A WAVE REFRACTION 'ANALYSIS FOR AN

AXf ALLY; SYMMETRICAL ISLAND

RONALD J. FORSI

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U. S. Naval Postgraduate »

Monterey, California

DUDLEY KNOX LIBRARY

NAVAL POSTGRADUATE SCHOOL

MONTFREY CA 93943-5101

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A WAVE REFRACTION ANALYSIS FOR AN AXIALLY SYMMETRICAL ISLAND

A Thesis By

LIEUTENANT RONALD J„ FORST

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UNITED STATES NAVY

Submitted to the Graduate College of the Texas A&M University in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

MAY 1966

Major Subject Oceanography

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A WAVE REFRACTION ANALYSIS FOR AN AX I ALLY SYMMETRICAL ISLAND

A Thesis

By

LIEUTENANT RONALD J0 FORST

//

UNITED STATES NAVY

Approved as to style and content by:.

ACKNOWLEDGMENTS

The author of this paper wishes to express very special thanks to Mr. R.0o Reid for his generous dir- ection of this work. This thesis was completed under sponsorship by the United States Navy Postgraduate School

_!._

Library

U. S. Naval Postgraduate 8e&f

Monterey, California

TABLE OF CONTENTS

Page

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CHAPTER

I. INTRODUCTION oo

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1 o Statement of the Problem oooooooooooo*. 1

C- o i~l iS LOrJ.Cd.1 i\ G v -L 6 W ooooooooooooooooooooo <--

3. General Description of the Island 6

-" o ASSUuip UXOnS ooooooooooooooooooooooooooe /

II. THEORETICAL ASPECTS OF THE PROBLEM ......... 11

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Oo Jr 11 Cl S G Xj a. CI oooooooooooooooooooooooooooo £L.\J

6. Relative Power Distribution o»«ooo<,o.o 22

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APPENDIX

I . CALCULATION SUMMARY .,<,., <,.><.o o «.« .0<, 000 ,<,„„„ „ „ . 38

II. GRAPHICAL REPRESENTAION OF RESULTS „«,„.,„.„ 0 . 41

IV

LIST OF ILLUSTRATIONS

Figure Page

lo Vertical Cross Section of Island .......... 10

3. Polar Coordinate Representation of

cLll \Jl. L*l _ O Cj OI- Q. J_ _>o_>*ooooooot?oooo0ooooooooo L_ -J

4. Schematic Diagram of the Island ........ . .„ 26

5. Angle Between Radius Line and a Ray ....... 2 7

o o ir na s e rcexerence ijiriG oooo..oo.ooo....ooo.»« <l i

I o WaVe ShadOW ZOne .000.. .000. ....... .ee.o.«o 2o

8. Schematic Diagram of Reflected Rays ....... 29

9. Initial Ray Angle vs. Azimuth ............. 42

10. Refraction Factor vs. Azimuth ............. 43

12. Relative Amplitude vs. Azimuth ............ 45

13. Refracted Power Intensity vs.

LlUCi. C«._ii L Ali^ 16 oooooooooooooooo'oooaeoooe

14. Total and Reflected Power Intensity

V S • I_».tL<2_LC[c__"C J\ Uvj ic ooooooooooooooooeoooo

V

LIST OF SYMBOLS

A Wave amplitude.,

b Distance between adjacent rays.

C ve phase speed.

E Mean wave energy per unit surface area.

g Acceleration due to gravity.

H tfave height.

h Mean depth of water.

I Travel time.

k ....ve number ( 2?T/L)„

L ...ve length.

m Slope of the ocean bottom.

n Ratio of group speed to phase speed.

o A subscript denoting values at the island shore

r Radial coordinate.

c A subscript denoting a critical value.

T The wave period.

t Time.

V Wave group, speed.

Tf 3.1415.

y Phase difference.

K Refraction factor r

K Shoaling factor.

s ■*

Vertical component of the particle displacement,

f\ Increment operator.

vi

1 A subscript denoting the outer depth contour or a deep water condition„

P Density of sea water (assumed constant) «

ds Arc length o

-9- Angular coordinate measured counterclockwise from a reference which is parallel to the deep water wave rays.

VII

ABSTRACT

This paper deals with the refraction of long gravity waves about a circular island with uniform slope below mean water level. The Fermat principle is used as a basis to derive an analytical solution for the ray paths. A refraction analysis is performed to determine the refraction factor, phase lag, and wave heig] at the island shore. The relative power distri- bution of rays in the far field is determined by con- sidering emergent reflected and refracted waves from the island.

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CHAPTER I INTRODUCTION 1. Statement of the Problem

The purpose of this paper is to study the influence on long gravity waves of an island with circular bottom contours and radially constant slope. Using the prin- ciples of wave ray theory, a refraction analysis is carried out to determine the modification of initially plane waves as they move into shallow water about the island. The refraction and direct shoaling effects upon incoming long water waves are considered. The re- flected and refracted rays emerging from the island are considered in order to study the divergent wave energy In the far field of the island. This refraction study is a contribution in part to a more exacting treat- ment of the subject by Vastano (1966), involving a numerical diffraction technique.

2. Historical Review

For centuries man has concerned himself with the effects of islands on surface waves. The early civiliza- tions of Micronesians and Polynesians utilized wave pat- terns to mark fishing grounds and aid in navigation among the islands. Winkler (1901) recorded some of their techniques through interviews with native chiefs. Krieger (1943) further illustrates this fact and states:

Every chief and sea pilot possessed elaborate charts based upon his own exper- ience and on knowledge handed down or gain- ed from others.. These sea charts are made with thin strips of the midrib of the leaf- lets of the coconut arranged on a frame usually rectangular in shape. The knowledge they record is indicated by the arrange- ment of the leaf strips relative to one another and by the forms given to them by bending and crossing. Curved strips in- dicate the altered direction taken by ocean swells when deflected by the presence of an island; their interactions are nodes where these meet and tend to produce a confused sea.

More recent interest on the specific subject of wave refraction was stimulated by the needs brought about by World War II amphibious operations. As a wave approaches shallow water, its velocity begins to decrease appre- ciably at a depth of about one-half the wave length and the wave height is influenced by the convergence and div- ergence of energy along the shore. As the wave bends around the island, "wave shadow" areas may be formed in

the island "wave lee," but often times not to the extent, that might be expected. This fact was painfully realized during the invasion of Sicily when the methods of Sverdrup and Munk (1947) were used to forecast wave conditions prior to the landing,, Despite the use of these acceptable methods, the wave heights in the lee of the island were greatly underestimated thus severely hampering the landing operation.

The graphical means of constructing wave refraccior. diagrams was first used by O'Brien (1942)« Application to the complex ocean bottoms of nature was made by Johnson, O'Brien, and Isaacs (1948) and Arthur, Munk, and Isaacs (1952) „ The wave refraction pattern for any bottom ropography assumes such that the waves change direction and are bent in such a manner that they tend to assume the slope of the depth contours. Aerial photographs (Munk and Arthur, 1952) qualitatively bear this out.. It was learned that the effect of bottom features upon waves depends not upon the absolute depth of water, but upon the depth relative to the wave length.. Graphical construction methods by hand are both time consuming and tedious*. Griswold (1963) has computer programmed the graphical construction of wave rays and Momoi (1964) has expanded on Griswold' s program to include the effects of wave refraction.

In some special cases, analytical solutions to the wave refraction problem have been developed to improve accuracy which can be critical in the investigation of the energy distribution especially on the wave lee side of is- lands o Analytical and experimental studies of refraction in the region of a caustic have been made by Pierson (1951),

method of conformal transformation was applied by Focinski (1950) » The actual cases treated are restricted by the mathematical complications that are introduced by a particular conformal transformation and by how well the transform simulates a beach contour. Arthur (1950) used Fermat's principle as a basis for an analogy between the refraction and the minimal flight path problem. The effects of current and depth distribution are considered,, Johnson (1947) also discusses the refraction of ocean surface waves by current for deep water waves.

Munk and Arthur (1952) applied ray theory to calcu- late the wave intensity along a refracted ray. This pro- cedure has certain advantages over the usual technique of computing intensity from measurements of the distance between rays ; namely in the areas of extreme convergence or divergence.

Analytical solutions for the refraction of waves around an island with concentric ci. cular bottom contours have been obtained by Arthur (1946) through the application

of Fermat's principle. Arthur (1951) determined that the important effects in the penetration of wave energy into the "wave lee" of an island are generally the result of refraction by underwater topography and variability in direction,,,

3. General Description of the Island

The island bottom configuration selected for t. present study is axially symmetrical (i.e., the depth contours are concentric circles.) At 5 km. radius from the island axis the topography drops vertically from mean sea level to a depth of 0.5 km,, then assumes a slope of 1:10 out to 41 km. radius, beyond which the depth remains 4.1 km. (see Figure Do The periods of the waves selected are 2,4,8, and 12 minutes to provide a variation in wave lengths within the tsunami class.

There was little attempt here to simulate the detailed underwater topographical features of any part- icular island. However, a computer program for the study of the interaction of monochromatic waves with an island of irregular shape is being developed by Reid and Vastano (1966). This program permits depth variation and the use of a special orthogonal system.

It is interesting to note that the selected island configuration lends itself to a fairly simple mathemati- cal treatment. However, if a case were chosen where the extended slope line intersects the mean sea level at other than the island vertex, then a somewhat more for- midable problem presents itself with regard to wave amplitude determination and is worthy of further in- vestigation.

4. Assumptions

In water of constant depth the wave celerity, C, for sinusoidal gravity waves of small amplitude, neglect- ing surface tension, is given by the classical relation (Lamb, 19 32)

(1) C2 = (g/k) tanh kh ,

where h is the mean depth of water, k = 2 7( /L the wave iber, and g the acceleration due to gravity,, A wave schematic is shown in Figure 2, where the wave height H equal to twice the amplitude, 1.e. the vertical distance between two successive crests, and the period T is the rime interval between the passage of two consecutive crests. For any wave greater than 0.1 feet in length, the effect of surface tension on wave speed can be ignored (Wiegel, 1964). The wave speed is related to wave length and period by

(2) L = CT.

There are two limiting cases of (1) which are appropriate to the so called "deep" and "shallow" water waves respectively. For deep water, h/L is very large and tanh kh tends toward unity. Under this condition (1) and (2) yield

On the other hand, under the condition h/L — 1/20, (1) becomes to a good approximation

C2 = gh „

There is, of course, no sharp distinction between "deep" and "shallow" water. The effect of depth of water on wave characteristics is gradual and waves in any finite depth of water are affected by that depth,, It is interesting to note that waves classified in the shorter period category are not refracted until near shore and shallow water, while long period waves are refracted further offshore in deeper water „ Tsunamis, for example, of periods 2 minutes to 1 hour would constantly undergo re- fraction effects in the open ocean. In a tsunami study, only waves that travel at a speed of (gh) 2 are considered*

As waves propagate into regions of changing depth, we consider that T remains unchanged while H and L vary.

Given an initially plane wave of a given period and wave height moving towards shallow water, the wave height in general can vary according to the following factors:

(a) Direct shoaling effects

(b) Wave refraction effects

(c) Reflection

(d) Diffraction

(e) Energy lost by friction and percolation

In this study, only the first two factors are considered over the sloping bathymetry, however, reflection is taken into account at the shore. The energy dispersion due to diffraction occurs in areas of appreciable convergence or divergence of energy and in shallow water zones such as behind breakwaters . The diffraction tends to spread the wave energy into potential shadow zones. Friction losses on slopes greater than 0o01 are considered neg- ligible (Walsh, Reid, and Bader, 1962) » Percolation is related to the bottom characteristics and is considered negligible with respect to friction (Reid and Bretschneider , 1953) . In addition, external wind influences are neglected, Kajiura (1964) discusses the partial reflection of water waves passing over a bottom of variable depth. Green's formula provides a very good estimate of the transmitted wave amplitude even if partial reflection exists on a sloping bottom (Reid, 1957)0

In the absence of energy losses by friction and percolation or dispersion by diffraction and reflection, the wave energy transmitted between orthogonal s (wave rays) remains constant.

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11

CHAPTER II THEORETICAL ASPECTS OF THE PROBL 1. Wave Refraction

Gravity wave refraction is analagous to the bending of light rays in geometrical optics. Since the wave speed decreases with decreasing depth (C = ( gh ) 2) , the wave rays are refracted towards regions of lesser water depth.

This can be put in quantitative terms by the relation

,£ ,

d s C D n

v/here /J is the angle of the wave ray measured counter- clockwise from a fixed reference (the x-axis), s is arc distance on the wave ray and n is distance measured parallel to the wave crest and to the left of the ray«> The above relation can be shown to be consistent with Fermat's principle (Munk and Arthur, 19 52). In the spec- ial case of straight and parallel depth contours, one can take the x-axis normal to the contours such that C = C(x) alone. In this case (1.1) reduces to

cos O dp 1 . n dC P -r~ = ~ sin P -r- J dx C r* ax

which is readily integrated and yields Snell's law-

12

(1.2 ) sin ,

— = — = constant

along the wave ray. Note however , that this is a special case of the more general relation (1.1).

In the general case, graphical methods are often referred to in describing the refraction pattern. The two methods most frequently employed are: first, the crest method, where successive positions of the wave crest are drawn by plotting the wave advance from point to point along the crest; second, the ray method, where each orthogonal is plotted directly by determining its shoreward deflection as it crosses successive bottom contours. Johnson, O'Brien, and Isaacs (1948) provide explicit instructions for this approach. In both cases a detailed large scale chart of the bottom topography is essential. For the advantages and disadvantages of each method the reader is referred to an excellent dis- cussion by Dunham (1950).

The refraction of energy in the lee of an island is difficult to determine by using a graphical approach.. For this reason analytical solutions for the rays have been developed, where the depth variation is expressed as an analytic function (Arthur, 19 51). Before proceed- ing with a discussion of this particular island case the relationship between wave height and energy will be

13

reviewed.

If it is assumed that the total energy transmitted between two orthogonals is constant, convergence of the rays denotes a concentration of energy and a correspond- ing increase in wave heights. To a good approximation, wave height is proportional to the square root of the energy provided that the waves are not near the break- ing point (Munk and Traylor, 1947). The wave height, however, will not usually be the same along a particular wave crest due to refraction effects over an irregular bottom.

The mean wave energy per unit surface area for progressive waves is given by

E = 1/8 PgH2

where 0 is the density of sea water, g the acceleration due ro gravity and H the wave height. The mean wave power, EVb is constant between two wave rays and it follows that the relationship for the wave height is given by

(1.3) H = K K K-

r s 1

where H is the deep water wave height, K is the

refraction factor and K is the shoaling factor.

s 3

The refraction factor is defined by

(1.4)

K = (b,/b)2

r 1

where b is the distance between adjacent rays measured normal to the rays and the subscript 1 refers to deep water.

The shoaling factor can be expressed by

(1.5)

Ks = (V;|/V>

where VF is the deep water group speed. In general

V = nC

14

where n is given by

n = h

4Tfh/L 1 +

sinh 47Th

L

For the wave periods considered in this study n is essentially unity for all depths and hence V = C, Thus

(1.6)

Ks = (hi/h) 4

For an incident wave of unit amplitude it follows from (1.3) and (1.6) that the relationship for relative amplitude is given by

(1.7)

A = K (hn/h)4

r 1

15

2. Fermat's Principle

Fermat's principle of least time is discussed in Joos (1934). It can be stated thus: the propagation of light always takes place in such a way that the actual optical path (e.g. length of geometric path multiplied by index of refraction of the medium) is an extreme value compared with all other paths which do not follow the law of optics. In a wave refraction analogy, we allow the light rays to correspond to orthogonals in a gravity wave refraction diagram.

The application of Fermat's least time concept to refraction by concentric, circular, bottom contours is discussed by Arthur (1946). The travel time along a pc from a point A to a point B of a refracted ray is given by

(2.1)

where C = (gh) 2 along the path and ds is the arc length along the path. Consider depth h = h(r) only; then for I to be a minimum requires an r(-G-) such that ol = 0. For r>r, , the depth is assumed constant and the or- thogonals are straiqht lines. For r <- r<r, the ravs

J o i

are curved. Using the coordinates as shown in Figure 3, (2.1) takes the form

16

(2.2)

I =

e-,

B

Je

V(dr/dG)2 -r r^ de C(r)

A

A path which makes the integral of (2„2) a min- imum is found by the methods of the calculus of varia- tions (Joos, 1934), which in this case must satisfy the Euier-Lagrange condition

/ 1 dr/de-i - -r + G/C2 dC/dr = 0 ,

de-

GC

2 ' 2 I ^

where G ^ j r + (r ) i 2 and r' = dr/dG- »

Let F = G/C(r)« For C independent of -©- , F depends only on r and r' then

(2.3)

d dG-

F - r

g>F 9r

= 0

r« ^ 0 ,

which implies that

F -r

, 2z

5>r«

= K, a constant

if the constant of integration is evaluated at r = r, the equation for the ray path takes the fc

(2.4) + dG = dr/ i rl/(r/rJ2(l/sin2 Gh. ) (h, /h(r ) ) -1

-

the sign being chosen according to the direction of the wave ray.

3. Ray Path

Consider the ray equation

17

(3.1)

de-

dr

v h

r2 -1

where

h

2 . 2 _ r, sin -6-,

Now h = mr for r <rr <:*?-, , where m is the island

o 1 '

slope. Thus

(3.2)

h

r, sin O

Substituting (3.2) into (3.1) and carrying out the integration, with the boundary condition -G- = -G-, at r = r, , gives

(3.3)

2 / Gn + e-\

CSC 1

2-

r

r, sin -©■..

The wave ray which enters at r, , -8-, will reach the

island boundary, r , at -9- = ■©■ where 2 ' o' o

(3.4)

-1

o . 2

— - sin

■e- = - -e-, + 2ctn ; i

o 1 \ —

sin ■©•-,

18

2

provided that sin -G-. 2 r /rn .

1 o 1

2

Consider sin -G-, = r /r., , then

o 1 7

_ft_ _ -TT" ft.

O /( 1

2 For sin ■©•_ > r /rn , the ray does not reach the island 1 o 1 '

perimeter. In this case, the ray becomes tangent to a circle of radius r given by

2 (.3.5) r = r. sin -9-n ,

C" 1 1

i.e. r > r at •©- = - ■&-, + 7T » A_ limiting ray exists at co 1 ' '

r maximum or when -©- = f( //2 (see Figure 4 ). Beyond this

maximum critical radius, wave rays are unaffected by the

island.

For r = r , the initial critical ancle can be deter- c o '

mined by

(3..6) sin -©•_

1c

For the particular island configuration considered he]

-©-., = 20.44 „ Hence rays will converge on the island

perimeter for -159.56 — -6- ^ 159.56 „

o

19

(4.1)

4. Refraction Factor

The refraction factor is given by the equation

0 r cO- cosfY -2 o o ^o

K

r, dG, cos -&%

wn

here r, cos -9-, dO, is the mathematical expression for the

ray separation at r, and r d-0- cosfY represents the wave

^ loo ^o

seoaration at r . From (3.4)

o

(4.2)

dO-

2 cos -G-.

dG.

- 1

1 r

sin -©-,

From Figure 5

tanCX =

rd^

dr

Substituting for dG/dr from (3.1),

tan CX

fr2 I o

- 1

3y further trigonometric substitution for ] (4.3) cosCX

"O

I r sin2 -e- \ 1- — -

o

Substituting (4.2) and (4.3) into (4.1) gives

K

-2

o

2 - sec -0-n

1

r o o . 2 „

— - sin -G-,

This is valid for

■©■-.

S -©■■

lc

20

5. Phase Lag

The water level displacement of a sinusoidal wave is given by

(5.1)

/ = A cos (U7t - (J) )

where (j) is the phase lag at any point on the ray rela- tive to some fixed reference. The change in (j) along a ray can be expressed mathematically by

(5.2)

d(j)

ds

U7 C(r)

where ds = (dr) + (rd©-)

The phase change along the path from r

r = r using (3.1) and (3.3) is given by

0 = => j.

(5.3)

4>

2 ^7

. 2

'gn.

Vri-risin ^i " Vro"ris

A correction ^ <$> must be added for the change be- tween the reference line (j) = 0 and r = r, (see Figure 6 )

where

(5.4)

A* =

uyr1 jl - cos ©^

VihY"

Thus for the total phase lag as a function of -G-.

If

21

2u/r.

(5.5)

A = 1 — 11 ^(i + cos ^ ) -"\/~ - sin2 •©-

22

6. Relative Power Distribution

The general relationship for the refraction factor

for r <: r<.r, is given bv o 1 3

K

-2

2 - sec -6-.

"A/— - sin2 -G-. - Vr1 1 ^

Substituting for r from (3o2) gives

■e- 4- -e- ;

(6.1) K = sin G-. esc = |;2-tan -G-n ctn ^

1 2 7

valid for -e-. ^ G- < ( 2 Jf - 3 "9-, ) • For -0- = -0-_ , K = 1. For

1 ; ' 1 1 ' r

■0- = 77* - -Gv , e.i, the critical angle for a minimum r of a ray, K~2 = 2 sin -G . For G- = 2 77' - 3 G- , K~2 = 3.

For direct rays characterized by a particular value

_2

of G- , K is given by (6.1) for any -6- along the ray

path. For those rays with -G-, \ <=- G-n , reflection occurs

at r = r , G- = G- • The reflected ray passes through r =

r. at G- = 2 G- - G\. (see Figure 8). It emerges straight

beyond r, and makes an angle 2 G- from the x-axis. Thus 1 J o

two rays G\. and -G' diverge at angular spread 2(G' - -G- ) and their intersection always lies within a circle of

radius r.. . Thus for r

'. at l-G-l = 2 ■©- and |e-n ; 1 ' o 1

'lc

23

r^cos ^d^

K. = ——————

r

2rd^- o

Substituting from (4.1)

~ r, cos -9-,

(6.2) Kd = i i

r

2 cos -6-, 2r| ± 1

H

r2. _ sin2 e. rl X

For those rays with 7T/2=>|-©-J > ■©-, no reflection occurs. The rays are refracted and emerge at r = r, , ■©■ = 2 Tf - 3 -0-, as a straight line and make an angle of 2 Tf - 2 O-. , from the x-axis. Thus two neighboring rays ■0\. and -B-' have a divergent angle 2(-©\j - -6-, ) and intersect within a circle of radius r. . For r=»=>r.. at |-G-| = 2 7/

" 2 ^i» "^ic*"!"0"^ ^/2# Then

0 r.. cos -&,

(6.3) K^ = -i^ i .

For any given radial distance, r, from the center of the island, the relative power intensity can be expressed in the form

(6.4) F(-e-) = §- „2

r1 Kr .

The form of this "beam pattern" for reflected and refrac- ted waves in the far field is determined by (6.2) and (6.3), respectively. The total, relative power intensity in the far field is the sum of that contributed by the

24

reflected and refracted waves along radial lines of -6- values (see Figures 9 and 10 in Appendix II).

CREST

T

FIGURE 8— WAVE SCHI

25

ORTHOGONAL

^- POLAR

FIGURE 5—

£6

t/v

CO H

i—'

o

O

H

O H

@

o

Ll.

27

dS

Fir-

*j>

a l-E

r=r,

REFERENCE LI I

FIGURE 6.

PHASE PEFE!

28

o

CO

1 J

c5

c

/

« Q

O

« O

H

O

CO

30

CHAPTER III DISCUSSION OF RESULTS

A calculation summary for the circular island appears in Appendix Io This is presented in two parts: (A) information pertinent to the rays and refraction on the island lee shore; (3) information pertinent to the far field power distribution,, The graphical repre- sentation of phase lag, refraction factor, relative amplitude, and relative power intensity for the island shore are given in Figures 9 to 12 (Appendix II), A ■-pie wave refraction pattern consisting of direct rays into the island shore is produced by the constant slope case. For those incident waves entering with azimuths within the range from -20 « 4 to 20 « 4 degrees, the waves converge on the island covering an angular range from 0 to - 159„56 degrees. A wave shadow zone of about 40 degrees range is formed in the lee of the island and is an area unaffected by refraction effects. The shadow zone is developed between the shore and the outgoing critical ray*

r those rays with j-0-..j<-0- , reflection occurs

at r = r . -0- - -9- and the ray emerges from r, at -0- = o ' o J 1

For those rays with 7f/2 ^j^-J > ^ , no reflection occurs, but the rays are refracted and finally emerge

31

- 2 -&-, from the x-axis. At bhe point of emergence K 1 r

= 1/ y 3, however, at ■©•_ = - ff/2, a discontinuity exi: Incident rays that are just outside r, at - Jf/2 are unaffected by the island, while those rays just inside r, undergo refraction„

For those emergent rays at large distance from the island (r=» ^r. ), the center of the island can be con- sidered as the origin of these rays and the relative

2

power intensity in the far field ( rK /r, ) is simply a

function of -G-.

The relative power intensity is found to increase gradually from J -©-J = 180 to an extreme at 2 -G-, (40.8 degrees) at which there is an abrupt decrease.. This discontinuity in the far field beam pattern of the power is caused by the shadow phenomenon inherent in this refraction analysis «, The contribution of the re- fracted waves in the far field exceeds that of the re- flected rays by more than a factor of 4 (see Figures 13 and 14).

On the other hand, when considering the wave

amplitude at the island shore, derived by use of Green's

Law, the shoaling factor is significant while refraction

effects are limited by the small -G-, value. Furthermore,

■* 1c ;

for a constant slope of 0.1, any variation in the

island contour parameters will not appreciably alter the critical angle., The refraction factor varied a small amount due to the slope and parameters of the island selected (see Figure 10 ).

The variation in relative amplitude, in this case, is governed by the refraction factor and drops off abruptly to zero at the point of ray tangency to the island. The energy for the near critical rays does not dissipate at a single point on the island shore as suggested by the refraction analysis. It is in this region of energy convergence that the assumption of constant power between wave rays is no longer valid and a down gradient flow of energy occurs across orthog- onals. The "shadow zone" of the island is no longer a region of constant energy flux, but would contain energy input lost by diffraction effects near the point of tangency of the critical ray. Thus, by conducting a re- fraction analysis alone, it is difficult to adequate- ly describe the energy distribution on the wave lee side of the island. This also applies to the effective shadow zone in the far field pattern.

On the far side of the island, where a diverging cross-over pattern of emerging rays is formed, the relative amplitude can be derived by combining of two outgoing wave rays at their proper phase relation- ship.

The phase lag on the island shore increases with -Q- (see Figure 11). The range of phase lag at the island shore varies from about 1 radian for waves of 12 minute period to about 6.5 radians for waves of 2 minute period. Thus the phase lag of waves in the lee of the island varies considerably with period, while the amplitude is unaffected by the period (within the tsunami range).

BIBLIOGRAPHY

ARTHUR, R.S. (1946). "Refraction of Water Waves by

Islands and Shoals with Circular Bottom Contours," Trans o Amer. Geophys. Union, Vol« 27, pp. 168-177.

ARTHUR, R.S. (1950). "Refraction of Shallow Water Waves; the Combined Effect of Currents and Underwater To- pography," Trans. Amer. Geophys. Unicn, Vol. 31, pp. 549-552.

ARTHUR, R.S. (19 51). "The Effect of Islands on Surface

Waves," Bulletin of the Scripps Institution of Ocean- ography of the University of California, Vol. 6, No. 1, pp. 1-26. ...THUR, R.S.; W.H. MUNK and J.D. ISAACS (1952). "The Dir- ect Construction of Wave Rays," Trans. Amer. Geophys, Union, Vol. 33, No. 6, pp. 85 5-86 5.

DUNHAM, J..W. (1950). "Refraction and Diffraction Dia- grams," Proceedings of First Conference on Coastal Engineering , Council on Wave Research, pp. 3 3-49.

GRISWOLD, G.M. (1963). "Numerical Calculation of Wave Refraction," Jour. Geophys., Research, Vol. 68, pp. 1715-1723.

JOHNSON, J.W. (1947). "The Refraction of Surfac res by Currents," Trans. Amer. Geophys. Union, Vol. 28, pp. 867-874.

~> JON, J.W.; M.P. O'BRIEN and J.D.. ISAACS (1.948).

"Graphical Construction of Wave Refraction Diagrams," Hydrographic Office, Navy DeptB Technical Report No. 2, K.O. Publication No. 605, 45 pp.

KAJIURA, KINJIRO (1964). "On the Partial Reflection of

,ter Waves Passing Over a Bottom of Variable Depth," "International Union of Geodesy and Geophysics ; -- j-oceedings of the Tsunami meetings associated with the tenth Pacific Science Congress, Monographic No. 24, pp. 206-230.

JOOS, C (1934). Theoretical Physics. 2nd ed„ New York; Strechert. (New York: Reprinted by Hafner Publishing Company, 1950). 748 pp.

KRIEGER, H.W. (1943). "Island Peoples of the Western

Pacific, Micronesia and Melanesia," Srrd isonisr. " - stitution War Background Studies, No. 16.

LAMB, H. (1932), Hydrodynamics 6th ed„ London: Univ- ersity Press. (New York: Reprinted by Dover Publica- tions, Inc., 1945)-. 738 pp.

MOMOI,, TAKAO (1964). "Construction of Refraction Dia- grams of Tsunamis," Bulletin of the arthouake Inst i- ■ tute, Vol. 42, pp. 729-739.

MUNK, W.H.. and M.A.. TRAYLOR (1947). "Refraction of Oce. Waves: A Process Linking Underwater Topography to Beach Erosion," Jourc. Geol., Vol. 55, pp. 1-26.

36

MUNK, W.H. and R.3. ARTHUR (1952). "Wave Intensity Along a Refracted Ray," in Gravity v/:.ves. U.S. Dept of Commerce, Nation Bureau of Standards Circ. 521, pp. 95-108.

O'BRIEN, M.P. (1942). "A Summary of the Theory of Oscil- latory Waves," c . S „ Army Corps of Lneers , Beach Erosion Board, Technical Report No. 2., 43 pp.

PIERSON, W.J., Jr. (1951). "The Interpretation of Crossed Orthogonals in Wave Refraction Phenomena," U. 5 Coros of Engineers, Beach Erosion Board, Technical Memo, No. 21, 83 pp.

POCINKI, L..S- (1950). "The Application of Conformal Transformation to Ocean Wave Refraction Problems," Trans. Amer. Geophys . Union, Vol. 31, pp. 856-866.

REID, R.O- and C..L. BRETSCHNEIDER (19 53). "Surface Waves and Offshore Structures," ach Erosion Board, Tech- nical Report, Texas A&M University, 55 pp.

REID, R.O. (1957). "Forced and Free Surges in a Narrow Basin of Variable Depth and Width; A. Numerical .pproach," Texas A&K Depto oi Oceanography, Tech. Report Ref. 57-25T, Research Foundation. 60 pp0

REID, R.O. and A.C. VASTANO (1966). "Orthogonal Coordinates for the Analysis of Long Gravity Waves Near Islands," roceedings of the Conference on Coastal aerir. .nta Barbara, 1965, American Society of Civil zlr.c-

3 7

ineers, (In Press),,

SVERDRUP, H.U. and W.H. MUNK (1947). "Wind, Sea and

Swell; Theory of Relations for Forecasting," Hydro- graphic Office, Navy Dept., Technical Report No. 1, H.O., Pub.' No. 601.

VASTANO, A.C. (1966). Personal communication with A.C. Vastano, Doctoral Candidate, ept. of Oceanography Texas A&M University.

..ALSH, D.E.; R.O. REID and R.G. BADER (1962). "Wave Re- fraction and Wave Energy on Cajo Arenas, Campeche Bank," Texas A&M Dept. of Oceanography Ref. 62-6T, 62 pp. 3GEL, R.L.. (1964). Oceanographical Engineering. New Jersey; Prentiss Hall Inc. 532 pp.

WINKLER, CAPTAIN (1901). "On Sea Charts Formerly Used in the Marshall Islands, with Notices on the Navigation of These Islanders in General," Annual Report of the Smithsonian Institution for the year endi. June 30, 1S99, pp. 487-508.

APPENDIX I CALCULATION SUMMARY

39

A. CALCULATION SI Y FOR THE

r = 5. km. rn = 41 km. h - 05 km. h, = 4.1 km, o 1 o

«■„ e- K $ <|)

T=2 T=4 T=3 T=12

ceq. deg.

J ^ mm mm mm

(rad) (rad) (rad) (rad)

00	00.00	1.3170	14.40	7.20	3.60	2c-	2.2284
10	2.10	1.3155	14..	7.21	3.61	2.41	2.2258
20	4.2	1.3136	14.51	7.29	3.65	2.42	2.2226
30	6.2	1.3110	14.69	7.35	3.70	2.45	2.2162
40	8.18	1.3061	14.89	7.47	3.78	2.49	2.2099
50	10.08	1.3000	15.12	7.56	3.81	2.51	2.1996
60	11.8	1.2935	15.46	7.70	3.89	2.59	2.1886
70	13.41	1.2856	15.85	7.89	3.96	2.62	2.1752
80	14.90	1.2768	16.25	8.07	4.06	2c	2.1603
90	16.18	1.2670	16.70	8.30	4.15	2.80	2.1438
100	17.30	1.2575	17.19	8.59	4.30	2.89	2.1277
110	18.30	1.2475	17.70	8.88	4.43	2.98	2.1108
120	19.06	1.2365	18.30	9.17	4.60	3.08	2.0922
130	19.68	1.2266	18.89	9.48	4.75	3.18	2.0754
140	20.01	1.2165	19.50	9.80		3.28	2.0583
150	20.32	1.2065	20.15	10.11	5.05	3.38	2.0414
159.56	' ■	1.1970	20.81	10.41	5o20	3 . 4 7	32 5 3

170

SHADOW ZONE ISO

40

3. CALCULATION S RY FOR TH , OF

THE ISLA1.

-©-

deg .

o

deq

deg

2

Xr

r _, 2

1 '11 total

reflo refr

ri

total

0	0	0
20	10	2.1
40	20	4.2
40.8	20.4	4.21
60	30	6.20
80	40	8.18
100	50	10.08
120	60	11.80
140	70	13.41
160	80	14.90
150	90	16.18
200	100	17.30
220	110	18.30
240	120	19.06
260	130	19.66
280	140	20.01
300	150	20.32
319.2	159.6	20.44
320	160
	170

.1057 .1054 .1032 .1032 .1089 .1128

L36 .1162 ,1168 .1177 .1186 .1177 .1168 .1162 .1136 .1128 .1089

,32 .1032 .1054

shadow

.468 .433 .383 .321 .250 .171 .087

0 .087 .171 .250 . 321 .383 .433 .468

. dow

.1057 o!054 .1032 .5712 .5419 .49 58 .4346 .3662 .2678 .2047

o2C

.2678 . 3662

;46 .4S .5- .5712

,32 • 1C

APPENDIX II

GRAPHICAL REPRESENTATION OF RESULTS

42

INITIAL RAY ANGLE -G-,

~0Tc = Z0-

FIGURE 9

REFRACTION

FACTOR (K ) r

1.20 1

1.190 \

1 \ 1 1 1 1 1 1 —

0 20 40 60 80 100 120 140 160

AZIMUTH (deq) -9-

FIGURE 10 — REFRACTION FACTOR vs. AZIMUTH

PHASE LAG (rad) (p

22 -

20

18 -

16 -

14

12

T = 2n

T = 4

T = 8 min.

T= 12

AZIMUTH (deg) -9-

o

FIGURE 11 — PHASE LAG vs. AZIMUTH

RELATIVE

•LITUDS (A)

2.26 1

2o22 1

2.06 " 2.02 " 1.98

X

\

zc

0 20 40 60 80 100 120 140 160 180 AZIMUTH (dec) -e-

FIGURE 12 — RELATIVE AMPLITUDE vs. AZIMUTH







-



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47

thesF59

A wave refraction analysis for an axiall

3 2768 001 95927 3

DUDLEY KNOX LIBRARY