Long period oscillations in basins of arbitrary shapes

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CHAPTER 7

LONG PERIOD OSCILLA TIONS IN BASINS OF ARBITRARY SHAPES

Fredric Raichlen

Assistant Profe" sor of Civil Engineering

W. M. Keck Laboratory of Hydraulics and Water Resources California Institute of Technology

Pasadena, California ABSTRACT

The equation of continuity and the equation of motion for two-dimen-sional shallow-water waves are transformed into a set of difference equations which can be solved numerically to obtain the fundamental per-iod and the periods of the higher modes of oscillation of a closed basin of arbitrary shape. From these periods and the difference equations the corresponding amplitude variations are obtained. By modifying the boundary conditions used, with certain restrictions, the modes of oscilla-tion of a fully open harbor can be determined. The data necessary for the solution of both cases are the distribution of the cross section areas and the surface widths along the talweg of the basin (the line of maximum depth).

The exact solutions which describe the lowest mode of oscillation of basins of simple geometry, e. g., circular and eliptical, are compared to the results of this approach. In addition the fundamental and the higher modes of oscillation of three actual harbors are discussed. The agree-ment obtained by this method compared to others is considered to be reasonably good. Some considerations of the case of three-dimensional forced oscillations are presented.

INTRODUCTION

The study of the free oscillations of closed bodies of water (known as seiche) has aroused the interest of investigators for many years. Considering first a closed rectangular basin of constant depth, the periods of the free oscillations of shallow water waves can be derived from the solution of the Laplace equation for the velocity potential of the fluid, continuity considerations, and the appropriate boundary conditions. Merian's equation developed in this way for the special case of a narrow basin having predominantly longitudinal long -period os cillations, is:

T -

21 - n.Jgh

(n= 1, 2, 3, .... N)

where T is the period of the oscillation,

1,

is the length of the basin, h is the depth of the basin, and n represents the harmonic order.

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Although this expression is sufficient to predict the modes of oscilla-tion of narrow basins which are approximately both rectangular and of constant depth, variations from this shape necessitate a more general approach. Numerous investigators have studied this problem, and only the results of a few will be briefly discussed here. However, the inter-ested reader is referred to Defant (1961) for a detailed summary of various analytical methods.

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OSCILLATIONS 117

Figure 1. D e f i m t i o n Sketch of A r b i t r a r y Shaped Basin

T a l w e g

Figure 2. Definition Sketch f o r N u m e r i c a l Analysis

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w h e r e i n the w a t e r s u r f a c e a m p l i t u d e T] i s now o n l y a f u n c t i o n of s. Since f o r the t w o - d i m e n s i o n a l c a s e , v a r i a t i o n s a r e i n the s - d i r e c t i o n a l o n e , t h i s can be expanded t o :

w h e r e X i s a c h a r a c t e r i s t i c v a l u e ( e i g e n v a l u e ) of the p r o b l e m and the e x p r e s s i o n : ^ m = ¥ T ^ , m = l , 2 , . . . . N (11) m r e l a t e s a p a r t i c u l a r c h a r a c t e r i s t i c v a l u e , X , to the c o r r e s p o n d i n g p e r i o d T ^ w h e r e the s u b s c r i p t m = 1, r e f e r ? to the p e r i o d of t h e f u n d a -m e n t a l -m o d e of o s c i l l a t i o n and o t h e r v a l u e s r e f e r to the h i g h e r -m o d e s of o s c i l l a t i o n .

The o b j e c t i v e of t h i s i n v e s t i g a t i o n i s the s o l u t i o n o f E q . 10 w i t h the a p p r o p r i a t e b o u n d a r y c o n d i t i o n s f o r an a r b i t r a r y v a r i a t i o n i n the s-d i r e c t i o n of b o t h the a r e a A ans-d the s u r f a c e w i s-d t h b . T h i s s o l u t i o n w i l l y i e l d the e i g e n v a l u e s w h i c h d e s c r i b e the p e r i o d s of the m o d e s of o s c i l l a -t i o n as w e l l as -the v a r i a -t i o n of -the a m p l i -t u d e i n -the s - d i r e c -t i o n .

F o r the case of a c o m p l e t e l y c l o s e d b a s i n the o b v i o u s b o u n d a r y c o n d i t i o n i s a z e r o w a t e r p a r t i c l e v e l o c i t y n o r m a l to the b o u n d a r i e s o f the b a s i n . F o r a v e r t i c a l b o u n d a r y i n the t w o - d i m e n s i o n a l p r o b l e m t h i s c o n d i t i o n (u = 0) can be r e p l a c e d by the c o n d i t i o n t h a t the s l o p e o f the w a t e r s u r f a c e i s z e r o at the b o u n d a r y , i . e. | I 1 = 0. I n o t h e r w o r d s the w a t e r s u r f a c e a m p l i t u d e i s a m a x i m u m o r a 4Tainimum at a s o l i d v e r t i c a l b o u n d a r y .

I n the i n v e s t i g a t i o n o f c l o s e d b a s i n s the c r i t e r i o n o f a m a x i m u m o r m i n i m u m a m p l i t u d e at the b a s i n ends was u s e d as the b o u n d a r y c o n d i t i o n (at s = 0 and s = ^ i n F i g . 1). A d m i t t e d l y t h i s t y p e of b o u n d a r y c o n d i t i o n b e c o m e s a p p r o x i m a t e w h e n the b a s i n b o u n d a r i e s a r e i n c l i n e d at a s m a l l a n g l e to the h o r i z o n t a l ; h o w e v e r , f o r the case of a c i r c u l a r b a s i n w i t h a p a r a b o l i c b o t t o m t h i s a p p r o a c h gave s u r p r i s i n g l y good r e s u l t s f o r the f u n d a m e n t a l m o d e . T h i s w i l l be d i s c u s s e d m o r e f u l l y l a t e r .

F o r a b a s i n w h i c h c o m m u n i c a t e s w i t h the o p e n s e a the b o u n d a r y c o n -d i t i o n at the e n t r a n c e u s e -d i n the a n a l y s i s was t h a t t h i s s e c t i o n i s a n o -d a l r e g i o n , and the a m p l i t u d e at t h a t s e c t i o n i s z e r o f o r a l l t i m e . H e n c e , t h i s a p p r o a c h n e g l e c t s the i n f l u e n c e of the e n t r a n c e a n d any d e g r e e of c o n -s t r i c t i o n o f the e n t r a n c e on the i n d u c e d m o t i o n w i t h i n the h a r b o r , and t h e r e f o r e , i t i s a p p l i c a b l e o n l y t o f u l l y open h a r b o r s w h e r e t h e b a s i n w i d t h i s s m a l l c o m p a r e d to the l e n g t h . T h i s a l s o w i l l be d i s c u s s e d l a t e r w i t h r e f e r e n c e to the a p p l i c a b i l i t y of t h i s m e t h o d to a c t u a l h a r b o r s .

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OSCILLATIONS

w h e r e n = 1, 2, 3, . . . N m = 1, 2, 3, . . . N

E q . 14a t h e n r e p r e s e n t s a set of equations w h i c h m u s t be s o l v e d s i m u l -t a n e o u s l y -to y i e l d -the e i g e n v a l u e s of -the p r o b l e m A (or -the m o d a l p e r i o d s T ^ ) .

C L O S E D B A S I N

A s k e t c h i s p r e s e n t e d i n F i g . 3 s h o w i n g m i r r o r i m a g e s a t t a c h e d to the b a s i n at S e c t i o n s 1 and N (the a r e a s and s u r f a c e w i d t h s at S e c t i o n 2 and N - 1 i n the i m a g e s a r e e q u a l to the c o r r e s p o n d i n g s e c t i o n s i n the b a s i n ) . T h e s e i m a g e s f a c i l i t a t e the a p p l i c a t i o n of the c l o s e d b a s i n b o u n d a r y c o n d i t i o n s p r e v i o u s l y m e n t i o n e d . B y m a k i n g the a m p l i t u d e at the c o r r e s p o n d i n g s e c t i o n s i n the b a s i n and the i m a g e s the s a m e , a m a x i m u m o r m i n i m u m a m p l i t u d e i s i m p o s e d at Sections I and N . U s i n g d i f f e r e n c e q u o t i e n t s the b o u n d a r y c o n d i t i o n s b e c o m e :

at s = 0 (n = 1) and a t s = i (n = N)

% - l = ^ n + l (1=^) The set of equations o b t a i n e d f r o m E q s . 14 and the a p p r o p r i a t e

b o u n d a r y c o n d i t i o n s i s : ^ 2 1 ^ 1 + a22Tl2 + a23Tl3 ^ 3 2 ^ = + ^ 3 3 ^ 3 + ^ 3 4 ' ^ 4 I I I I I I I I " ^ N N ^ N - 1 ^ N N %

The e i g e n v a l u e s , _/\_ , o f t h i s set of e q u a t i o n s can be e v a l u a t e d b y f i n d i n g the e i g e n v a l u e s of tEe m a t r i x w h i c h i s f o r m e d f r o m the c o e f f i c i e n t s of the t e r m s on the l e f t h a n d side (see H i l d e b r a n d , 1958). The m a t r i x of c o e f f i c i e n t s i s :

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r e s t r i c t i o n s i t can be a p p l i e d t o the case of a f u l l y open h a r b o r c o n n e c t e d to the o p e n - s e a . I p p e n and Goda (1963) have s h o w n t h a t f o r a f u l l y open r e c t a n g u l a r h a r b o r w i t h an a s p e c t r a t i o (the r a t i o of the h a r b o r w i d t h t o the h a r b o r l e n g t h ) l e s s t h a n 0. 1 the b a s i n a c t s e s s e n t i a l l y as a q u a r t e r w a v e l e n g t h r e s o n a t o r . T h i s m e a n s t h a t f o r these d i m e n s i o n s the f u l l y open h a r b o r w i l l have a node at the e n t r a n c e f o r a l l r e s o n a n t m o d e s of o s c i l l a t i o n . A s s u m i n g t h a t an i r r e g u l a r l y shaped h a r b o r of a n a s p e c t r a t i o l e s s t h a n 0. 1 a c t s i n a s i m i l a r w a y , t h i s t h e n i m p l i e s t h a t a s i m p l e m o d i f i c a t i o n can be m a d e to E q . 16 s t i l l u s i n g the s a m e c o n c e p t of m i r r o r i m a g e s . T o i m p o s e a c o n d i t i o n of z e r o a m p l i t u d e at S e c t i o n N , the b o u n d a r y c o n d i t i o n : ^ N - r - ^ N + 1 is c h o s e n ( w h e r e S e c t i o n N + 1 i s the f i r s t s e c t i o n i n the m i r r o r i m a g e a t t a c h e d to the e n t r a n c e ) . T h i s a p p r o a c h e s s e n t i a l l y f o r c e s t h e a m p l i -tude at S e c t i o n N to be ze r o . I n c o r p o r a t i n g t h i s c o n d i t i o n i n t h e s o l u t i o n the l a s t e q u a t i o n i n the set E q . 16 b e c o m e s :

- ^ N ' N % - 1 + ^ N N % = A ^ ^ N

w h e r e ^ ^ N - r ^ N

N N

T h e r e f o r e , the m a t r i x p r e s e n t e d as E q . 17 can be m o d i f i e d s i m p l y b y s u b s t i t u t i n g the c o e f f i c i e n t - a j | j . ^ f o r the c o e f f i c i e n t -aj.^|.g i n t h e N t h r o w (N - l ) t h c o l u m n k e e p i n g a l l o t h e r c o e f f i c i e n t s the s a m e a n d t h e n s o l v i n g t h i s f o r the e i g e n v a l u e s .

W i t h the o r i g i n o f s e c t i o n s at the c l o s e d end of the b a s i n t h e r e c u r -s i o n f o r m u l a -s E q . 18a and 18b u -s e d to d e t e r m i n e the a m p l i t u d e v a r i a t i o n r e m a i n the s a m e . E q . 18c b e c o m e s : 'IN 1 1 1

^IIN^'^NN

N N ' N - 1 ₍₂₁₎ The c o n f i r m a t i o n of t h i s a p p r o a c h l i e s i n the r e s u l t s o b t a i n e d - w h i c h w i l l be d i s c u s s e d l a t e r . A L T E R N A T E P R O C E D U R E FOR O B T A I N I N G E I G E N V A L U E S F o r an e q u a t i o n of the f o r m : 0 (22) | _ ( p g ) + (q + X r ) y

the R a y l e i g h - R i t z m e t h o d of d e t e r m i n i n g the e i g e n v a l u e s can b e u s e d (see K o p a l , 1961). T h i s m e t h o d i s s o m e w h a t s i m i l a r to the m e t h o d o f

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OSCILLATIONS

12

L e t : F = A (4^)= {26a) G = b'n2 (26b) T h e n by S i m p s o n ' s R u l e the n u m e r a t o r of E q . 24 can be e x p r e s s e d ; F ds = A£ { F , + 4 F , + 2F3 + + 2 F j ^ _ 2 "1 (27) X 4 F + F • ^ N - 1 ^ N J

A s i m i l a r e x p r e s s i o n can be w r i t t e n f o r the q u a n t i t y G. O n l y the s o l u t i o n f o r the c l o s e d b a s i n w i l l be c o n s i d e r e d , s i n c e the o n l y d i f f e r e n c e b e t w e e n the open and c l o s e d b a s i n i s the s u b s t i t u t i o n of E q . 25b i n s t e a d of E q . 25a i n E q . 24. S u b s t i t u t i n g E q . 25a i n t o E q s . 26a and 26b one o b t a i n s :

F . r ^ J ' A ^ s i n ^ ["(n-l) ^ A s I (28a) G^ = Tli=^ b ^ cos^ | - ( n - l ) T r A s J (28b)

T h e r e f o r e , f r o m E q s . 28a, 28b, 26, and 23, as a f i r s t a p p r o x i m a t i o n the s m a l l e s t e i g e n v a l u e f o r a c l o s e d b a s i n i s ; = 1^ { 4 Ag s i n (TT^ ) + 2A3 sin2 ( 2 T T ^ ) + . . . . + 4 A j ^ _ ^ sin= [ { N - 2 ) ^ ^ ] } - { b , + 4 b , cos== { ^ ) + . . . + b ^ } (29) T h i s e x p r e s s i o n can be set up q u i t e s i m p l y f o r r a p i d s o l u t i o n b y m e a n s of a desk c a l c u l a t o r . A b e t t e r a p p r o x i m a t i o n to as w e l l as a p p r o x i m a t i o n s t o t h e n e x t two e i g e n v a l u e s , c o u l d be o b t a i n e d by u s i n g the f o l l o w i n g e x p r e s s i o n f o r the a m p l i t u d e ; ^ n = ^o 9n + ^ ^ n + """^ ^^^n , „ ( n - 1 ) T T A s w h e r e 9 = ^ —-— (30)

£

H o w e v e r , i t can be seen f r o m t h i s e x p r e s s i o n a n d the p r e v i o u s d e v e l o p m e n t t h a t t h i s m e t h o d r a p i d l y l o s e s i t s s i m p l i c i t y and ease of c o m p u t a -t i o n as -the o r d e r of a p p r o x i m a -t i o n i n c r e a s e s .

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E a c h of the b a s i n s d e s c r i b e d i n T a b l e 1 was d i v i d e d i n t o 21 s e c t i o n s w i t h a s p a c i n g b e t w e e n each s e c t i o n ( e x c l u d i n g the f i r s t a n d l a s t s e c t i o n s ) of As = 10. 0 f t . Due to the t r u n c a t i o n s h o w n i n T a b l e 1 the d i s t a n c e b e -t w e e n -the f i r s -t a n d -the s e c o n d s e c -t i o n s and -the n e x -t -to l a s -t a n d -the l a s -t s e c t i o n s was s o m e w h a t l e s s t h a n 10 f t . I n the t h r e e cases w h e r e the s u r -f a c e w i d t h o-f the -f i r s t s e c t i o n was 1 -f t . t h i s d i s t a n c e was 9. 999 -f t . and -f o r the case w h e r e the s u r f a c e w i d t h of the f i r s t s e c t i o n was 5 f t . the f i r s t i n c r e m e n t a l d i s t a n c e A s , was 9. 969 f t .

The p e r i o d s of the l o w e s t mode of o s c i l l a t i o n , the " s l o s h i n g m o d e " , f o r t h e t h r e e d i f f e r e n t shapes d e s c r i b e d i n T a b l e 1 o b t a i n e d b y the m a t r i x m e t h o d ( E q . 17) a r e p r e s e n t e d i n T a b l e 2 c o l u m n one. The e x a c t s o l u -t i o n s , w h e r e a v a i l a b l e , a r e p r e s e n -t e d i n -the second c o l u m n , and -t h e p e r i o d s o b t a i n e d f r o m R a y l e i g h - R i t z a p p r o a c h ( E q . 29) a r e s h o w n i n c o l u m n t h r e e of t h i s t a b l e .

Table 2. Fundamental Period of Test Basins

B a s i n C o n f i g u r a t i o n T l (sec. ) M a t r i x T l (sec. ) E x a c t T l (sec. ) R a y l e i g h - R i t z C i r c u l a r C o n s t . D e p t h 18. 40 19. 00 18. 45 C i r c u l a r P a r a b o l i c B o t t o m 2 4 . 4 5 * 24. 50=:<* 24. 75 24. 0 E l l i p t i c C o n s t . D e p t h 18. 38 18. 58 R e c t a n g u l a r C o n s t . D e p t h 22. 30 22. 30 The f u n d a m e n t a l p e r i o d of a r e c t a n g u l a r b a s i n 200 f t . l o n g a n d 10 f t . deep i s i n c l u d e d i n T a b l e 2 o n l y f o r c o m p a r i s o n w i t h the p e r i o d s o f the o t h e r b a s i n s w h i c h have the s a m e l e n g t h .

I n the cases w h e r e e x a c t s o l u t i o n s a r e a v a i l a b l e the a g r e e m e n t b e t w e e n t h e s e f u n d a m e n t a l p e r i o d s and the s o l u t i o n s f r o m the m a t r i x o f c o e f f i c i e n t s i s w i t h i n 3%. The a g r e e m e n t b e t w e e n the p e r i o d s o b t a i n e d by t h e R a y l e i g h - R i t z m e t h o d i s b e t t e r , w h i c h w o u l d be e x p e c t e d s i n c e b o t h a r e a p p r o x i m a t e m e t h o d s w h i c h i n c o r p o r a t e the same a s s u m p t i o n s .

The case of the c i r c u l a r b a s i n w i t h a p a r a b o l i c b o t t o m i s of p a r t i c u -l a r i n t e r e s t b e c a u s e o f the shape of t h e b a s i n , the t w o - d i m e n s i o n a -l a p p r o a c h , and the b o u n d a r y c o n d i t i o n s e m p l o y e d . The c r o s s - s e c t i o n

* B a s i n t r u n c a t e d at 1 f t . s u r f a c e w i d t h . * * B a s i n t r u n c a t e d at 5 f t . s u r f a c e w i d t h .

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a r e a and the s u r f a c e w i d t h f a l l o f f q u i t e r a p i d l y w i t h d i s t a n c e f r o m the c e n t e r of the b a s i n . H o w e v e r , i t i s seen t h a t the f u n d a m e n t a l p e r i o d o b t a i n e d by t h i s n u m e r i c a l m e t h o d i s w i t h i n 2% of the e x a c t v a l u e . P r o b -a b l y even b e t t e r -a g r e e m e n t c o u l d be r e -a l i z e d b e t w e e n the e x -a c t -and the a p p r o x i m a t e a p p r o a c h b y t a k i n g a l a r g e r n u m b e r of s e c t i o n s . I t i s a l s o seen that the t r u n c a t i o n of the b a s i n does n o t a p p e a r to h a v e a s e r i o u s e f f e c t upon the n a t u r a l p e r i o d f o r t h i s case. T h i s i s seen by the a g r e e -m e n t b e t w e e n f u n d a -m e n t a l p e r i o d s of the t w o b a s i n s w h i c h a r e i d e n t i c a l e x c e p t f o r the d e g r e e of t r u n c a t i o n .

F o r the t h r e e cases f o r w h i c h the m a t r i x m e t h o d was e m p l o y e d 21 e i g e n v a l u e s w e r e o b t a i n e d . H o w e v e r , i t i s f e l t t h a t f o r the c i r c u l a r b a s i n s and f o r an e l l i p t i c a l b a s i n w i t h a r a t i o of the l e n g t h t o the w i d t h of t w o i t i s n o t r e a l i s t i c to e x p e c t a t w o - d i m e n s i o n a l a p p r o a c h to y i e l d the c o r r e c t e i g e n v a l u e s f o r a m p l i t u d e v a r i a t i o n s w h i c h a r e h i g h l y t h r e e -d i m e n s i o n a l , as a r e the h i g h e r m o -d e s of o s c i l l a t i o n f o r t h e s e b a s i n s . A m p l i t u d e d i s t r i b u t i o n - The v a r i a t i o n of the a m p l i t u d e i n the c i r c u l a r b a s i n s w i t h a h o r i z o n t a l b o t t o m and w i t h a p a r a b o l i c b o t t o m f o r the f u n -d a m e n t a l m o -d e of o s c i l l a t i o n was e v a l u a t e -d f r o m E q s . 18a, 18b, a n -d 18c. T h e s e a p p r o x i m a t e s o l u t i o n s a r e c o m p a r e d to the f o l l o w i n g e x a c t s o l u -t i o n s f o r -t h e s e cases (see L a m b , 1945): 1. C i r c u l a r b a s i n , c o n s t a n t depth Tl _ J j J k r ) , , w h e r e Jj^ i s the B e s s e l f u n c t i o n of the f i r s t o r d e r , k i s the w a v e n u m b e r , R i s the r a d i u s of the b a s i n , and kR = 1. 84 c o r r e s p o n d s to the p e r i o d of t h e " s l o s h i n g m o d e " .

2. C i r c u l a r b a s i n , p a r a b o l i c b o t t o m

The r e s u l t s of the a p p r o x i m a t e m e t h o d and the e x a c t s o l u t i o n a r e p r e s e n t e d i n F i g . 4 w h e r e the r e l a t i v e a m p l i t u d e tl/ripj i s p l o t t e d as a f u n c t i o n o f the r e l a t i v e r a d i u s r / R . The n u m e r i c a l c o m p u t a t i o n p r o -ceeds f r o m r / R = + 1 . 0 t o r / R = - I . O . The r e s u l t s of the a p p r o x i m a t e m e t h o d f o r the c i r c u l a r b a s i n w i t h c o n s t a n t d e p t h a g r e e s q u i t e w e l l w i t h the e x a c t s o l u t i o n t h r o u g h o u t the f u l l r a n g e of r e l a t i v e r a d i u s . H o w e v e r , f o r t h e c i r c u l a r b a s i n w i t h a p a r a b o l i c b o t t o m the a g r e e m e n t i s n o t n e a r l y as good i n the r e g i o n f r o m r / R = - 0. 6 t o r / R = - 1. 0 as i n o t h e r r e g i o n s . The m a j o r p r o b l e m i n a d i f f e r e n c e e q u a t i o n a p p r o a c h s u c h as t h i s i s t h a t e r r o r s a r e c u m u l a t i v e (see E q s . 18a, 18b, and 1 8 c ) , h e n c e any e r r o r i n the e i g e n v a l u e i s of i n c r e a s i n g i m p o r t a n c e as t h e d i s t a n c e f r o m the f i r s t s e c t i o n i n c r e a s e s . I n a d d i t i o n , the b o u n d a r y c o n -d i t i o n w h i c h i m p o s e s a m a x i m u m o r m i n i m u m a m p l i t u -d e at t h e en-ds o f the b a s i n i s a n a p p r o x i m a t i o n w h i c h does n o t a p p l y as w e l l to t h i s b a s i n as i t does to t h e one w i t h a v e r t i c a l b o u n d a r y .

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1.0 X I04 8.0X I03 . 6 . 0 x | 0 3 4.0X I03 2.0 X I03 0 1.0 X I06 8.0 X 10^ ^ 6.0x10^ "^"'U.0x,0S 2.0 X 105 0

Figure 6. Variation of Surface Width and Cross-Section Area, Apra Harbor, Guam, M.I.

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OSCILLATIONS 133 100 5 0 c ' E Q O cr a. 10. / / •. / / / / / / / / / / / / / / / / LEGEND Q MATRIX METHOD I MERIAN'S FORMULA 0 OBSERVED IN MODEL 7 / y / / / / / / / / / / / / / / / / / / / / MODE OF O S C I L L A T I O N

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1.0 X 10"* 8.0 X 103 ^ 6 . 0 X 1 0 3 4,0 X I03 10 15 20 25 30 D I S T A N C E IN l O O O f t I N C R E M E N T S 35 40 — SUPERIOR ENTRY

Figure 9. Variation of Surface Width and Cross-Section Area,

Duluth-Superior Harbor, Minnesota-Wisconsin CO

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I n F i g . 7 the p e r i o d s o b s e r v e d i n the m o d e l a r e i n d i c a t e d f o r the 2nd, 3 r d , and 4 t h m o d e . I n a d d i t i o n the p e r i o d s o b t a i n e d f r o m M e r i a n ' s f o r m u l a :

= 3, 5, . . . . ) (33) a r e s h o w n . T h i s e x p r e s s i o n a s s u m e s , as does the m a t r i x m e t h o d , that

a node e x i s t s at the h a r b o r e n t r a n c e . I t i s seen t h a t the o b s e r v e d v a l u e s a g r e e r e l a t i v e l y w e l l w i t h the c o m p u t e d p e r i o d s and t h a t the m a t r i x m e t h o d i s i n f a i r l y good a g r e e m e n t w i t h the p e r i o d s o b t a i n e d f r o m E q . 33.

The p e r i o d s shown i n F i g . 7 a r e p r o b a b l y too s m a l l f o r t h e case of a t r u e f o r c e d o s c i l l a t i o n because of the e f f e c t of the a s p e c t r a t i o of the h a r b o r ( r a t i o of w i d t h t o l e n g t h ) and the e f f e c t of the r e s t r i c t e d e n t r a n c e . B o t h of these e f f e c t s t e n d to cause the a m p l i t u d e at the e n t r a n c e to be d i f f e r e n t f r o m z e r o , and the h a r b o r w i l l r e s o n a t e at a l a r g e r p e r i o d t h a n t h a t i n d i c a t e d b y the a n a l y s i s . I n f a c t , u s i n g the r e s u l t s of I p p e n a n d Goda (1963) and c o n s i d e r i n g A p r a H a r b o r as a r e c t a n g u l a r b a s i n of an a s p e c t r a t i o of 0. 41 a n d a r a t i o of e n t r a n c e w i d t h to m a x i m u m w i d t h of 0. 19 the p e r i o d of the f i r s t h a r m o n i c s h o u l d be 74% g r e a t e r t h a n shown and t h a t of the second h a r m o n i c s h o u l d have a p e r i o d 32% g r e a t e r . A l t h o u g h t h i s a p p r o a c h does not a p p l y d i r e c t l y since the h a r b o r i s not r e c -t a n g u l a r and -the e n -t r a n c e c o n d i -t i o n s a r e no-t e x a c -t l y -the s a m e i n -the -t w o cases i t does i n d i c a t e s o m e of the p r o b l e m s i n h e r e n t i n the a n a l y s i s w h e r e the e n t r a n c e e f f e c t s and the a c c o m p a n y i n g r a d i a t i o n of e n e r g y to the o p e n - s e a a r e not c o n s i d e r e d .

The a m p l i t u d e v a r i a t i o n a l o n g the t a l w e g o b t a i n e d f o r the f u n d a m e n t a l and the second h a r m o n i c a r e shown i n F i g . 8. I t i s seen t h a t the i m p o s e d e n t r a n c e b o u n d a r y c o n d i t i o n i s m e t i n the t w o c a s e s .

D u l u t h - S u p e r i o r H a r b o r - The second case c h o s e n f o r a n a l y t i c a l s t u d y was D u l u t h - S u p e r i o r H a r b o r . The p r o b l e m of s e i c h e i n t h i s b a s i n w a s t r e a t e d p r e v i o u s l y b y H o u s l e y (1962) u s i n g the m e t h o d of D e f a n t to c o m -p a r e -p r e d i c t e d a n d o b s e r v e d -p e r i o d s of o s c i l l a t i o n . Since H o u s l e y t r e a t e d the p r o b l e m as the f r e e o s c i l l a t i o n s of a c l o s e d b a s i n , a s s u m i n g t h a t t h e D u l u t h and the S u p e r i o r e n t r i e s , and St. L o u i s B a y do not s i g -n i f i c a -n t l y a f f e c t the o s c i l l a t i o -n s of the h a r b o r , t h i s was a u s e f u l case to c h e c k . D u l u t h - S u p e r i o r H a r b o r is a p p r o x i m a t e l y 4 3 , 0 0 0 f t . l o n g and 4000 f t . w i d e w i t h an a v e r a g e depth of a l l s e c t i o n s of 1 2 . 4 f t .

F o r t h i s a n a l y s i s 35 s e c t i o n s w e r e t a k e n s p a c e d 1250 f t . a p a r t and the v a r i a t i o n of the s u r f a c e w i d t h and c r o s s - s e c t i o n a r e a w i t h d i s t a n c e o b t a i n e d at t h e s e s e c t i o n s a r e shown i n F i g . 9 w h e r e the o r i g i n i s a t the S u p e r i o r end of the H a r b o r . The p e r i o d s of f r e e o s c i l l a t i o n of the b a s i n o b t a i n e d by the m a t r i x m e t h o d a r e p r e s e n t e d i n F i g . 10 f o r the f i r s t t h r o u g h the e i g h t h m o d e s of o s c i l l a t i o n . I n a d d i t i o n , the p e r i o d s of the f i v e m o d e s o b t a i n e d by D e f a n t ' s m e t h o d and t h o s e o b t a i n e d f r o m M e r i a n ' s f o r m u l a , E q . 1 a r e a l s o shown i n F i g . 10. I t i s seen that the p e r i o d s o b t a i n e d f r o m the t h r e e d i f f e r e n t m e t h o d s a g r e e w i t h i n 10% e x c e p t f o r the 2nd m o d e w h e r e the d i f f e r e n c e i s s o m e w h a t g r e a t e r . Some of the d i f f e r e n c e s b e t w e e n the r e s u l t s of t h i s m e t h o d and D e f a n t ' s m e t h o d c a n

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OSCILLATIONS LEGEND Q MATRIX METHOD lOx-SECTIONS I MATRIX METHOD 20 x-SECTIONS 2 3 4 MODE OF O S C I L L A T I O N

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p e r h a p s be a t t r i b u t e d to the d i f f e r e n t n u m b e r of c r o s s- s e c t i o n s u s e d . I n the m a t r i x a p p r o a c h 3 5 s e c t i o n s w e r e used to d e f i n e the m a t r i x of c o e f f i c i e n t s c o m p a r e d to 154 s e c t i o n s u s e d by H o u s l e y .

M o n t e r e y B a y The t h i r d b a s i n i n v e s t i g a t e d was M o n t e r e y B a y , C a l i f o r -n i a . T h i s i s a l a r g e bay w i t h a l e -n g t h of a p p r o x i m a t e l y 56, 000 f t . f r o m the s h o r e to the e n t r a n c e , a m a x i m u m w i d t h of 138, 000 f t . and an

a v e r a g e depth w h i c h d e c r e a s e s f r o m 581 f t . at the e n t r a n c e to e s s e n t i a l l y z e r o at the s h o r e . The l o n g i t u d i n a l v a r i a t i o n of the s u r f a c e w i d t h s and the c r o s s - s e c t i o n a r e a s a r e p r e s e n t e d i n F i g . 1 1 .

T w e n t y s e c t i o n s w e r e used (2960 f t . a p a r t ) i n t h i s a n a l y s i s and the bay w a s t r e a t e d as a f u l l y open b a s i n w i t h a node l o c a t e d a t t h e e n t r a n c e to t h e bay. ( F o r c o m p a r i s o n , the b a s i n was a l s o d i v i d e d i n t o 10 s e c t i o n s . ) T h i s a s s u m p t i o n i s p r o b a b l y c o n s i d e r a b l y i n e r r o r due to the l a r g e a s p e c t r a t i o of the bay and the r e s u l t a n t e f f e c t on the w a t e r p a r t i c l e v e l o c i t y at the e n t r a n c e of the e n e r g y r a d i a t e d to the open sea. N e v e r t h e l e s s t h i s t r e a t m e n t does i n d i c a t e s o m e i n t e r e s t i n g t r e n d s .

T h e p e r i o d s of t h e f i r s t f i v e m o d e s of o s c i l l a t i o n o b t a i n e d f r o m the e i g e n v a l u e s of the m a t r i x of c o e f f i c i e n t s a r e p r e s e n t e d i n F i g . 12. I n a d d i t i o n the p e r i o d s of t h e s e m o d e s o b t a i n e d by u s i n g 10 c r o s s s e c t i o n s (a 10 x 10 m a t r i x as o p p o s e d t o a 20 x 20) a r e s h o w n . The a g r e e m e n t b e t w e e n the r e s u l t s o b t a i n e d f r o m the t w o cases i s f a i r l y g o o d i n d i c a t i n g t h a t as a f i r s t a p p r o x i m a t i o n a c o a r s e s p a c i n g of the c r o s s s e c t i o n s i s p r o b a b l y s u f f i c i e n t f o r d e t e r m i n i n g the e i g e n v a l u e s .

The a m p l i t u d e v a r i a t i o n o b t a i n e d f o r the f i r s t f o u r m o d e s u s i n g the f i n e r cr os s - se c t i o n s p a c i n g (20 s e c t i o n s ) i s p r e s e n t e d i n F i g . 13. The shape of the f u n d a m e n t a l l o o k s r e a s o n a b l e c o n s i d e r i n g the c o n f i g u r a t i o n of the b a s i n a n d the a m p l i t u d e v a r i a t i o n s shown i n F i g . 4 f o r t h e s l o s h i n g m o d e s of c i r c u l a r b a s i n s w i t h a h o r i z o n t a l b o t t o m and w i t h a p a r a b o l i c b o t t o m . H o w e v e r , due to the p l a n t e r m and the d e p t h v a r i a t i o n of M o n t e r e y B a y , i t is f e l t t h a t the o s c i l l a t i o n s of the h i g h e r m o d e s w o u l d be of a m o r e t h r e e d i m e n s i o n a l n a t u r e . I n a d d i t i o n , as m e n t i o n e d p r e -v i o u s l y , s i n c e the bay c o m m u n i c a t e s d i r e c t l y w i t h the o p e n - s e a i n an u n r e s t r i c t e d s e n s e , the e f f e c t of e n e r g y r a d i a t i o n f r o m the b a y o n the m o d e s of o s c i l l a t i o n w o u l d no doubt be m o r e i m p o r t a n t t h a n t h e case of A p r a H a r b o r d i s c u s s e d p r e v i o u s l y . T h e r e f o r e , the shape of t h e a m p l i -tude d i s t r i b u t i o n f o r the h i g h e r h a r m o n i c s s h o u l d be c o n s i d e r e d to be m o r e q u a l i t a t i v e t h a n q u a n t i t a t i v e . S O M E C O N S I D E R A T I O N S F O R T H R E E - D I M E N S I O N A L M O T I O N T h e p r o b l e m w h i c h i s of p r i m a r y i n t e r e s t i n the study of f o r c e d o s c i l l a t i o n s of h a r b o r s i s t h a t of t h r e e - d i m e n s i o n a l m o t i o n s . I n f a c t the m o t i v a t i o n f o r t h i s t w o - d i m e n s i o n a l study was to d e v e l o p a m e t h o d w h i c h c o u l d be a p p l i e d to the m o r e r e a l i s t i c p r o b l e m of d e t e r m i n i n g the r e s p o n s e c h a r a c t e r i s t i c s of an a r b i t r a r y shaped h a r b o r c o n n e c t e d to the o p e n - s e a . A n a p p r o a c h to t h i s p r o b l e m i s p r e s e n t e d i n t h i s s e c t i o n ; h o w e v e r , no s o l u t i o n s to s p e c i f i c cases have b e e n a t t e m p t e d as y e t .

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W i t h r e f e r e n c e to F i g . 14 f o r t h r e e - d i m e n s i o n a l l o n g p e r i o d f l u i d m o t i o n s , the c o n t i n u i t y e q u a t i o n i s :

| - ( u h 6 y ) 6 x + | - ( v h 6 x ) 6 y = . A - [ (TI + h) öx 6 y l (34)

ox öy Ö T; L J

w h e r e u i s the w a t e r p a r t i c l e v e l o c i t y i n the x - d i r e c t i o n , v i s the w a t e r p a r t i c l e v e l o c i t y i n the y d i r e c t i o n , h i s the d e p t h w h i c h i s a f u n c t i o n of X a n d y , and 5 x and 5y a r e the d i m e n s i o n s of an e l e m e n t i n the h o r i z o n t a l p l a n e . A p p l y i n g the s m a l l a m p l i t u d e , s h a l l o w w a t e r w a v e a s s u m p t i o n s , the e q u a t i o n s of m o t i o n i n the x - d i r e c t i o n and the y - d i r e c t i o n a r e :

1? =

-^!^

1? =

-^!^

A s s u m i n g that Tl i s p e r i o d i c w i t h t i m e and of the f o r m :

Tl = Tl (x, y) cos a t (36)

T h e n E q s . 34, 35, and 36 can be c o m b i n e d to g i v e the f o l l o w i n g e x p r e s -s i o n f o r the v a r i a t i o n of the w a t e r -s u r f a c e a m p l i t u d e r] i n the x a n d y d i r e c t i o n s w h i c h i s a m a x i m u m w i t h r e s p e c t t o t i m e : h r | ! ^ + | ! ^ l + + | h S l ] + x ^ = 0 (37) LSx2 By2 J gx Bx 3y S y ' w h e r e X i s a p a r t i c u l a r e i g e n v a l u e and is e q u a l to . U s i n g d i f f e r e n c e q u o t i e n t s s i m i l a r to E q s . 12 and the n o t a t i o n of F i g . 15 w h e r e A = A = A_{x y " ' ^} , E q . 37 r e d u c e s t o : - 2 h + h , + h , Tl + h Tl , L m , n m - l , n m , n - 1 J ' m , n m , n ' m + l , n (38) + h , Tl , + h ri , + h , T i , + X A^ T i =0 m - l , n ' m - 1 , n m , n ' m , n+1 m , n - 1 ' m , n - 1 ' m , n w h e r e i n h i s the d e p t h at the i n d i c a t e d m e s h p o i n t .

I n s t e a d of the s q u a r e m e s h a r e c t a n g u l a r m e s h can be used(A i A )

so t h a t a b e t t e r d e f i n i t i o n of the a m p l i t u d e can be r e a l i z e d i n e i t h e r ^ the X o r y d i r e c t i o n . I f t h i s i s d e s i r e d the x c o o r d i n a t e or the y c o o r d i -nate c a n be t r a n s f o r m e d i n such a w a y t h a t the m e s h is s q u a r e once a g a i n and the f i n a l s o l u t i o n can t h e n be t r a n s f o r m e d b a c k to the a c t u a l c o -o r d i n a t e s y s t e m . H -o w e v e r , f -o r the sake -o f t h i s d i s c u s s i -o n E q . 38 i s s u f f i c i e n t .

T h e s a m e b o u n d a r y c o n d i t i o n s can be e m p l o y e d h e r e f o r t h e c l o s e d b a s i n as w e r e u s e d f o r the t w o - d i m e n s i o n a l o s c i l l a t i o n s . E q . 38 t h e n b e c o m e s an m t i m e s n set of e q u a t i o n s , w h e r e the p r o d u c t of m a n d n

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OSCILLATIONS 145 R E F E R E N C E S C h r y s t a l , G. , "Some R e s u l t s i n the M a t h e m a t i c a l T h e o r y of S e i c h e s " , P r o c . of R o y . Soc. of E d i n b . , V o l . 4 1 , 1904. D e f a n t , A . , " P h y s i c a l O c e a n o g r a p h y " , P e r m a g o n P r es s, V o l . I I , p p . 1 6 7 - 1 6 7 , 1961. H i d a k a , K . , " A p p l i c a t i o n of R i t z V a r i a t i o n M e t h o d to the D e t e r m i n a t i o n of Seiches i n a L a k e " , M e m . I m p . M a r . O b s . , K o b e , J a p a n , V o l . 6, N o . 2, 1936. H i l d e b r a n d , F . B . , " A d v a n c e d C a l c u l u s f o r E n g i n e e r s " , P r e n t i c e H a l l , I n c . p . 74, 1958. H o u s l e y , J . , "Seiches and C u r r e n t s i n D u l u t h S u p e r i o r H a r b o r J u n e -N o v e m b e r 1958", U n i t e d States A r m y C o r p s of E n g i n e e r s WES M i s c . P a p e r N o . 2 - 5 0 2 , June 1962.

I p p e n , A . T . and Goda, Y . , "Wave I n d u c e d O s c i l l a t i o n s i n H a r b o r s : The S o l u t i o n f o r a R e c t a n g u l a r H a r b o r C o n n e c t e d to the O p e n - S e a " , M . I . T . H y d r o d y n a m i c s L a b o r a t o r y R e p o r t N o . 59, J u l y 1963.

K n a p p , R. T . and V a n o n i , V . A . , " M o d e l s t u d i e s of A p r a H a r b o r , G u a m , M . I . " , C a l i f o r n i a I n s t i t u t e of T e c h n o l o g y , H y d r o d y n a m i c s L a b o r a t o r y , R e p o r t N o . N - 6 3 , June 1949.

K o p a l , Z . , " N u m e r i c a l A n a l y s i s " , John W i l e y and Sons, 1 9 6 1 .

L a m b , H . , " H y d r o d y n a m i c s " , S i x t h E d i t i o n , D o v e r P u b l i c a t i o n s , 1945, p p . 2 8 4 - 2 9 3 . P r o u d m a n , J. , " F r e e and F o r c e d L o n g i t u d i n a l T i d a l M o t i o n i n a L a k e " , P r o c . L o n d . M a t h . Soc. (2), 14, 1914. R a i c h l e n , F . , and I p p e n , A . T . , "Wave I n d u c e d O s c i l l a t i o n s i n H a r b o r s " , J o u r n a l of the H y d r a u l i c s D i v i s i o n , A S C E , M a r c h , 1965.