.

' a n d -, (see F ig u r e 2.11). th e in teg ral in eq. (2.22) c a n be w r itte n by

2.3 The S o lu tio n M eth o d 2 8 1 f

hi

dk\ ~ I ^0 4 - / j 4 - 4 - 4 - / , = ~ 5 1 residues of in V (2.28) T H k , . k - , )

Here, the in teg rals d e n o te d by Iq. /[ . A. /:}. a n d I, a r e co rre s p o n d in g ly taken

R . - p l s u i e branch cut ik. - R Figure 2.11: I n t e g r a t i o n p a t h i n t h e u p p e r h a l f p l a n e . T h e d irection of th e in teg ratio n is show n by th e arrow s alo n g th e closed c u rv e T>.

along the p a t h s ~o. ' i . '•>. " i- a n d respectively.

.A. fu rth e r e v a lu a tio n show s t h a t th e integrals ly a n d A t e n d to zero as R ^ yc.

a n d I, also te n d to zero as e —> 0 + . .\o n -zero c o n t r i b u ti o n s therefore only com e from the in te g ra tio n along th e b ra n c h c u t an d from t h e poles. T hese poles are

sim ple poles since Tk^ # 0 (su b sc rip t d e n o te s a p a r t i a l d eriv a tiv e ) a t these poles.