A WAVE REFRACTION 'ANALYSIS FOR AN

AXf ALLY; SYMMETRICAL ISLAND

RONALD J. FORSI

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U. S. Naval Postgraduate »

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A WAVE REFRACTION ANALYSIS FOR AN
AXIALLY SYMMETRICAL ISLAND

A Thesis
By

LIEUTENANT RONALD J„ FORST

>/

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

u.

1\.

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

OOOOGOOOOGOOOOOOOOOOOOOOOOOOO

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

X o ViaVc -V 6 X" X Ci ^ C-LO** oooooooooooooooooooooo XX

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* o i\ C -1- X Ci (_> U X CJ XI t. Q ^— *— O X oooooooooooooooooooo X _7

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

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

XXX* XJXo^-UOOXvJiM Oi' r\.i_^Oo -_j 1 J oooooooooooooooooooooo Jv

O X Xj Xj X \y O i \ /\ it i i X 00000000000000000000000000000000000*0 0*~x

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.

Vlll

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

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

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

2 -O- -•©-_. making an ancle of 2 -0- with the x-axis.
o 1 ' o

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

31

at r = r1? -e- = 2 Jf - 3 -©% and with an angle of 2jf

- 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

-

LO

-en

X,

V]

47

thesF59

A wave refraction analysis for an axiall

3 2768 001 95927 3

DUDLEY KNOX LIBRARY