N PS ARCHIVE
1967
FARRELL, C.
1 .!*,; i ra
:.-./■ i'
IP
'I IWM J;V!SL AND BOTTOM RUCTION SWCTS ON
WAVE REFRACTION AS DETERMINED BY
NUMERICAL WAVE WEFRACTlON PROCEDURES
CHARLES AUGUSTUS FARRELL
IB I Hill ■
', i " V.)
AW.
Jill
HI
■MP
mMwWMW 111
-'!• -.'•■' '■••'■ ;'•
-■•■.,■';■-'-■.'''■■.
11111
nih'-f.
IJAVAL POSx lTE SCHOQb
warruutf. calif. sk<q
'■•■
t n
TIDE-LEVEL AND BOTTOM-FRICTION EFFECTS
ON WAVE REFRACTION AS DETERMINED BY
NUMERICAL WAVE REFRACTION PROCEDURES
by
Charles Augustus J'arrell, Jr.
Lieutenant, United States Navy
B.S., Naval Academy, 1961
Submitted in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE IN OCEANOGRAPHY
from the
NAVAL POSTGRADUATE SCHOOL
June 1967
I i\ <l '.» ! - i ■* ■' y j> u. /
ABSTRACT
Numerical wave refraction programs permit a detailed study of the
transformation of wave energy as waves move from deep water to shallow
water. By eliminating the subjectivity that is present with hand drawn
diagrams the effect of small variations in the initial assumptions and
wave conditions can be investigated. The effects on wave refraction of
tide level changes and bottom friction are investigated here. It is
demonstrated that a uniform increase in water level as would be caused
by tidal fluctuations can cause a significant change in the wave re-
fraction pattern for a given nearshore region. A computing procedure
is developed to permit numerical refraction programs to account for
bottom friction. The reduction in wave height caused by bottom friction
depends primarily on bottom slope; this relation is shown in tabular
and graphical form.
TABLE OF CONTENTS
Section Page
1. Introduction 11
2. Computer Program 13
Depth Grid 13
Input Data 13
Computer Output 15
Computer Operations 15
3. Effect of Tide Level Changes on Wave Refraction 17
Differences in Shoaling Factor, Wave Speed, and 17
Ray Curvature at Specific Depths
Differences in Wave Refraction Patterns for Specific 21
Locations
4. Bottom Friction Effect on Wave Height 30
Introduction to Bottom Friction 30
Computing Procedures 31
Results of Bottom Friction Computations 33
5. Conclusions 38
6. Recommendations 39
7. Bibliography 41
8. Appendix 1 Computer Program 42
9. Appendix 2 Sample Input and Output 56
10. Appendix 3 Summary of Equations 59
.
LIST OF TABLES
Table Page
I. H/H ratio at low tide and high tide for specific 23
coastal locations near Point Sur, California.
II. H/HQ ratio at high tide and low tide for specific 23
coastal locations near Eastern Head, Maine.
III. . Wave height reduction due to bottom friction as 34
determined by Putnam and Johnson and by the
numerical wave refraction program.
IV. Percent reduction of wave height for various values 35
of TIME.
V. Wave height reduction for bottom slopes of 1:100 35
to 1:400 and wave periods of 8 to 16 seconds.
LIST OF ILLUSTRATIONS
Figure Page
1. Effect on wave speed of varying tide levels for 18
8-16 second waves.
2. Effect on ray curvature of varying tide levels for 20
8-16 second waves.
3. Low tide refraction diagram for Big Sur, California. 24
4. High tide refraction diagram for Big Sur, California. 25
5. Low tide refraction diagram for Big Sur, California. 26
6. High tide refraction diagram for Big Sur, California. 27
7. Low tide refraction diagram for Eastern Head, Maine. 28
8. High tide refraction diagram for Eastern Head, Maine. 29
9. Expected wave height reduction at breaking point due 37
to bottom friction for bottom slope of 1:10 to 1:400.
TABLE OF SYMBOLS
A approach angle of wave ray
fi ray separation in shallow water divided by ray separation in
deep water
C wave velocity
C0 deep water wave velocity
f dimensionless friction coefficient for the bottom.
FK curvature of ray
g acceleration due to gravity
H wave height at position x
HQ deep water wave height
H^ wave height at position x]^
H/Hq wave height at position x divided by deep water wave height
HHFR wave height at position x, when bottom friction is included,
divided by deep water wave height
K|: wave height reduction factor due to bottom friction alone
K refraction factor
K.c. combined friction and refraction factor
tr
Ks direct shoaling factor
L wave length
LQ deep water wave length
s wave ray
T wave period
x distance measured along the wave ray in the direction of
propogation of the waves
Ax x - Xl
^ 3.1416...
Note: All values are assumed to be in the English System of Units.
1. Introduction
Preparing accurate wave refraction diagrams is a necessary first
step in many coastal engineering and amphibious operations. The advent
of the numerical wave refraction programs |_Griswold (1963) , Stouppe
(1966), Wilson (1966) J has considerably reduced the time required to
construct a wave refraction diagram. The speed of the computer opera-
tions permits more parameters to be included in the calculations and
improves the accuracy of the output. Of equal importance is the fact
that numerical products provide a completely objective basis for evalua-
ting the cumulative effect of various assumptions on wave refraction.
The present study has investigated the effects of tide-level
changes and bottom friction on wave -re fraction patterns for several
environments. The investigation first compared the wave refraction
programs of Griswold (1963) , Stouppe (1966) , and Wilson (1966) . Since
there was no way available to the author to compare program accuracy
on an absolute scale, the comparison was based on method of computation,
simplicity of use, and presentation of results.
The program written by Stouppe was considered to have the most
potential and was chosen for use in this investigation. The major
advantages of Stouppe' s program are these:
(1) The program uses water depth directly as the interpolation
surface for evaluation of wave speed, thereby limiting the
number of computations and increasing the program versatility.
(2) The program uses a second-order non-linear differential equation
to determine the refraction coefficient at each point along
the ray. Thus, the wave height at each point can be computed.
11
(3) The program uses constant time steps in the computation rather
than constant distance steps. This reduces the distance be-
tween computations as the shoreline is approached. Thus for
the same number of computer operations a greater percentage of
computations are in the shallowest area where refraction,
friction, etc., are most important, and are changing rapidly.
(4) The program includes a graphical output that plots the wave
orthogonals, wavecrests and the shoreline.
The following modifications were made to Stouppe's program:
(1) The program was made to recycle, permitting more effective use
of computer time when more than one diagram is to be constructed,
(2) Provision was made to add a constant value to the depth field
to investigate tide level effects.
(3) A subroutine was added to account for bottom friction in the
computation of wave height.
12
2. Computer Program.
Depth grid.
The first step in the utilization of this program is the construction
of a grid of water-depth values for the desired area. The grid must be
sufficiently large so that the starting point of all rays to be followed
is in deep water, i.e., the ratio of water depth to deep water wave
length (d/L0) is greater than 0.5. By convention the x-axis is positive
increasing toward the shore while the y-axis is positive to the left of
the x-axis. The grid interval is selected so that the bottom contours
are reasonably parallel to one another within a given square.
In constructing the depth grid it is advantageous to record the
average value of all depths within that grid square, rather than the
value of water depth at the intersection point. This procedure tends to
correct for small variations in the depth of water within the depth grid
without necessitating the use of a smaller grid size. All actual depths
are made positive; extrapolated depth values are continued on land for
two grid units from the shoreline and are made negative. Beyond two
grid units from the shoreline any arbitrary negative depth may be used.
For depths on the shoreline itself zero is used.
To reduce the time required to obtain the depth grid a clear plastic
overlay with a black dot at each grid intersection point was prepared by
initially locating the grid dots on white paper and producing a viewfoil
transparency with an Ozalid copier, Model 400.
Input data.
Stouppe's order of data input was modified to permit the program
to recycle. The present data deck consists of three sections. The first
section contains only the value of MM and NN which designate the number
of points in the depth grid in the x and y directions respectively. The
13
second section contains the water depth grid. These values are arranged
so that the computer reads all the y values for each successive x-grid
position. Water depths are read in fathoms and are converted to feet
within the computer program. The last section contains one card for
each set of input parameters for which it is required to construct a
wave refraction diagram. The input parameters include:
X,Y: The grid position for the starting point of the first ray
(feet) . All starting rays must be more than two grid
intervals in distance from the edge of the depth grid.
FCF: The coefficient of bottom friction.
TIDE: The tide level, in feet, above chart depth for which com-
putation is required. This value is subtracted from the
depth grid after each refraction pattern is completed.
HINT: The deep water wave height (feet) . This is required for
the friction subroutine.
T: The initial wave period (seconds) .
Al: The initial wave angle (degrees) . The smaller angle
measured between the positive Y direction and the initial
wave crest. The angle is made negative if the slope of
the wave crest is negative.
NOR: The number of rays to be followed.
TIME: The time interval between computations for advancing of
the wave front (seconds) .
DIST: The distance between rays (feet) .
GRID: The grid interval (feet) .
This order of data input permits wave refraction diagrams to be
computed for many combinations of input variables while requiring the
computer to assemble the program and read the depth field only once.
14
Computer output.
The output from the computer consists of both a printed output and
a graphical output. The printed output variables are these:
X,Y: The x and y coordinates for a given wave crest and ray
number (yards) .
COREFR: The coefficient of refraction (Kr) for the wave at the
point x,y.
HHO: The ratio of wave height to deep water wave height
neglecting bottom friction.
HHFR: The ratio of wave height to deep water wave height when
bottom friction is included.
NGO: Indicates that the ray has terminated (1) , or it is
continuing (2) .
DEPTH: The water depth for grid position X and Y (feet) .
The graphical plot of wave crests is programmed for the Calcomp
160 system utilizing the DRAW subroutine programmed by J. R. Ward for
the Fortran 60 system. See Appendix II, Stouppe (1966) , for subroutine
listing. The first and each succeeding third wave crest are plotted.
The zero water depth points are used for contouring the shoreline on
the graph.
Appendix 2 contains a sample of the printed and graphical output
for Monterey Bay, California.
Computer operations.
Since the details of the computer operations are discussed by
Stouppe (1966), only the main features will be discussed here. The
formulas used in the computations are listed in Appendix 3.
15
The water depth at the first point is computed by fitting the
closest nine grid-point depths to a quadric surface by the least-squares
method. An iterative procedure is then used to solve for wave velocity.
It is assumed here that wave velocity is a function of water depth and
period only.
The wave ray is then moved to the next point by solving for the ray
curvature and projecting the ray forward a distance equal to the product
of the wave speed for the point multiplied by TIME. At this new point
the value of ray separation (/£), coefficient of refraction (K ) ,
shoaling factor (Ks) , H/H0, and HHFR are calculated.
This procedure is repeated until all orthogonals have been advanced
one time interval. The new wave crest is plotted and the entire pro-
cedure is repeated until all orthogonals have either gone off the grid
or reached the shoreline.
16
3. The Effects of Tide Level Changes on Wave Refraction Patterns.
Differences in shoaling factor, wave speed, and ray curvature at specific
depths.
The effect on wave refraction of a uniform increase in water depth
caused by tide level changes is cumulative and is dependent on the bottom
depth contours. Therefore, to evaluate the effect properly, a wave ray
must be followed along its entire distance from deep water to the shore.
However, by calculating the value of shoaling factor (Ks) , wave speed (C) ,
and ray curvature (FK) at both high tide and low tide it is possible to
evaluate the order of magnitude of the differences in the basic parameters
of wave refraction caused by tide level fluctuations.
The value of Ks, C, and FK were calculated initially for 25, 50,
and 100 foot water depths; second calculations were made for a water
depth equal to the initial water depth plus a tide level increase of
6, 10, and 16 feet. To evaluate these differences on a relative scale
a percent difference was calculated by using the formula:
7o difference X =
Xht
(1)
where
X = variable representing C, Ks , or FK
X^t = value of X at high tide water depth
X^t = value of X at low tide water depth.
The percent difference of Ks was generally small and under certain
conditions went to zero (for those values associated with the inflection
point in the curve when Ks is plotted against water depth) .
Figure 1 shows the percent difference in wave speed corresponding
to the low tide water depth for a specific tide range and wave period.
17
en
O
cj>
te-
en
vO
I— I
I
00
fa
en
l-J
w
>
w
w
Q
o
M
Oil
Q
W
w
fa
en
[25
O
H
cj
fa
fa
fa
fa
3
60
•H
fa
fa
o
fa
w
Q
fa1
fa
H
fa
fa
fa
I
I
I
I
w
o
I— 2
fa
33N3tf333I(I JLN30S3J
It is apparent that the difference is often greater than 10 percent and
increases with increasing wave period, increasing tide range and decreasing
low tide water depth.
The values of ray curvature (FK) were determined using the equation:
FK -- {■ C^-fi fjj. - /6#*?i£ 3 (2)
where
A = approach angle
Q-=~ , % — = derivative of wave speed in the X and Y directions
Z4- ; ir-
respectively.
Figure 2 shows the percent difference in FK versus tide range (computed
using equation (2) with the following initial values: A = 30°, — — = 0,
2J=i. = a C for a twenty foot change in water depth starting with an
initial depth of 100 feet) . The values of percent difference plotted
are representative of the percent difference for all values of
A, r— 7 , =» > and water depth; this occurs because the terms inside
the brackets in equation (2) have nearly the same value for high tide
and low tide and tend to cancel out in the percent difference calculation.
This reduces the percent difference calculation for FK to
C/.7- ^ H T
% difference FK =
H7
(3)
The fact that the difference calculation is primarily dependent on tide
range and wave speed is clearly shown in Figure 2.
It is concluded that the change between high tide and low tide in a
refraction pattern is primarily determined by the change in ray curva-
ture (FK) .
19
H
W
w
15
w
Q
aDNaaaaaia iNaouaa
20
Difference in wave refraction patterns for specific locations.
To investigate the combined effect of the changes in C, Ks and FK
caused by a uniform increase in water depth, wave refraction diagrams
were constructed for several locations using both the low tide water
depths and the high tide water depths.
The first area considered was the California coast near Big Sur,
between latitude 36° 14' 30" and latitude 36° 17' 42". This area was
chosen because eight-second waves arriving on the coast will be refracted
by the offshore bar (shown by the shaded area in Figures 3 and 4) at low
tide but not at high tide. Thus tide level changes will cause selective
refraction by the offshore bar. Figure 3 shows the refraction pattern
for 8 second waves approaching from 270 degrees at low tide; Figure 4
shows a different pattern for the same condition at high tide.-'-
Figures 3 and 4 show two major differences in the refraction pattern.
First, orthogonals 6 and 7 cross approximately halfway to shore at low
2
tide but they do not cross at high tide. Secondly, at low tide there
is a convergence of wave orthogonals at Point A; this convergence is
not present at high tide.
To make the changes in the orthogonal pattern more prominent
Figures 3-8 are plotted with an expanded ordinate scale. Also, the
computer was not used to contour the wave crests but rather the wave
orthogonals were contoured manually; the shoreline was indicated by a
triangle plotted whenever water depth was zero. TIME equalled 30
seconds for all diagrams.
o
Interpretation of the effect of crossed orthogonals is beyond the
scope of this paper. In this discussion we will only be concerned with
the difference in ray paths between low tide and high tide.
21
Table 1 shows the value of the wave -height ratio (H/K0) computed
at four wave crests before the shoreline is reached for specific points
along the coast. This table clearly shows that the value of wave height
can be considerably different at low tide and high tide.
Figures 5 and 6 represent the same initial conditions as Figures
3 and 4 except that the initial wave direction is from 310 degrees.
The change in the orthogonal pattern is clear. It is noted that in
Figures 3 and 4 the primary change in the orthogonal pattern (between
low tide and high tide) is in the vicinity of points B and C, whereas
in Figures 5 and 6 the greater change occurs near Point A.
Wave refraction diagrams were also computed for the coast of Maine
between latitude 44° 39' 35" and latitude 44° 44' 45". This area was
chosen in order to determine if the combination of a large tide range
(16 feet) and deep water depths (greater than 200 feet a short distance
from the shoreline) would cause a change in the refraction pattern.
Figures 7 and 8 show the refraction patterns for 16-second waves arriving
from 070 degrees at low tide and high tide. The difference in refraction
patterns is evident. Table 3 shows the H/H0 ratios (calculated at four
wave crests before reaching the shoreline) for specific locations at
low-tide and high-tide conditions.
In total, about 20 sets of refraction patterns were analyzed and
it appears impossible to predict what changes in wave height and re-
fraction patterns will occur as the result of tide level changes. The
only generalization that can be made is that the curvature of an or-
thogonal is generally greater at low tide than at high tide. It is
concluded that precise water-level information, including tide state,
should be incorporated into wave refraction procedures.
22
TABLE I
H/H0 RATIO AT LOW TIDE AND HIGH TIDE FOR SPECIFIC COASTAL LOCATIONS
NEAR POINT SUR, CALIFORNIA.
LOCATION
f
H/I
lo RATIO
(SEE FIG.
3)
LOW TIDE
HIGH TIDE
A
1.1
1.2
B
1.3
.9
C
3.1
1.7
D
.8
.8
TABLE II
H/HQ RATIO AT LOW TIDE AND HIGH TIDE FOR SPECIFIC COASTAL LOCATIONS
NEAR EASTERN HEAD, MAINE.
LOCATION
(SEE FIG.
7)
H/HQ
LOW TIDE
RATIO
HIGH TIDE
A
2.8
3.3
B
1.0
1.2
C
.7
.8
D
.7
.7
23
24
u
CD
•
tfl
£t
U
o
e
en
•r-4
Pn
H
o
H
o
o
•
<]-
O
o
CD
r>
^H
CM
TO
<J
II
en
<
rf3
4-1
•
3
U
o
0)
co
CO
1
-C
oo
4-)
5-i
II
O
Z
H
.
X
to
u
■H
C
C
•H
M
~\
o
•
<4-j
4-1
•H
Ph
i— 1
TO
o
u
o
o
•*
'.O
M
3
01
W3
r— 1
TO
60
a
•r-l
CO
CO
4J
u
CO
o
<U
M-l
!3
|
S
■U
TO
CO
u
TO
60 W
TO
•H
•
-a
Z
Oi
c
«tf
0
-
■H
r^
4J
^H
0 o
TO
^D
U
ro
14-1
CD
O
5-i
4-1
QJ
z
TJ
o
■rl
co
4J
~
<r
jz
r— 1
MO
•H
t£>
EC
ro
.
0)
<t
X)
CD
3
S-i
4-J
=3
•H
00
■U
•H
TO
PL,
*1
25
o
CO JZ
o
o c
co -h
H fa
M O
H O
O
• <J-
O
o cu
I— I I— I
CO CD
O
II co
< -c
• 3
u o
CU 00
CO I
00 -u
II o
H
• £1
co o
•H C
C -w
H ^
O •
4-1 4-1
•H |JL,
r-l
CO O
u o
o
3 cu
w i-i
CD
oc o
•r-l CO
PQ
}-i co
O cu
CM J2
E 4J
CD CO
U CD
00 w
CD
•H
T) =
CN
o -
•h r-»
■u .— i
u O
CD v£>
J-i co
M-l
CU O
U 4J
CU r
X) O
•H CO
4J -
o o
CO
LO cu
QJ 3
!-< -U
3 -H
60 -U
•H CO
fa »J
A
o
C
•H
>
O 4-1
<U fa
CO
o
o o
en o
■o-
II
0)
H •" '
"i
g ro
■n
M o
■■''
H w
• J3
O -u
o d
r-l 0
ro co
1! 4=
4-1
57
< U
\ /
o
:"j
• :s
u
cu •
co x:
o
oo d
•H
ii --~
H -u
KJ
fa
CO
•
si
CD O
•H O
C O
U v£)
O
M-l CU
•H i-H
i—l CO
CO O
---1
CJ CO
CM
« 4-1
CS)
U CO
3 cu
CO Is
00 4J
•H CO
PQ CO
_
W
J-i
o •
v '_,
4-1 r
«M
CN
03
e <t
CO -
Jj r~-
60t-H
co o
•i-l vO
T3 on
C cu
O T3
•H 3
4J 4-1
(T?
O -r-l
CO 4-1
!-l CO
Ct-I .— 1
CU
U O
4-)
CU
XI z
•H O
tV/>
4-1 CO
: .7
4= <f
u
60 i-l
•.-IO
W vo
co
%£> CU
XI
cu 3
U 4-1
3 -H
f'.'1
60 4J
. "^
■H co
-1
fa »J
27
.'\ia
cas
QY<L
tm
o
C
u
fa
<u
co
o
o
o
o
en
vD
II
OJ
1— 1
w
eg
p>,
U
M
CO
H
X.
•
4-1
O
0
o
o
1^.
CO
o
1
J=
II
4-J
u
<
o
S3
u
.
a>
43
CO
U
C
vO
•H
O
o
CD
o
C
oo
•H
CD
a>
S
<— i
CD
u
13
cfl
CD
0)
4J
HJ
en
01
C
12
J-i
i
0)
j-i
4-1
CO
CO
CD
CD
W
W
u
~
o
m
14-1
<T
—
e <
CD
<t
^O
&o<t-
CO
<t
•r-l
13
cu
-a
ti
3
O
4J
•H
•H
4-1
4J
o
TO
CD
1— 1
H
14-1
0
CU
4J
U
cu
m
T3
en
•H
»
4-1
o>
en
S O
o
<t
hJ
<*
.
cu
r^
-a
CU
3
U
4J
3
■i-l
M
U
•H
TO
fa
l-J
28
■iia
n
o
o •
OJ 4J
o o
CO o
o
II ^o
M <u
JS H
M CD
H U
in
o' &
o -u
O O
<z>
II I
4=
45 £
• O
u Z
<u
CO •
4=
Fn
QJ
o
c
o
•H
o
CD
oo
s
CD
f\
r-l
TJ
CO
CO
o
QJ
en
EC
4J
a
w
u
01
OJ
[2
4-1
i
CO
4-1
CD
W
W
CO
W
S-i
O
•
<4-J
~
LO
e <
CD
—
5-1
<r
wxj-
COO
•H
<t
ID
<t
C
QJ
O
-a
•H
3
4-1
4-1
O
•H
TO
4-1
l-i
CO
LW
.— 1
0)
S-4
o
4-1
0)
T3
r
•H
uO
4J
co
—
-C
cr*
00 CO
•HO
w
-tf
<f
CO
01
T3
<u
3
u
4J
0
•H
00
4-1
•I-f
CO
Ph
,-J
29
4. Bottom Friction Effects on Wave Height.
Introduction to bottom friction.
The present method for calculating wave height assumes that the
propagation of energy between two adjacent orthogonals is conserved as
the wave proceeds from deep water to shallow water. This implies that
the addition of energy by surface winds, dissipation of energy by bottom
friction or internal friction, and the reflection of energy by a steep
slope are absent. In nature this is not the case - the wave energy
between adjacent orthogonals may be changed by all of these means.
According to Lamb (1932) , in fluids of small viscosity (such as
sea water) , the velocity of the wave and the relationship between depth,
length, and period are unaltered by internal friction. This is based on
the assumption of a sinusoidal wave form but the result appears to in-
dicate a very slow rate of damping of all waves in deep water, which is
in accord with the limited observations in nature (Putnam and Johnson,
1949) .
At the sea bottom, however, friction can cause a loss of wave energy.
It can be demonstrated both theoretically and experimentally that the
particle velocities of shallow water waves are finite at the bottom and
actually approach the theoretical orbital values at a short distance from
the bottom. Therefore, a boundary layer in which mechanical energy is
dissipated through the distortion of fluid elements must exist at the
bottom. The effective roughness of a sandy bottom probably is determined
by the size of the ripples in the sand rather than by the size of the sand
grains. The ripple size, however, appears to be determined by the size
and sorting coefficients of the sand in addition to the height and period
of the wave and the depth of water.
30
Putnam and Johnson established the general magnitude for the dimen-
sionless friction coefficient for the sea bottom to be 0.01. (This value
was computed assuming initial conditions of 12 second waves, 4 feet high,
traveling over a sandy beach with a ripple spacing of 5 inches.) Putnam
and Johnson point out that since the friction factor only enters the
equations for wave height to the first power, the friction factor
could change by + 50 percent without appreciably changing the order of
magnitude of wave-height reduction.
Based on the following assumptions Putnam and Johnson were able to
develop a procedure to calculate wave height when bottom friction is
included.
(1) The oscillatory motion at the bottom is sinusoidal and
corresponds to the wave theory based on small amplitudes.
(2) The friction coefficient does not vary with velocity and
is approximately independent of water depth. This assumption
amounts to choosing an average friction factor for the entire
region over which the sea bottom affects the wave motion.
(3) Perpendicular flow at the bottom resulting from percolation
in the bottom material is negligibly small.
(4) The sea bottom has a constant upward slope from offshore to
the breaker point.
Computing procedures.
Bretschneider and Reid (1954) extended the original work of
Putnam and Johnson and developed new equations to solve for wave
height which included simplified equations for the special cases of
constant water depth and constant bottom slope.
31
Bottom friction is of importance only on gently sloping beaches
and in shallow water; thus it is possible, when using numerical wave
refraction procedures, to calculate the reduction to wave height due
to bottom friction by making the assumption that the water depth is
constant between successive computations along the ray path from deep
water to the shoreline.
The wave height at each step is computed by the equation
H = H, [ I" {('-KfWi-K^)-*- •• + 0-K4 )}] (4)
where
Hi = the calculated wave height, neglecting bottom friction, for
the point;
Kf = the refraction factor between point n-1 and n.
■"■n
The coefficient of friction (Kf) is calculated by the equation (from
Bretschneider and Reid)
h - ~W* 7 '' (5)
ft' =
f H,
9 L
K = direct shoaling factor;
s
A ^ = incremented distance.
Kf is computed at each ray point with .Ax equal to the distance
along the wave ray between point n and n + 1. It is assumed in the
computations that the water depth between point n and n + 1 is constant
and equal to the depth at point n.
32
Results of bottom friction computations.
The accuracy of the approximation that water depth is constant can
be seen by a comparison of wave height obtained by this procedure to
the wave height obtained by the more sophisticated and accurate pro-
cedure used by Putnam and Johnson. This comparison is shown in Table 3
(computed for wave period = 12 seconds, bottom slope = 5 feet). The
comparison is very favorable along the entire distance of travel from
the point that the wave first feels bottom to the breaking point. The
difference is certainly within the order of magnitude of the error in-
troduced by other assumptions. The difference in values is caused by
the approximation to constant depth steps in the calculating procedure.
Since this program uses constant time steps in moving each ray point
toward the shoreline, the horizontal distance over which depth is assumed
constant can be reduced by shortening the computing time interval (TIME).
Table 4 shows the reduction (in percent) of wave height at the breaking
point for different computing time intervals (with wave and bottom con-
ditions the same as shown for Table 3 above). As expected, there is a
slight increase in wave height reduction as the computing interval is
decreased. It can be concluded that the value of TIME will affect the
wave heights calculated by this procedure, but the effect is small.
Realizing the limitations imposed by the initial assumptions of
Putnam and Johnson and the calculating procedure employed here, the
computer program was used to compute the wave height (with bottom friction
included) for varying initial periods and bottom slopes. The results are
shown in Table 5. (In arriving at these results it was assumed that the
initial wave height was 5 feet and the friction coefficient was 0.01.)
The reduction in wave height (in terms of percent of the wave height
33
TABLE III
WAVE HEIGHT REDUCTION DUE TO BOTTOM FRICTION AS DETERMINED BY PUTNAM
AND JOHNSON AND BY THE NUMERICAL WAVE REFRACTION PROGRAM.
T = 12 sec. HQ = 5 FT. f = .01 TIME = 30 sec.
Bottom slope = 1:300 Breaking depth = 11.5 Ft.
DISTANCE FROM
WHERE h = Lo/2
TO BREAKING POINT
PCT
0.0
50.0
62.0
70.3
80.5
86.8
90.8
93.0
95.0
97.1
100.0
PERCENT REDUCTION WAVE HEIGHT
FARRELL
CONSTANT DEPTH STEPS
PCT
0.0
0.0
0.2
0.4
1.0
1.9
3.1
4.2
6.0
8.7
18.5
PUTNAM & JOHNSON
CONSTANT SLOPE
PCT
0.0
0.0
0.2
0.4
1.1
2.1
3.2
4.5
6.2
9.3
20.8 •
!']
34
TABLE IV
PERCENT REDUCTION IN WAVE HEIGHT FOR VARIOUS VALUES OF TIME
TIME (SEC) WAVE HT. REDUCTION (%)
20 19.0
30 18.5
40 18.0
50 17.0
TABLE V
WAVE HEIGHT REDUCTION FOR BOTTOM SLOPES OF 1:100 to 1:400 AND WAVE
PERIODS OF 8 to 16 SECONDS (PERCENT)
WAVE
■■
BOTTOM
SLOPE
PERIOD
1:100
1:200
1:300
1:400
8
5.4
14.0
21.5
28.6
10
5.2
13.5
20.2
27.0
12
5.2
13.0
18.5
24.8
16
5.0
12.2
18.2
24.2
AVERAGE
5.2
13.2
19.6
26.9
35
without friction) is shown for the theoretical breaking depth (from
H.O. 234). To obtain the percent reduction at the specific breaking
depth, it was necessary to fit a smooth curve to the plotted points of
wave height reduction at the three closest water depths and interpolate
the wave height reduction at the breaking point.
Figure 9 shows the average reduction (for all periods from Table 5)
of wave height versus bottom slope. The curve is extended by a dashed
line to the reduction amount found by Putnam and Johnson for a 1:10
beach slope.
The quantitative value of Figure 9 is controlled by the input
accuracy of the coefficient of friction, the initial assumptions and
the computation procedure. However, Figure 9 does show, qualitatively,
the effect of bottom slope on wave height reduction due to bottom
friction. Figure 9 is considered sufficiently accurate to be used in
surf forecasting on sandy beaches to provide a first approximation of
wave height reduction due to bottom friction.
36
Figure 9. EXPECTED WAVE HEIGHT REDUCTION AT BREAKING
POINT DUE TO BOTTOM FRICTION FOR BOTTOM SLOPES OF
1:10 to 1:400.
B
M
H
O
S3
Q
H
!S
W
c_>
OS
w
CM
.Of
L _
JL
r
|
|
1
(
>
i
or
L..
£V
t
i
(
1
|'
|
i
jl
i
... Jv.
1
1
!
!
f
i
i
U"'i
>
I
r
' s
^ i
o-t>
^PlJTNy
vNH
JOI
HNS
ON
i
1:11
t»
ir:
too
1«
L
JCO
1*4
00
BOTTOM SLOPE
^7
5. Conclusions.
The numerical approach to wave refraction is a powerful tool in
that it eliminates human subjectivity and permits the refraction com-
putations to account for more factors than is possible with hand drawn
techniques.
A computer program that uses constant time intervals between com-
putations has more suitability in accounting for the effects of friction,
wind, reflection, etc., than a computer program that employs constant
distance steps.
Tide level changes can have a significant effect on wave refraction.
This is especially true in areas where there is a large tide range or
an offshore bar that can cause selective refraction.
The changes shown by the tide level effects also point to the im-
portance of having an accurate water depth grid. To avoid significant
errors in the refraction pattern a great deal of care must be exercised
in the preparation of the depth grid.
Bottom friction can and does have an effect on the wave height.
The numerical refraction program provides a convenient way to account
for this reduction.
38
6. Recommendations.
The course of this investigation has permitted a close look at wave
refraction from many points of view. The numerical refraction program
is capable of yielding wave refraction information with great speed and
accuracy. (Computer time is approximately 5 minutes for a 20 ray diagram.)
However, before the advantages of the computer product can be fully
realized the author believes further investigations and improvements are
needed in the areas of input data, computing procedures and field in-
vestigations .
The need for accurate depth information is essential to the compu-
tation of accurate refraction patterns. It is anticipated that the present
efforts of the Naval Oceanographic Office and the Coast and Geodetic Survey
in the area of color photography will improve the water depth information
especially in areas where the bottom contours change considerably and
in the shallow areas where it is difficult for hydrographic vessels to
proceed.
The computer program can be expanded to account for more variables
and to provide additional output information', in particular:
(1) The program should account for the fact that in nature there
usually exists a spectrum of wave periods and directions,
rather than the single period and direction that is considered
here.
(2) For amphibious operations it would be advantageous if the
program determined the breaking depth of the waves, printed out
the height of the waves at the breaking point, and then contoured
the breaking point on the output refraction diagram.
39
(3) Another desirable output for amphibious operations is a computer
plot on the output diagram of the area where the wave height is
a minimum, as this would be an ideal area to anchor large ships.
(4) The various subroutines should be checked to ensure that each
section is providing comparable accuracy.
Field work is necessary to determine the value of the bottom friction
factor for various type bottoms. This is essential before accurate wave
height information can be obtained. Field investigations are also needed
to determine the accuracy of the computer program; until now the numerical
wave refraction procedures have been checked only by comparison with hand
drawn results.
.
40
BIBLIOGRAPHY
1. Breakers and surf, principles in forecasting. U. S. Naval
Hydrographic Office, Pub. 234, 1944.
2. Bretschneider, C. L. and Reid, R. 0., Modification of wave
height due to bottom friction, percolation, and refraction.
Beach Erosion Board, Tech. Mem. 45, 1954.
3. Griswold, G. M. , Numerical calculation of wave refraction.
Journal of Geophysical Research, v. 68, 1963 : 1715-1723.
4. Griswold, G. M. , and Nagle, F. W. , Wave refraction by numerical
methods. Mimeo Rept., U. S. Navy Weather Research Facility, 1962.
5. Lamb, H. , Hydrodynamics, 6th Ed. Cambridge University Press, 1932.
6. Putnam, J. A. and Johnson, J. W. , The dissipation of wave energy
by bottom friction. Transactions of the American Geophysical Union,
v. 30, 1949 : 67-73.
7. Stouppe, D. E. , Ocean wave crest and ray refraction in shoaling
water by computer. Master's Thesis, Department of Meteorology
and Oceanography, United States Naval Postgraduate School, 1966.
8. Wilson, W. S., A method for calculating and plotting surface wave
rays. U. S. Army Coastal Engineering Research Center, Tech.
Mem. 17, 1966.
9. Wiegel, R. L. , Oceanographic Engineering. Prentice Hall, 1964.
41
APPENDIX 1
Computer Program for Wave Refraction using Fortran 63
on the C.D.C. 1604 Computer
Subroutine Title: FRICT1
Variables of Subroutine:
HHQ.... Height of wave at point X, Y after bottom friction
has been included.
RLO Deep water wave length.
RLQ Wave length at point X, Y.
* CFF Friction factor between point n and n+1.
RCFF (1 - Kf)
TRCFF Total of (1-Kf) terms to point n.
TCFF. Total friction factor to point n.
Summary of Subroutine:
The subroutine calculates the value of Kf and wave height reduced
by bottom friction (HHFR) for each point along wave ray.
See Stouppe (1966) for discussion of remainder of program,
42
o o o o o o o
0 0
O
0
O
O
O O
OOO
oh ryj ro <t mvoi*-
co 0 0 <—
1 cm Cl
1 >*
IT
> no r-
oc
1 a
1 OH tS
1 CO ,*
in vo Is- 00 o>
O <H
CM CO 4"
m
oooooooo
0 0 •— ■ •—
1 w-* 1—
1 rH
rH 1— • i-H
1 — 1
1 1—
1 (N rg r\
1 <M r\j
CM f\l (\J rg «M
CO CO
CO CO CO
CO
oooooooo
00000c
1 0
OOO
0 c
1 O O O O O
00000
0 0
OOO
0
oooooooo
00000c
1 0
OOO
0 c
• O O O O O
00000
0 0
OOO
0
ooooooooc
1 0 0 c
1 O O
0
OOO
c
> c
1 0 0 c
1 O O
c
1 0 0 0 0
0 0
OOO
0
oooooooo
c
1 0 0 a
1 0 O
0
C
1 0 O
c
1 0
, 0 0 a
> O O
c
> 0 0 0 0
0 0
OOO
0
oooooooo
G
1 0 0 a
pt
0
0
O O
m
<
X
X
0
c
1 0 0
c
1 c
'OOO
' O 0
c
' 0 0 0 0
0 0
OOO
0
<r> lu
r-4
•
•>
V — X
—
X
i—
•
O < l-
iZ
1—
z
X
0 ex 2:
u.
a
>-
*-H
<N
0 < z :*
u.
UJ
>-
X
»
r-t O. O C£ — <
»
0.
Q.
•>
1—
z h- e> • or
— •
•>
Q
UJ
Z
or < onoo
— 0
z
O
0.
^H
UJ *IX UJ-U
0 0
z
Ol
»-H
X
co t- r- a. h- _i tjj
O H
■1
*
r-
X
O to vo to t- x
."H »
2:
X
•
<* — 0
t— ui 0 ui >- —• i—
— CO
2:
X
u.
O
» 0 ft
UKHll/Jh
>- —
■>
Q.
u
►-<
x -. a:
O U H- Z
U. CO
or
Q
u.
CX
HKO
* V 0£ X O
• «
0
O
*
0
* O *
UJ U. < • Ui Q.
— . r~
z
Q.
>-
*
UJIh
a. 0 2: _i 1— < a.
0 0
*
■1
1—
O <f l/>
a »UJ DQC 3
0 0
X
>-
_l
to
r-l «K 1— 1
d z > a. 0 uj
i-H rH
<
a
UI
■— )
r- X O
O O -J UJ 21 h-
«~ w
2:
u
0
0
X CO •>
1- « J J QUI M
X 0
*
0.
•>
»
-^ •> UI
t/> f- uj u x or
u. X
iZ
•>
X
UI
• H- 21
< or uj h- 5
* X
u.
X
2: ~
X tO *-"
• H-KQ -J
•~ •*
■)
a
_l
—• 0
rH I-H f—
QD<-<oq:uj
0 —
Q
u
UI
a
h- •
» O *
a u_ t- vo 0 x
0
*-*
a
0
UI
• r-
IlICC
•2! <H U. (—
rH
or
m
*
1—
cx u.
V <t 0
h- 0 • a
— » r-H
0
>
h-
i/>
O (M
u. • z z
_j u u z u 10 z
V ~
*
Q.
•— 1
3 —
z *
X X * z
< O UJ -<
> a:
UJ
O
2:
-> 2:
• i-H
CO f>J i-H *
>■ UJ • u 5:
>- u.
2:
Q.
•>
O 21
rH •
•> »• *X T-4
CDII-Z < O
» UJ
1—4
•
—•
< •
< iTl
X UI • 11
r- _l O UJ Z UJ
— X
h~
X
^0
.—»
• U.
^- 2: 1- ~>
CX — X 33
0 0
**
Q.
— r-t
H- II
h- »
»■ 1 — 1 »
uj or >- t- h- 0 — 1
O KJ
ce
Q
U X
X —
* tT\
> 1— 1— ^o
1- Oau uj or
<~t •>
<
a
» r-
O *
I— >-H
>- X z ^
UJ
UI
< u. - a a u
»— — >
co
*
0 a.
I— « *-^
z •
X >* «-i in
0
0
3 qq: 0- 10
X 0
*
*—
_J UI
QC Z
—1 rH
<N ♦ X
•— 1
H*
2J LU U_ U. l/) Ul
X 0
a:
0
Q Q
z
X •
* X • O
rvj r-
1—
KD < — 1 JJ Q
X r-t —
<
0
• •
UJ »
• in
X (N UJ O O 1
+
Z CC U. U_ CO 0
» — O
a.
t— *
— 0
CO i-H
UJ u.
lf\ • &
m 0
0
•—> 0 •— • 0 O tO
— V 0
Wk
•>
0 X
II
O <f
•■ ex —
* •
•
-J O O -U <
rg » <-h
> 1—
0
u
O X
I— -)
■ — • «
> 0 1-
VO V0
vO
< or 0 1- z x
— t -~ ^_
X lO
0
a.
i-H •
I/) •
1— rg
XZ »w
i-H *
*
0 a. 2: 0 < (— 0
~ O <
^> »-
H
Q
- 0
3 —
• •
X X u.
in ~
x uj or »—
oJO<
* Q
~-
0.
iZ 0
2: -) —
u, m
CM CO VJ
• ->
->
o;ioztou_r-arz
_l r-l <
< *
Q.
*
^Z _l
* 0
u u,
• •> u.
r-t »
•>
U. O < U, Ct aJ UJ
i— -~ •>
• 0
UJ 1
u
u. •
z — •
u. •
X X »
0 -*
t—l
UJ Z « 3 11J O I- 5:
— X 0
h- _J
0 u
u. <—
z
Z ~ lT>
* 0
iNfM > 2
m —
w
K — • »- liZil
1— X O
"N. *
V. 1
a.
\ 0
z
—
a. u.
> •
• • > 2:
— a
a
> UZ UJ UJ I—
•
K-< —< Z
(M <
cn 1
0
<f 0
*
0
QUJ-t
>- r-
H H * •
-. UJ
UI
< Z < < T Z <
-J
v •
±£ CX
iZ 0 iZ -•
2
l—i
Z O rH
» u.
X < > -H
-> 0.
Q
•i O a: ct: i- « >- 1- ;
z
Z -1 —
-J O
_l 1
X
_l —
2
^— 1
< ;
> fM
rH X X II
• II
.0 II
— < U_ O < 1/) '
O 3D O
CO z
CO
9k
CO CX
*
(N
•—
X — 1
c*
— (^J •> >— 1
t-M -r^
rH UJ
5IHUJOUJZZ 1
—
— < \ O
\ *
\
>-
\ u.
<f
■—
2: •
•
rH
» (M
-^ ~>
m -> 3
< u ar or a ui ui :
D
lOZH
z <
z
X
Z X
[*»
1—
2: -H r-
r\J 1—
r- X vO O. »
• Z 1
o
ar < a. z> 1- to _i 0
z 0 ~
O u.
0 a.
0 X
<
<
< 1
—
< >t P- rH
UJ I-H
0 •— • •—•
•
e? Q£ UJ 11 — or: 1 — :
X
ui 2: 0
2: UJ
2: a 2: •
Q
2:
UJOZ
Q 2: :
z
2: * z m 0 — > 1
r- — (— i-H
OU. >UJU - < aj •-< :
□
2 21 O
2: ex
2! Q 2: U.
<
CX
1— < or
< CX 1
CXI- —
— 0.
0- z
II
q£lu<xzq:i/)h-:
D
— OO
0 0
O Q.
0 X
UJ
O
0 UJ 0
ui 0 1
a:
O X CX O U. UJ
O UI 0 <
-H
na 3H-. j^ m.
n
.2 <J z
u u
^
#■
U X
X
u.
z ar u.
ex u. a,
U. rH 0. Q
r- • Q 1
JOUCQ
rH
r-l
-H
i-H
rH
m r\j
CO
CO
rH
i-H
0
in
CM vO
O rH
in m
UUUUUL^UO
u
43
o
vO r~ co ct»
co co co co
o o o o
o o o o
o o o o
o o o o
o o o o
o
cm co
vO MO O
-a- ** «* >*
o o o o
o o o o
o o o o
o o o o
o o o o
o
O rH f\| CO -J- lf\ >0
in in in in in in in
o o o o o o o
o o o o o o o
o o o o o o o
o o o o o o o
o o o o o o o
o
r- co on o
iT\ in in vO
o o o o
o o o o
o o o o
o o o o
o o o o
o o
rH C\J CO -3"
\0 vO *0 o
o o o o
o o o o
o o o o
o o o o
o o o o
in vO
o o
o o
o o
o o
o o
o
r- co
vO vO
o o
o o
o o
o o
o o
o
on o
vO r-
o o
o o
o o
o o
o o
r-
o
o
o
o
o
UJ
or
CO
o
in
CO
r— t
CO
CO
CM
•
-o
II
dc
<
CO
in
00
rH
CO
CO
CM
cm
•
CM
CO
II
DC
<
CL
fer)
on
CM
CO
in
o
rH
<
II
II
o
tf>
<
II
o
<
t*
>
<
X
CO
II
r*>
S
<
-1
Q£
_J
O
UJ
o
DC
QC
DC
a
<
u.
X
X
CO
CO
II
II
. —
-~
<f
i— i
—
—
aJ
UJ
_l
o
UJ
<
II
<
II
<
DC
O
>
II
>
>-
tr>
>
— X
o »
UJ X
— i X
<
II
<
CM
CO
II
o
>
>-
II
>-
>-
br>
>
X
II
X
X
u_
l/J
o
u
*
I—
I/)
I—*
+
>
>-
<
DC
o
tr>
CO
co cm «-« -~
O — <
o
o
tr>
br>
UJ X
tr>
•-' II
f— 1
rH 4-
l— -J
II
II II
II UJ
X
a a
O T3
<
o —
U. <
CO 2 JZmJ
m • -4" -4" W
tr) >t • •
tr> » O II O M
2 a _l O • • CO
O UJ > > • II O Q O II cO
— < r- < _J O < O I— —
r— Z CO U- UJ •— Ir-lr-trHI^O
UO OJrHjwQiw .HI-r-OtD-
<S>-0 — OQUJQtN •
QC UJrHQZUQ-U- nOO
U. r-U<CDXC0COrH(J3
fMiniimi«coM »
rHC0CO0OCOCOr-.GOII II llkr»mtr>
• II II II tt t« fl )■ — — ^
rH -~ ~ — . — . — — .r-»OrHf\|rH — O
II^COLnvOr— 00(7»rHrHrHCO • •
^^.^.^^^^~^,^^. || c> o
LUUJUJUJUJUJUJUJUJUJ — <HcO
00_J_l_J_l_J_l_»_J_l_JrH I
rH(— I— I— I— I— r-r-l—l— I— • I— II
„HHHHMMMMMH->-
Or-r-r-H- r-r-r-f-h-r- — U_X
QmmhhmhhhmmiQhU
o
I/)
• o < —•
o
in
II
>•
X
<
o
<
<
<
II
<
fcft
<
DC
o
>-
II
>-
>-
hr>
»
o
DC
o
z
I
<
DC
o
u.
I— 4
br>
+
<
o
z
II
< <!■
Qi cvj
o •
Z m
u u
Or Q
O I rH
Z > ll
• X <
rH II QC
- > o
CO X z
— — fcr) m
<
or
o
UJ rH
Z X
o
ON
ON
*
*
*
o
Ch
ON
•>
•
*
r\j
CVJ
CM
•>
*
•
rg
(\l
f\J
•>
*
*
O
o
o
»
•
«
o
o
o
•>
•
•>
•
•
•
o
o
o
o
o
o
o
o
o
o
o
o
<-i
r-i
rH
•>
9k
#•
•
•
•
o
o
o
o
o
o
o
o
o
o
o
o
UJ
UJ
:0
<
CM
UJ
UJ
co
<
CM
rg
in vO
CM cm
» •>
O vO
CM CM
* *
O rH
CM CM
X CM
•
DC O
O H-
z
— o
<
or **
o
u — i
u. u. u.
< <
CO
m \0 Is- m
co -j- co
cm cm
O <f m o
rH cm cm
<
U
CM
UJ
UJ
CO
<
_l
o
*
CM
>
X
»
DC
O
<
<
u
CM
44
o o o
o o
o
o
rg co
^•mvors-cooo<-»rvj
co 4- m v© r- oo
O O r-t
m m 4 m >o
r- r-
r^r-r-r--r-i,-oococo
00 00 00 00 CO 00
CO 0^ o
o O^ 0^ o^ o*
o o
ooooooooo
o o o o o o
o o o
o o o o o
o o
ooooooooo
o o o o o o
o o o
o o o o o
o o
ooooooooo
o o o o o o
o o o
o o o o o
o o
ooooooooo
o o o o o o
o o o
o o o o o
o o
ooooooooo
o o o o o o
o o o
o o o o o
o o
r^cooo<->rgrOstif>vor-
ooooooooooo
OOOHrtHHHHHH
ooooooooooo
ooooooooooo
ooooooooooo
ooooooooooo
X t-
1—
a.
— »
o>
a
LU
-)
*
LU
Q
«-
0
a
X
O ♦
«
X
m
0 —
C\J
m
»
O rH
•
»
X
Z +
og
X
m
• "J
*
m
*
— * M
O
*
DC
""> rH
rH
»
ct
a.
~ X
-1 1
O
u_
X
rH (—
co 2;
*
X
X
x a
» II
•
X
X
r- LU
0 z
O
X
<*
a. q
CO
O
^*
m
LU •>
» t*
O
m
X
O —
rH
O
•
X
m
» rH
CO rH
rH
CO
in
•
— +
— +
•
•n.
•
O
3 ~)
— 2:
•
«■»
0
X
•— «—
-> II
O
Z
X
X
cc ct
• 2:
O
-»
X
X
Lu Lu
<—<
O
>-
X CO
X X
rr
«- br)
O
II
CO
m
X X
O
a.
rH
— >
*
X
• •
z
LU O
*
Z
X
CO
-^ -»
*
Q H
LU
i«
CO
•>
""> rH
rH
— Q£
_l
>
— »
•>
<
— +
II
li. O
r-
rH lj_
1
\
<*
<
u.
u.
LU
O ->
X —
^
<f
•— • *
1— <
X *A
*— 1
LU
a.
X 0
—
rH
rH
W rH
r-4
<
oc
O
* X
V
rH
*
1
•
z
S •
II
0
u
- X
— .
O
>t
z ->
-J
z
II CO
u
X
~> •
>t
O
rH
z —
LU
* 0
0
Z *N.
H-
X
0
^« — w
►— «
•>
» II
00
rH •
•
►— 1 — »
lO
vO
*
DC rH
•>
«A
CO
rH —
<
II vO
*0
2: 2:
UJ
•>
X
Lu +
rH
CO
r-t
II 21
_J
~> V.
^
— ■
a:
X
vO
LU -)
•
»
O
~) —
*
vO —
<-.
w X
u
vO
«
cr —
O
eg
II
-—
rH >
O
rH f\J LU
LU
n
•
>-
0 a:
u.
11
O
<x
CO >
#>
vO O O
Q
rH -.
lu
>
X
u u_
«
~)
O
rH
0
0 >-
CO
OvO«
»— 4
+ 51
0
X
(N
• LU
<t
z
rH
z
Q
*
O • r-
r-
X —
rH
•
— cc
•
O
z
»
1
fcA
>
v£> +
1
< X
6c
•>
X
-> 0
CO
CNJ
0
br>
V
<— 1 **
2: u_
LU
X
r-
— u
u_
O y*
rH
0
O
>-
0 ->
~)
II
CO
0
*
> •
CO
Q
■>
z
Z w
*
tr) » »
•>
X t»
2:
01
X
LL —
•
— .
r\j
z
2: a:
X
•-H r-l
•—1
<
3
X
X
CM
• rH
<-t
tr>
►— 1
rH
~-
* «J?
X
0 —
•a>
2: cr:
z
X
rH
m
— +
•
•>
—
u_
<H *
X
nO q.
0.
O
z
X
rH
«
QC
-> ~)
0
oc
~i
rH
^■^
II — •
*
2: - LU
LU
*» z
--H
r-
•>
X
O
•~ —,
u_
0
—
1
r— (
•— 1 rH
z
2: - 0
Q
•
2:
r-»
X
vO
z
X >-
f\J
z
CO
■r*
+
W=>
rH 1
-_
• ~> —>
w«
rH r-t
•-
^
.*
♦ — -
•>
u. u.
—
—
•>
II
^
O
CO ■— '
3
rH » II
sO
II CO
+ II
LU
CO
\
\
0 0
r-l
* •
1— 1
Oi
LU
rH
— 1
^—
(D
LU
CO
LU
0 —
LU
<
II •-< ~
rH
— LU CO
Q. 21
Z>
r\l
«_ VO
— »
0 0
II
CO <—
+
«—
3
II
•— •
O
Z
3
rsi
3
0 II
D
QC —
►"I—-)
>JD
o Dm
— .
*-*
z
(—
h—
z z
~)
rH
"">
h-
Z
•— 1
«
O
Z
Z
Z
^>
Z
Q r-
O. »
• z
r- rH
Z fM
— •
1—
< 1-
<
X X
r- +
•-"
<
1 — *
f-t
O
M
M4
O
»— 1
♦A 2:
t — »
1/)
vO UJ >-<
O
M ~ O
lO II
II CO
h-
Z
2: z
T.
CO CO
s
Z ~)
O 21
r-
O
z
O
^-
r-
h-
M>
r-
-J <
rH Q ~
^-
— r- r-
< 0l
a
Z
— >
Q£ -.
ac
* «
»— 4 ■*— i
0 cc
Z
~>
—
O
z
Z
rH X
Z
_l _J
-O — Q_
0. Z
_J — ■
-■ 0
O CC
0 a:
0
X X
O ct X
OOO
O
— .
u.
z
O
0 c
H X
O
< •
O Lu LU
O u
Z
Z (2k
u
a
u. a.
u_
rH f,
aa u. .
z
u_
u
Q
CO
1 — t
2
u 0 u :
u
U rH
a —* a
O
OUifl
i— t
r-i CM
rH
rsj
rH
rH CO
cm
<r
CO
-0
CO
r~-
0
r\j
rH
CO
^
O
rH
rH
r\j vO
f\J CO
CO
ro
CSJ
r-t
rH
rH
rH
CO
CO
O
O rH
45
oooo o ooo oo
cc^0^fNJ<^4'^>or^co<>Or-»(Nrn^-invOf*-oo<>0'-<fvJcO«i'ir\vOr,~co<>0'-<rgro
OOHHHHr-iHHHHHrgrsicMt>4r>j(NfV((MtvjP>J(^rri<^fOCi(0(^(fi(fi(ri^,4'^«t
ooooooooooooooooooooooooooooooo ooo
ooooooooooooooooooooooooooooooo ooo
ooooooooooooooooooooooooooooooo ooo
ooooooooooooooooooooooooooooooo ooo
1XJ
— <
o x
o X
r-H »
— X
u.
o
o
CO
CO
o
o
o
X
X
or
o
u
o
>-
>
X
X
<
or
LlI
z
— o
> o
in o
\ o
Z -H
o -
2 o
2 O
o o
<J Z
a
LU
2:
2
<
<
2:
—
— o
OT LU
lu 2:
>
a
a
Q
a
*
X
X
a
a
a
a.
«
>-
a
u
a
«
x
a
u
a.
<
<
<
0-
o
or
<
m
at
<
> t—
X <S)
u —
< •
• o
X
a.
a
CL
o
o
_J
CO
\
z
o
2:
2:
o
LU
or
o
LU
Q
lu
u
LU
LU
Q
a
> 1—
Q. —
o 2:
a »
vo
—~ H
U X
• I—
o a
_J LU
O U
O CL
rH Q
— a
a *
lu u
0 u
\ Q.
co o
_J CL
CD
< Z
u_ O
2;
2
O
U
o
X
X
o
00
O
O
O
o
ON
o
o
o —
- o •
V: _l r-t
Lu • 1— t
\ O
■4" O
^ -«
_l ~
co or
\ LU
Z X
O X
2: -
o
ao
O irt
O
O X
U X
o <
•- or
o
o z
e? —
o
o
rvj Z
11 11
O O
O O
z z
X ~
X <M
«- o
o
o
r-
O <A
in V
O Lu
rH »
• <
O
a.
LU
a
o
11
<
o
u
<
or
o
in
o
co
o
< O
11
o
LU •
OT CO
00 —
Z «i» CO Lu
O
•
O
II
LU
O
o
>-
X
X
U OT ~
>
o
2:
<
II
<
or
o
<
<
<
>
11
<
or
o
z
< X
Lu X
LU II
QT —
O <
U OT
II O
— z
< —
OT X
o
z t*
QT
Lu
LU
OT
o
X O Lu
X O —
ll _J
^ <
Lu V
X >^
X Lu
— II
< —
r- <
LU Q£
CO O
z
_l —
_) >^
< LU
U U-
o
z
II X
— h-
< a
or lu
O o
z 11
o <
o or
o o
o
o
Lu
O
o
CO
CO
o
o
o
— X
O X
o •
o or
O Lu
.— 1 LU
— or
o o
>-
>-
o
o
LU
X
X
II
<
or
o
X ~
1- or
0. Lu
LU X
<£> X
or
X — ~
X vO V
Z LU —
Z> • O
LU -> O
hvOH
CL • —
LU vO X
a — \
lu v
Z Z _J
•hO CO
H- — . S
Di^Z
o z o
or lu 2:
a co 2 2:
z 3 — o
anflou
O r-i 00
--H rH O
O
cnj ir» r~- vO o> co
OOOOOO
46
o o o o ooooooooooooo o o
oooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooo
*
■-■ ►-• » »<J-|l CO —
— >
<
— Q. — II «* II ■* »-~l|wvO
2:
X
OUJO— «-— — vOiT.-»0~
•>
X
+ Q+NQmQ- *Lnt«uj
z
•
*
Q
— +-^~-W^«rtOfNJ •~^»
X
f-
<— 1
h ~ (V LU - LU ^- 11 - 0 rg <f
0
r-
z
K
*rH »Wr-Htf»f\|— Q«- * *
—
a
>-
*«^
e)
^■^-^■rg »fO * -t tf» Q >fl *
U-
LU
>-
X
*
— LU •— * vO * vO * ~ w ^ — .
10
Q
0.
«
~
O HQ*-*-OH^Q-)
CO
*
a
LU
i~i
11 —. h — O — » O -* * * 11 —
<
z
Q
O
1
— 1-1— ->ii ~> 11 Otr>* -~q.
1
z
a
»—i
~>
^-t ^, rg — ■— — ^»t««— ~vO +
rsl
•
m
1—
—
• LU *Q.<nQ.<t'->Q-' »-~
«•
2:
X
»
n
4"W^* *+ *~)ll— PvlvO
u.
2:
X
u_
-—
^04 — — •cO^fO — —O— »
I— 1
*
a
u
~)
Q* Qw-mwain+Qifl
a:
Q
u.
—
W*t»— Q»Q* * — t#» —
fcA
0
Q
«
a
-^'•O^^^^Hin^Q
z
a
>
-)-)rH + — .— ~*— »^H II
>o
•
•
m
m»
«— «- »~HQr(j* am » —
•
X
>
_J
•
Q-Q.>^"OJ*ll • -^ tf) — vO vO
2:
<
a
LU
i—»
Q
+ + — •^--^^►_-.Q»- »
II
2:
u
O
+
•-^
— — QvO— CO— — '-) II QvO
z
*
a
•
or
ce
H H II - Q »QO • — II —
rg
^
*
X
>-
0
• • ~ Q II vO II + — ' m ~ Q
r-l
u.
X
11
*
0
*
a
-I
_i
— »
— — •^mQ<l"^"Q.ir» •-»
0
— .
0
u
UJ
r-l
QQf^rg *(A * » LU — r-i <J-
->
^4
CL
0
tf>
1
11 11 — «(\j~r\jmQQ- »
«A
•
oc
•
*
•— *
— Q<0- <N- — * & d <*
a.
0
V
1—
Q
•—
rHi-lW— O •OQ- — tA-
0
u
•
Q.
•-«
•-*
II
•> ».~Qy»<J-tft 11 ~"> <f- — Q
•
ID
a
2:
(X
— ■
0
Q
21
a.
•>
»
1— 1
+
1—4
QOin* »u • • 4c M »_i «o
II
II
O
CO
r-
X
«o
>-
r~-
0
J.
Otf»tf»~~*m~-4-Lr\-~Q— »
-*
•
*
a
•—
.— 1
>-
•>
*>
II
— r\iQ^ — ro— — i-i 11 o_m
«A
«
O
cr
0
u
X
II
h-
vO
<
lA
•-■* M«-hO «Q Q-~uj w
O 2
in
<
a
*
(—
a:
«
«
Z
— * — — u m 11 y»OinoQ
O
<N
«m
»
rO
•
0
a
>-
<M
tN
<0 — (MO — • CVJ + * ^ tf*
II
*
O
O
»
— -
_l
LU
sO
O
*A
Z
+ — 1 » + mCl<J* --JIN'"
a.
O
in
ae
0
a 0 y»
—
»-•
♦
~^r\j— ' • <A »* -^- — * >^-
^
r\J
tA
—«.
— <
0
•>
*
<!•
•»
0
1— 1
z
H O " IN H N H — — O* »~
•
rH
0 a.
rH
~-
0
m
^
_J
*
1
II
» + Q »— *— ~}LU«» — vO~}
tA
z
CO
Mi
O
0 •>
*
0
X
•
1
1
— 1
*!
HN
rg-~*Ainci*Q— ii-»— .— •
II
•-*
~>
O
r-* >- h~
0 u
0
X
r- 1
— «
— »
sO
11
II
— ~« — — fcfl — t^CL'-tNJ- Q1-1
vO
a
— -
«
O
-XiO
0 a
r— 1
m
+
r-l
H
*
-^
Z
O
Q •r-iQ^->-~* <l- »o ti-
•>
^
a 2:
O
< U -•
■H O
—
0 ce
1
1
rH
t— 1
ll m •>ll_>— -)<N — <f + ~Q_
1— 1
u
^^.
•
< • Q
>_ a
^
O X
21
z
II
ao
«A
O
-^^.C\l~ ♦Q. ► * LU — ~ v£) LU
II
&
1
O
1
< < •
Q. •
^
_i
11
51
z
1— <
O
Q
•-« Q — f\i»-<*'-<*t(»Q1n *o
21
X
II
• • 0
LU U
u.
■1
^
—
—
O Q
e— 1
• HO •» — - — ■ — ■ — - CO M • — <J" 4c
>d-
—.
tsl
0m,
Or- -J
Q U
u.
«-*
—
O
+
M
(\J— II mo_«-«CL'-i* -^ LU — c\i
m
m
u_
fcr>
2:
O *v *
>» a
\
0
tA
*
*
Q
M
_l
— r-l— — LU — LU — * m II Q*
2:
»— 4
•»
ZN<
CO O
-fr
0
•-»
—
II
<
Q •niQQOQO— • — «♦ *
O
•t
■^-
2:
• ^ a:
^ Q
*:
r-l
Q
O
0
tf)
O
<
21
w in »t«* +* + ~> ro in — > —
Q «Ni
z
tr> Q
— _» 0
_i a
_i
—
»~N
•
•
•
Z
•
rH
«af
Z *J
O
ouz
ao •
CQ
ex
a:
.— 1
rH
-£>
O
O
S
00 — ")«pn <r a Quj » —
LU
tr> •
Q
LU
• II
O >> *
\ >-
\
u_
0
1
1
#>
11
•
t^
11
II «A Q — — »— »*t«y»sOCL
z>
«NJ
•M
32:-
LL.
hZ<
Z X
Z
X
^
X
>
r— 1
•■»
O
~>
'•-rtQ.O^H't^ — — — +
z
rH
u_
z
M ~>
LO
- Q U.
0 a
O
X
X
X
>
II
~)
11
i-l
'H~)-»* + — +— — «(M>J-Q —
• — 1
r- •
lO
►— «
~> •
03
OZUJ
2: Q
21
•>
X
—
—
~J
«
—
•>-~)(M~Q-<Q- » »||vO
►—
II r-l
CO
►—
<t Q.
<
os a:
2: d
2:
u.
11
—
w
O
• — 1
►— *
-J
r-l
h a **c\JN^HOin^"-» —
z
2: rH
<
z
rH V
000
O Q.
O
X <x
IX.
U_
O
—
—
II
O
— * •-<*— -^— — + ^-^vOLU
0
a —
It
00 —
LL.
z u u
U *
V
X X
t— <
HH
a c
LU2
QQ — — - UJ (^lil -t - Q a » 1
w
z —
N4
u
Q Q
»-H
rH rH
1— 1
f-«
r~
^0
O
CD
HiMm^ifivor^aj^HCMm
0
rH
rH
rg
rH
fi rH
O
<N
47
o o oo oooo o o
o^cNic^-4-ir\vor-oo(>0'-tf\j(^^-invor»-oo(>o«-<fvic0>>4"U^vor-ao(>0'HCNirn^-ir\
G000C00000C0C0a000C0OO0,*OOO^0,<OOOOOOOOOOOO'— I i— 1 <— (rHrHi— l
HHHHHr-tHWHHHHHHHr-tHHHHf\JOjrg(MNN(\|(\IN(Mrg!M(NNOgN
oooooo oooooooooooooooooooooooooo oo
oooooo oooooooooooooooooooooooooo oo
oooooo
oooooo
oooooooooooooooooooooooooo
oooooooooooooooooooooooooo
o o
o o
co
o
o
I
CM
co
CO
*
CM
co
o
•
o
II
i/>
II
2:
a
— a.
Q _J
vO
II
~)
cO -)
co •>
o 2:
Q
& *
— o
U_ ~) —
«}
a. 3: '- —
I
II
II
O 2
• CO
o o
•>
rH fcf*
co
2: <
O I
— a
~H Q II
*C + II --
HOT
2; II —• •
*- r-i ►— Z
u. a < —
—• -lJ "X. Q
t#>
— v0
in
o
o
o
II
UJ
tA
vO — •
tf> 1—1
I
CM
<*•
o
Q
— 0.
z 3
w *-■
U — 1
* —
fer> ~ —
r-t • «-
+ -- II
UJ <
3 <- • >— 1 1— 1
Z II >— —
-. -. II Q-
cm
UJ
X
«
4-
00
•
o
UJ
Q.
>-
>-
X
m
X
co
•
CO
o
CM
UJ
a.
UJ
>-
>-
■I
X
X
•k
m
^M
M II Z
a
1-1 II ^
hJ)U
or
<
X
X
X
m
• X
X -*
co oc
(X
Z UJ
I/) —
co
m
•
CM
m
CM
*
*
vO
•— '
+
CM
*
*
X
X
m
u
+
>-
*
X
X
*
+
— < w
I- <
•
CO
vO
*
co r-
m o
• r—
o
o o
_l
fcfl
— <
CO \
•~ X
u
+
X
X
CM
fM
II
o
2:
CM _l
— 00
u
+
3
z z
on —
3 h-
Ui OC O
^ a u.
O Ui o
o at u
CM tft _l
-* I— tr>
X *
i— a: at
Q- < <
uj a. a
Cj II II
~- O >-
U IX
-- < KJ
X *
Z r-l
< vO
o
o
I
CM
II
o
X
X
*
in
tf> x — m
X
u
I
X
o X
in u
-J u
fcr» CM
+
CM >-
II >
o *
*
CM
II
>-
>-
a.
o
o
a
> —
>- m
— u
— CM
II
X
X
a.
Q
o
a.
— Z <t <t kA
kpl
X (A U U —
<J + + •*
• ^XUJCMrou>t
'HU.XZ) — — II 0
II i^U
21 CQ -
< II
vO -~ >-
OU.XOQOQO
0—<UUQ-aQ.O
D — —
z u u
t-H II II
I- X >
2 Q. Q.
O Q O
00
U.
u.
o
o
CO
CO
o
o
o
X
X
a:
u.
UJ
or
o
— o
^ o
U. rH
< >
I— o
u o
Z r-i
3 —
U. X
UJ ^
O -t cc
I- h- 3
Z r-
O O UJ
3
O
3
tQ
O
Si
2:
o
u
O rH
CO CO
CM CO ^" O
CO CO CO -3-
(M CO O
<t >* m
in
rH CO cm
in in in
in vO co
\Q vO *0
O rH 1^-
vO vO vO
CM
<t rH
^) 00
48
o
O
000 0
O
O
O
O
O
O
O
O
sO r-
oo o
0
h rg en •*■ ift nO
l"»
00 a
1 c
1 rH
c\
1 m
<*
lT»
vC
1 r- 00
c*
0
rH
(M
i m
<*
lf\
g3
r- c
O
O r-i
rH rH
i-l t-H
CM
rg rg rg rg rg rg
rg
rg c\
1 fr
1 m
cr
1 m
rr
1 CO
CO
1 co ro
m
^*-
^
<t
-*
^
4" 4" «♦ ■*
«*
\c\ tn
rg f\j
rs
1 rg
rg
rg rg rg rg rg rg
rg
rg c\
1 f\
1 rg
rsj
1 rg
t\
1 rg
fM
t rg rg
(M
rg
rg
rg
rg
r\
1 rg
rg
rg rg
rg
rg rg
o o
o o
0000000
O
O
O O
O
■ O
O
1 O
c
1 O 0
0
O
O O
O
c
1 O
O
O O
O
O O
O O
o o
0
OOOOOO
O
O
O
1 O
O
O
O O
0
1 O O
0
O
O
0
O
c
1 0 O
OOO
O O
O O
o o
0
G
1 0 0 0 0 0
O
O
O
O
O
O
O
O
0
O O
O
O
O
O
O
c
1 O
O
OOO
O
1 O
O O
o
1 O
0
c
• 0 0 0 0 0
O
O
O
O
O
O
0
O
0
O O
0
O
O
0
O
c
O
O
O O
O
c
1 O
<
0
•
•
X
*
— •
<
•
X
*
O
X
O
4>
«
or
O
X
11
a
>
X
<
rH
•
*
OC
r-
co
—
X
h-
<
_i
_J
0.
*
^
r-
Z
GO
UJ
UJ
UJ
>-
r»
u.
a
>-
•-<
u
0
O
o
>
*
u.
UJ
>-
X
u.
11
+
*
a
*
•
0
0.
•>
X
>■
z
Q
i-t
^»
#-
Q
UJ
*r>
11
z
0
or
0
z
O
0
_J
>-
•
a
11
0
z
a
l-H
•
UJ
>-
2
*
<f
rH
•>
•>
1—
O
a
2
X
oc
*
x
X
*
11
fer>
•»
X
CO
X
X
u.
^
«r>
a:
a.
<A
w
»
0.
u
^
X
o
a
CO
ct
Q
u.
U.
<
Z
Q
•*
»
O
Q
•
_J
•>
(X
*
—
z
CL
>-
«r>
-1
UJ
X
«
*
0
•>
•
UJ
0
<
>
ce <t
0
X
>-
_l
<
a
+
2
a
<
<-\
<
CL
UJ
N
*
X
»
u
co 0
M
3:
u
Q
<
m
II
>
a.
* h-
0
•>
CL
*
<
•
X
O
>
*
in
X
V
»
X
+
X
O
•
X
* O
X
u.
X
tr>
<
Z
Q
Q.
* O
»
•
a
_J
11
tA
— »
f^
U
rH •
-~
a
u
UJ
-~
or
O
Q£
a
or h- >- t*
1-^
CL
0
<
<
M
rH
O
*
II ^ X
cc
«
m
■^r
co
cr:
•
*
>-
m<fu >
rH
0
>■
»—
u.
<
<
00
UJ
a
OS CC \ X
— »
*
a
•— ^
z
CO
CO
2
Q
* — u
cr:
UJ
0
2:
1—1
-A
<
Mi
•— i
0.
br> • vt U "V
u_
2:
CL
»
10
«M
0
t—
•
00 cc a —
UJ
•— 4
•>
—
4c
<
LL
r-
*
X
rg + * Q ~
cr:
r-
X
0
a
z
cr:
CL
* • • CL <
O
•
a
— rH
_i
_J
*— 1
O
<
Q
* lTN 00 >. ~
u
a:
0
U X
UJ
UJ CO
O
CO
a
QJ ^ + >- U.
•
<
a.
• f—
a
3
41
m
«
< rO ro CL 1/)
— »
CO
»
O CL
+
+
0
«f>
OL
—»
CO or Q£ Q 0
0
•
—
_l UJ
>-
<
<
0
* * * a kj
0
cr:
0
O Q
11
II
_J
— a.
0
m • • 11 *
T-\
-~
<
0
• »
>-
<
UJ
o »
rH
* vO »o > >
«—
O
a
rH
— (3
>-
<
0
o >-
r-
#-
* + + Q. a
«■»
>
O
*
•>
O X
11
— i X
CO
0
u
r-4 • (M U U
>•
*
r— t
>
r-
O
KJ
O X
»f>
*A
>-
mm
*- u
•— 1
0
CL
Cd co q£ a Q.
•
*~
—
X
<n 0
a
r-t •>
>■
< •
a
rH
Q
II s* 1
X
O
<
u
>— •
rH
a
- O
r-
a
_i
>-
< 13
«
*"»
a
rg rg • t* -.
»-
O
<
«
0
—
CL
i^ O
— ■
<
UJ
r-
»
< 3
o
0.
■1
cr: o: >t <
UJ
rH
<
<
•
CL
M
^ _l
<
0
«—
_i
a
»—4
X
•> »
-J
UJ
u
* + 0 —
>
»~
«
•
0
UJ
Vj
U_ •
CL
<f-
>-
_J
u_
UJ
2
X
O »-
•>
Q
u
0
fcr> IHO.U.
O 3
O
r-
_i
Q
u
u. -»
O
•
X
«-»
10
a
«a
o x
<
•V
CL
-J
<f itQZ
5:
\
O
V.
»
\
CL
^ 0
z
r-4
u
rH
0
O
*
•O
z
Z fNJ
cl
CO
0
— 1
CNJ + * CL —
r-i
z
rg
<
m
Q
<t 0
^-r
•
41
rg
u
en
cc
«-.
•1
D
• *:
o
^
0
vO
• ■— t • v» (_0
UJ
*:
■1
\i
or
V
Q
^ rH
^
rH
UJ
•
♦
•
<
Ct 1
— <
u.
— _j
z
_J
CL
«
rg Cr: rg X *
z
_i
«-.
_l
0
_)
CL
_l —
^
—«
2
rg
0
rH
CO
< 1
D
1—
O CO
*
CO
*
uf\
«mh -ax
•— <
CO 0
CO
z
CO
■1
co a:
u.
1— 1
\L
i-h
rg
_l
II
^
CO 1
— (
a
O v.
<
\
>-
-'
>* • II O CL
1—
^
0
\
•
\
>-
V> u.
u.
1
u_
t—
—
UJ '
sO
1—
ti-
< ■
UJ
-i z
u_
z
X
>- rg u a u
Z
3
z
rH
Z
<
z
X
z x
II
X
II
11
0
r— •
ll
w
0
>- o
UJ o
CL
O
x 11 u 11 a
•
cr:
O O
—
O
LL
0
CL
O X
CL
<
cr:
1— 1
O
O
+ 00
<
U. 1
D
o 2
QC
2
Q
1—
u u a x —
Z)
azotius
OS*
<
5:
<
11
r-
X r-
rg
(-0 1
_i
o 2
o 2:
0
II Q. Q CL II
II
r- O
cozozas:
d 2: U.
CD
~-
CO
i-
_l
II
_l
0
_i
o o
u 0
CL
O
rH 0 0 u ^r
^
UJ z
D O O O
O O
CL
O X
^
u.
*i
r— 4
ujoxooujuo
<
z u
*
u
•>
ID ce Q. CL Q. u. 1
j_
en uu
1/)
u z
u
u
lor
«
KO X
u.
1— t
U. 2
0 0
X IT
£> O
0
u
rH
r— 1
1— (
m
vO 1
r\
4-
'
r-H
rH
r-t
rH
rH
<r
eg
rg
rH
rg
o>
rH
rH
rH
O
49
o
o
o
O
o
o
o
o
o o
rg m
>*
m
n0 r-
00
t> o
r-t f\l CO «^
in
o
r-
00 O
o <-*
rg
m
nJ- iT\
g3 r-
oo ^
o
rH C\
i <r\ *? tr» gj r-
in u>
iT\
in
m m
iTi
\f\ v0
sO v0 vO gb
vO
o
o
^0 gj)
r- r-
r»
r-
r- r»
r~ r~
p» r»
oo
oo oe
i 00 00 00 CO 00
f\i rg
(N
<\J
rg rg
<N
rg rg
rg rg (\
I rg
rg
rg
rg
rg rg
rg rg
rg
rg
rg rg
rg rg
rg rg
rg
r\
I rsi
i rg rg rg rg rg
o o
O
o
o o
o
o o
o o C
i O
O
O
O
a
o o
o
O
o o
o o
o o
O
o o o o o o o
o o
O
o
o o
o
o o
o o o o
O
O
O
o
o o
o
o
o o
o o
c
• o
O
o o o o o o o
o o
O
o
o o
o
o o o o o o
O
O
O
o
o o
O
o
o o
o o
o o o
c
i o o o o o o
o o
O
o
o o
o
o o
o o o o
O
O
O
o
o o
O
o
o o
o o
c
i o
O
c
i o
O O O O O
•>
*
<
o
X
o
X
*
l—t
•
*
•
•
w
X
r-
l-t
rg
^
h-
z
II
1
Ll
Q.
>-
1— <
CO
X
Ll
UJ
>-
X
I
X
»
O
a.
•
X
a
^«
•>
Q
LU
u
o
z
o
a
(A
u
o
z
a.
»— <
a
r-i
•
»
i—
•
*
*
£
X
■h
i— t
rg
CO
£
X
LL
II
*
—
*
CL
U
CO
*
CO
OH
Q
LL
LL
-«
•
o
Q
•
LU
<
—
z
a
>-
cr
— »
o
»
•h
O
LL >-
O
X
>-
-I
u
Z X
i—t
<
a.
LU
— u
w
2:
u
a
<A
tO \
o
•>
a
*
»— — .
c*
X
^
•h
X
-~
II >-
«-»
r-t
X
LL
X
<
o >-
^
•
*
a
_l
Q£
Q.
\£
o
-»
O
u
LU
O
tf» u
ll
1—1
h-
or
0.
z
«
a.
*:
»
o
ft
o
V
i—
rg
> *
u_
o
00
o
—
*
a.
i— •
—
X rg
~
CNJ
rH
o:
UJ
o
21
CO
U *
*
•
»
«*>
Ll
s:
CL
•
II
v. + ~-
lT>
vO
00
LU
^-H
«
— .
-~
— — rg
•
•
r-H
rg
Q£
1—
X
vO
<
>- < *
11
sO
—
ii
O
•>
0.
>— r-i
(X
a. — *
a:
r-
rH
i—
\£
u
a:
o
U X
o
-~
uu. -
<
r-t
o
•—I
^
■t
<
0-
» H-
z
u
u u a. to >-
en
o
o
s:
LL
— .
CO
m
O 0l
ik
u
u u * o
^
o
o
II
o
•s
~*
_l LU
m
a
a a — u _i
u.
o
o
o
•
br)
Ll.
^
o
rH —
cr:
<
o
o
CO
QQO< + UJ
Q Q Q — >- O
w
•
1
1
a
«A
>-
>-
— O
>- o
a.
rH
•
— o
O X
<A
a
+
a. a ll x w
+ + z a +
•~
"■>
0£
v:
•
» ft
>-
I—
o u
O X
u
U U >- U rg
*:
oc
<
^
<
X
~- >—
X
to
o a.
rH •
•
a
a. o. to u *
*£
<
33
Ll.
<
X
o <
u
►— 1
rH Q
— a
rH
a
q o i a *
ll
CD
i/.
+
H
or
— »
o <
*
a
— a
^ o
II
a
0. CL x * —
» t>
t> i^
ll
or oo
<
<
<
rH <
<
•
0. •
*£ _J
<
Ml
— ~ CL — X
< •
o
• LL
1 •-«
<
CO
1—
%* «
*
o
LU u
Ll •
X
V.
>* N. U <
< r-
*
O 1
Q. II
CO o
*r>
*:
LU
x o
1—
-I
Q u
LL —
O J
X
>• >- CL — _l
— m
r~
(M 0.
CL O
^ I'-
Li.
CO
"v (JJ
\
•>
v. a
X o
_l
X
>- X * Ll LU
i— •
a:
■
• u.
*: O
LL.
V
N
rH Z
rg
<
en o
<J" o
. —
tf>
CL
CL a — tO Q
u m
<
r-
o ^ or
*: z
— o >-
-^
LU
i^ •
U
or
^ Q
*: rH
gD
Q Q Q < O <-.
z —
CQ
—
rg ll
<
u. t*
* o
II
<
z
_l —
_l
o
_i a
_l —
•
•
Q Q Q — U —
Z> <n
v:
.— i
r~ — .
CO
«—
oo m
>-
UJ
ac
r^
CO o
CO Z
CO •
CO ct
m
rH
0.
0. CL LL * Ll
ll <-•
Ll_
I
rH U_
^
Ll
<n • *r»
z> o
1—
N. O
\
•
\ >-
\ Ll
—
II
O
r-
II
II II tO — ■ 1—
V 1
II
X
1 to
u_
to
ii
*r>
z
z
Z
3
Z rH
z
<
Z X
Z X
<
•
X
>- > O < CK
h-
a
<
1— CO
II
CO ro
OK M
•— «
—
Cl
o
O-
O
Ll
o a.
O X O LL .
— t
o
X
> x u — o
_J •—
a
:e
-. <
0.
< II
1— < H
X
r-
^
D
ct
z o s:
LU
s: o s »
i—
LU
ll
r-
a.
Q. CL LL tO
_l ~-
^
~~
~- w
^
«~ I'-
CO o X
z
^
1— i
Q CO
r o
S. cc
s: q
T. LL
Ct 1
-L
u
U U — Z II
< u_
^
u_
Ll u.
^
ll _•
o ^ o
II
o
LL LLl Z 3
o o
o
O
O 0-
O X
o c
X
o
u
U U II ""I Q
u —
u_
•— i
>— 1 t— *
u.
— 2:
t£> LL. Z
X
LL!
u. ac 1
jj i/)
OZ
u
U
o *>
U X
ii?
u :
r
t£»
a.
CL CL CL tO Q
r- 1
r-l
rH
1—t
rH
o
r»
lA
r- o
o
>* r~
CO vO
CO
CO
m
vO
r— t
m
>-\J
rg r-t
rH
m
50
o o o o
ooo* OHfgm ^
00 CO o & o &> &>
rg eg f\i cm rg cm eg
oo o o o o o
o o o o o o o
o o o o o o o
o o o o o o o
o o o
OOOOOOOOOOOOOOOr- IrHrHrHrHrHrHrHrHi-irgegegeg
o oooooooooooooooooooooooooo
o oooooooooooooooooooooooooo
o oooooooooooooooooooooooooo
o oooooooooooooooooooooooooo
u
u
*
h-
m
«
rg
or
f>
o
— .
•
z
—»
f>
»
o
rg
<
*
1 •
or
rg
rg (X
o
*
* O
z
*
* Z
*
a
O •
u.
o
u <
X
*
u a:
X
•
* o
•
rg
rg Z
V-
1
rH •
*
•
r-i LU
ex
>*
ao X
<
w
• X
a
*
oo •
•
«-»
rg k-
o
-~
<
+ _J •
o
-»
IX
ooq:
_l
rg
O
U X <
•
*
z
U tO Q.
X
*
—
•
* X •
r-
rg
•
rg
o a
GL
•
rH
M
m * 3
UJ
rg
+
ao
— •
•h _J
O
en
—.
+
—»
co < •
«
—
w
_J
—»
-~
• Lu x
LU
O
*
— .
— o
•
<
N tU K
Z
O
O
O m x
rg
1
o
hq:q.
i o lu
1—
f— 1
•
ex
* X
Lu
LU
Q
z
o u a
•>
u_
~^
N
-» ^
U
Q
•
i-» M •
>-
u.
\
X
rg Q
or ~
*
rg
n < lu
X
u
—>
r-
r-
+ <
CL
— ■
rg X X
u
or
m —
Q.
LU _l
— or
— -
35
• I —
•
r-
* <
LU
X LU
< o
*
—
CO 1-
<
•»
* ct
a
1- O.
ex z
~
U.
II IA •
X
»»
in
rg o
♦
\ *
o -
<
tO
_l >
X
O
rH
On Z
in
_l <
Z Lu
tx
CO
o <t x
—
o
h-
in —
r-l
O H-
— Lu
o
<
X * u
h-
1—1
•k
—1 LL
m
X LU
LU U
z
— '
i/) * •
u
w
rH
^- u.
CO
IS) X
LU or
■k
u_
X o <
~
•—4
—
u.
o
r-« U
a
rg
X r-
<J r-
r— (
r~
U X
_l
(X
_J
u_
>
r-
• 1—
u.
•
— *
CC 1
—
(X
** u x o
Lu
o
u
X
♦.
CO *
r-
vO
— a:
r- •
OQ
o
* ~
X
X
i—
u
in
or
o
«- r-
\
«—
* a.
II rH
1—
~-
00
a un
I/)
l/)
a:
*
.-»
O O
o
•
* z
o
o
Ii.
-* <
U-
~ II
z
H —
N»
< HH
X
LU
X
<
LU
r»
z •
•
o
o -«
u
X
X
i
U.
< —
l—l
— Q
•
0.00 0
••
z
»
z
a
s:
-»
•> r-l
t-H
II
• X
r-
u
X
z
LU —
VJ
or <
X
< a
rH
S vO —<
h-
"— •
1—
o
"S,
— i
rH
rH II
II
—
LU
<J-
>* *
*
*
*
t— 1
X v.
1
O ex
\
a *
It
>- • IX
Z
r-
z
►-M
>
i—
1
II -~
'—
>— 1
D
*
o <
ct
o u.
CO
1— o
•
z o
o
oa
<
X HU.
1-^
z
Z>
•—I
</>
X
ii
o
•— 1 l—l
»— i
—
z
*
II X <
-I
u
II
II •
r-4
— z
X z
z +
u_
U 'H
X
DT
O
X
z
u o a
-»
^*
u.
I— I
r-
r-l I Q.
ex
u.
rg
< rH
II
U. «-
X m or
• •
LU
II _l
*
3
or
•
LU
ii
_l
CO u.
u_
u.
r-
II
r- II
II
II
ii
1—
I— M
u.
Lu LU
II II 3
on rg
ex
o en _J
u_
h-
o. co
Lu
S
o
_i
«—
u,
LL.
u
z
a
LU o o
o a: lu lu u.
Lu
U LU
LU O r- O
—. _•
o u * <
u
UJ
Z Z3
u
»-M
u
LU
u.
o v
u
(X
o
u_
X X
_J
_i
a
X
X U_
U
or u
iomz
CO \
u
u * u
Lu
(X
LU i/>
Li.
o
U <Z
»— •
Q r-
t—
r-
vy
r-
h- X Qi
a
<
r-
r- LJ
a:
r- 1—
x _i or lu
^-«
rH
H
r-
.— i
H
•-I
o
ao
51
■-ic\ic^,3-inv©r-ooo
f-t(\jcn-3-tr\vOf,~oo^Oi-«f\icri«$-ir*vor— ooct^
f-t cm en ^- m >o
oooooooo
oooooooo
U. U. U. U. U. U. U. Ll
xxxxxxxx
OrHrHr^t^t^rHrHi^rHi^cMCMCMCMCMCMcMCMOsjcMcococococnrnro
oooooooooooooooooooooooooooo
LuIlLlU_LlU-LlU-U.L_IuU.L_U-LuU_IlLlU-IuLlU-U.LcIlU.U.U_
XXXXXXXIXXXXXXXXIXXXXXXXXXXX
zzzzzzzzzzzzzzzzzzzzzzzzzzzz
i/)i/)t/)(/)t/)i/)l/)iOi/)i/)(/)(/)»/)t/5i/)i/)i/)(/)t/)i/)(/)i/)i/)i/)l/)i/)(/)iO(/)tOtOi/)i/)i/H/)i/)
to
to
UJ
QC
Q
a
X
X
►—
<
z
•£.
TL
LlI
z
o
Z)
D
X
<x
□£
5:
S.
3
D
UJ
•— <
►— <
O
r-
M
X
Z
(X.
LU
<
"— 1
<
q:
Lu
s:
X
LU
LU
^
^
a
a:
(—
u
u
o
o
<— i
LU
LU
H-
(—
X
X
X
lO
tO
LU
u
u
X
tO
in
r~-
m
h-
r-
fM
O
i-H
r-
o CO (\|
CD N ^
u. o • m
X • o o
z o o o
— i O If- o
in -3" — «
CM CM rH
N OO
<-* <t r-
m ^- r-
vO
m
o
to r>-
r- co vO in
O vO o o
m ^- vo
cm in m
in ^- r>-
cnj m co
o r- o
O I- o
(M
vO
in
m
m
r-
o
o
o m
CM f\J (NJ rH CM CM
o
I
m Li.
X X
z z
A •—'' *-~<
* to to
O CM t0
ro CO QC Qi 2 cm
x x o ■-« ONUjmimaa.
z zacracMara-t-azarEX
— ■*-«xa:x + ccxtox'-'XLULu
LO*tOLULUUJ*LULUM LUlOLUr-t—
3:
t— LU >
zi-a:
LU tO t— U U
r-r-r-hr-r-r-UU")
Q>ZliialUUUUUUUlilUJJ
3 Z> < tO D
-•iOLUQQOOOOOOOOOtom_J_l<tO— •
_i<Q.i^-)io-)->><aD_jQ(r)<
--Q~1X-JX_l-JQr-t0<<t0r-
tO_l<r-tOr-tOtOLLLOLi_LOli-Lt_lO
Ll.
HfMfn^iAvO^CO U. I
aQ.o_aaaa.acLLu_iz
xxxxxxxxxz<— ■
LULULULULULULUUJLUOX-0
X
z
tO
52
p- ao 0> i-Hrgco^ir^vOf^cOCT* rHrgm^-m^r^coo i-Hr\irn^tn\Or"-co<> •-< r\j
oooooooooooooooooooooooooooooooooooo
U,U,t±.U.U>U.U,U_U-U.li-ll-U_U-U_U.U-U-U-U-U.U-U-U_U.U.U.U_U.U-U-U-U.U_U.U_
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
ZZ ZZ Z ZZZ ZZ ZZZZ Z Z2 ZZZZZZZZ ZZZ Z ZZ ZZZZZ
i/H/)i/)(/)(/)i/)i/)i/)iOi/)(/)i/)i/)i/)i/)i/)i/)(/)*/)t/)tOi/)t/)i/)i/)^i/)i/)i/)iOi/)t/)i/)i/)i/)ifl
Q, LU
UJ i/)
t— II
r-
X
UJ
CM
a
UJ
I—
iO
II
r\l
vo a.
CL 2:
X UJ
0.2:0.2:2:2:2:2:2:
XLUXUJUJLiJUJUJUJ
UJI— l±J»— I— (— »— ^-h-
co or
o O
rH o a:
+ HQOK
* I C\J CO UJ
a a 00 a.
2: uj 2: a. 2:
uj z uj x uj
i- O 1— UJ H-
cn
U. X
_l z
< —
X <y)
CO Z) to
O O I-
O — 1 Z UJ >0
a: o — « o ir>
ct ao <s> 00 <t-
UJ Oh O h
f\j
CO
<o
3<Q<OQDQ<Q<<CO
ZK<hQQ<I<h<hQifl
lLl/)lL(/)JlLll.lLlLl/)ll.iOJU.
>tO<Q.<<tOCL<<>
Q0iZ-)ZZ_J-)^OO
QNJw<MwJ<l/)JH
2: Q 3 -> —
U<Z J2
tO U. U. tO UJ 1/) -J
a -j q _»
Q < U < «-»
ucoum
X
tO
CO cc
X O
z a:
•- or
to UJ
CC
O
C£
ct
UJ
53
m .4-
o o
li- li_
X X
o o
o o
o o
o o
o o
o o
o o
o o
m so
o o
o o
o o
o o
o o o
MO^OHNm4tf\\OMoo>OH(\ifn^invONooo<OHMw
OOOHHHHHHHi-tHr-l(\JN(NJ(NJf\JN(\JC\((\J(\jmfOf'l(«>
ooooooooooooooooooooooooooo
ooooooooooooooooooooooooooo
ooooooooooooooooooooooooooo
^ to
o
o
o
il
00
or
LU
Z 00
CD
o
II
CD
<*
U. ~> z
X • z
Z X »
-- z —
t0 — 00
o. —
u z —
co z x
•- z
z —
Z O U-
O •"-• to
Q — tO CD
m Z h- Z <
sO LU U LU X
Z> — h-
U.QZ
o
II ~
O CO
LU «
o
o
o s:
II X
OT II
LU 0.
*- z
l/>
trt LU
u
<
_J
co a.
o
f\i »-
O X
r\J O
o -•
f\l LU
o
r\J
o
CM
o
f\l
o
CM
MflO
in in r\j
• * II
hhQi/)
in m U — '
CD
<
X
UJ
QC
o
21
O .-•
LU LU
vO
in
I — t — (Ni
-> -) u
•~ ~ Q£
li- U- LU
•— < •— • >— I o
LU
(NJ ro X
in in u
o
o
ON
on
o
ON
ON ON
On On
• On
I- On
O ON
• ON
fNJ
ON
On
On
o
On
On
On On
I On
a
LU
U
X
LU
z
<
-I
CO
• — <
z
LU
^)
ao
*
LU
r- i—
U. X <-l X
^ u_
— lD — li
-, Z HH Z
.-i r-
On — I O
On U I—
Ct
LU O
i— i if)
X
z
— II
.-H Z
I It
« z
or lu
— Z ►-" U.
o
or
«- u_ lu r- u
u. ■— fsl * z
O. U. i-i < CO •
O I— in tO II _i fNJ r-»r-l
£ZNmmU]h rvi il
«-• ll •-> il il — •
■-•«■* n Oh-o >r^O
-h ~)Zl— D^—iaoili— ro
on — lu a «■» — _i
O M Zi-OZOZZCOOO
a")ZLnD«QZZTl£)Q
ao
ii
ao LU
Z
<
_l
II
3
Z
in cm
x\
U. vO o
— -4-
m
CM
— Z
z o
z u
u
u
54
m
in
^■mvor-aoo^O'-^osjm^'mxor^oooO'-'rgcn
oooooooo oooooooooo
OOOOOOOO 0000000,000
oooooooo oooooooooo
CO
*
—t
II
o
oo O
II —■
r-t * *L
I —
•-• t-t z
II 4" Z
CO
~)
II
_J
~i co r~
CsJ ~) CM
* *
o o
— >0
O CO <N
• O —
vf -J
II CO
II 3 • — ->
— Z O _J I
^ — i uj co a
5^ h- Z ~) Z
CO
• o
O I-
m
o
o
UJ LU
3 •
o
UJ
fcA
a
z
II
— Z ll
roou-ZOu.zu.au.->
_i +
CO
~)
II
CO
*
z
N
o
ft
^ z
— z
z -> II
z * ~
-. ^ ^
~) H I
i^ ^ 00
I +
^ <!• V
«~ en «-
00
f\J r-l
o <h r~- >o o <-* (\j
rsj r\l cnj cm m co m
z
z
en
Z UJ
•4-
m
z
z
o
u
CO
o -»
m •— i
• cc
oo oo
UJ I—
o <
o z
UaQQ
z o z z
Li. UJ UJ
o o
55
APPENDIX 2
Sample Computer Input and Output
Sample Input
52 56
440,
, 460,
515.
505.
510.
, 470.
425.
355.
300.
330.
370.
360.
340,
, 345,
, 350,
365.
375,
i 390.
455.
560.
700.
850.
950.
865.
550,
, 490.
. 470.
460.
415.
, 380.
335.
400.
460.
520.
560.
595.
510,
, 460,
, 400.
315.
275.
, 300.
220.
140.
80.
55.
53.
51.
360,
, 405,
, 450.
435.
445,
i 380.
330,
i 270.
235.
250.
280.
280.
290,
, 297,
, 300.
305.
315.
■ 360.
420,
, 490.
655.
810.
930.
870.
540,
, 500,
, 515.
515,
480.
, 445,
i 440.
. 500.
540.
593.
630.
630.
610,
, 560,
, 515.
400,
, 315,
. 275,
, 2 34,
, 190.
110.
63.
53.
50.
295,
, 370,
. 370,
, 325.
, 325,
. 307,
, 260,
, 190.
, 170.
, 175.
190.
190.
240,
, 250,
, 250,
, 265,
, 275,
. 305,
, 405,
i 470,
, 570.
750.
900.
885.
560,
, 540,
. 550.
, 560,
• 550,
. 540,
, 550,
t 605,
, 620,
, 640.
610.
, 570.
540,
, 590,
■ 570.
, 430,
. 300,
, 310,
, 280,
i 240,
, 170.
70,
55.
50.
240.
250.
235.
195.
225.
200.
190.
160.
130.
140.
160.
170.
190.
195.
210.
230.
250.
280.
350.
427.
490.
670.
820.
860.
610.
590.
590.
615.
600.
610.
620.
660.
650.
620.
560.
505.
480.
520.
560.
515.
405.
370.
320.
260.
190.
80.
60.
49.
150.
195.
200.
110.
125.
120.
115.
95.
80.
98.
105.
110.
162.
170.
180.
205.
210.
210.
290.
380.
455.
580.
740.
785.
660.
650.
640.
660.
660.
670.
690.
685.
650.
585.
510.
480.
450.
490.
550.
525.
450.
380.
325.
270.
215.
120.
65.
50.
80.
80.
90.
75.
73.
70.
72.
70.
70.
70.
72.
80.
110.
120.
140.
160.
175.
195.
240.
290.
390.
475.
610.
670.
760.
720.
715.
725.
730.
710.
680.
655.
620.
540.
480.
445.
440.
410.
550.
490.
415.
370.
325.
285.
235.
170.
85.
65.
61.
60*
60.
58.
60.
60.
61.
63.
60.
60.
65.
65.
80.
85.
100.
120.
130.
120.
160.
230.
280.
380.
460.
540.
770.
750.
735.
715.
665.
635.
620.
610.
570.
485.
440.
415.
425.
490.
550.
500.
445.
375.
315.
295.
245.
190.
110.
80.
47.
47.
48.
49.
52.
54.
53.
54.
56.
59.
60.
60.
63.
65.
70.
77.
82.
85.
110.
150.
185.
250.
340.
395.
560.
610.
590.
600.
570.
540.
525.
520.
500.
460.
410.
380.
420.
510.
515.
500.
465.
410.
283.
265.
235.
210.
145.
105.
4500. 2 . .01 0. 5.0 18.0 10.0 10 30.0 1500.0 1500.0
4500. 2 . .01 6. 5.0 18.0 10.0 10 30.0 1500.0 1500.0
56
o
o
z
I
a.
l j-
o
M J -J ' vj \J
ff « » ^ ^
(M H 1ft H >
<\j »r -o in -x>
prf »4 H N ^
lo o o o o
xo o o o o
o o o o o
o
(J
z
M
•\j
rg
iM
i\J
X
t—
Q.
LU
p-
•e
o-
ff
Q •
•
•
•
#
m
-o
00
LP.
.-4
-t
ff
in
vO
—*
>f
O
pg
.-*
■f
— «
-H
rg
rr\
2£
LL
XvO
o
o
O
o
I-«
o
o
O
o
to
( )
o
O
^j
ff
o
O
Z^
X
*—
a.
a
-vj r\| rg fM 'N
IT -r.
O H CO (O M
J H O S ^
r*> nj >r it r>-
— « pg
XO ■* ■£> O O
x ^ o> o o
PJ *> f"> C~l
o
IS
Z
rg 'M -g 'M
a.
a
r»- o --t * LO
m p- I— in oo
pn #<\ p^ QD gD
XO o o o o
I o o
o o
XO o o o o
xo o ~> o o
o o o o o
•. ' r> o <■*> o
x-o o o o o
x-* o o o o
pp> O O C O
ff ,-> <"1 "* '"1
<1 _, _4 _i ,h
XV3 ■* -O O O
x>r *-• ff o o
X PM <*> O O
ff ff ff o o
x<n « oo o o
xao ff i«i o o
oo o o o o
o a> ■"*> '" <~>
aco o o o o
Ot O o o o
oco o o o o
OO o o o o
U ) •-> O • o
oroc o o o o
opp o o o o
jj
eeff w oo o o
o^ o f\ r> o
pn -j ff p-
>ff -t ao f
(fl *^ (\J N N
s :T> ff o -4
I — rvi — < ct- I—
vx »r ff <-p a
r- -o J- m — <
* <T <* ■* •*
p- ao ff . 5 — i
00 >0 ff O ff
» • • • • •
>-^- M N * 53
•c so in in in
P- 00 ff O — *
in fM ff Pg O
» • • • • •
>or (M o gj- o>
*- -o >f -»^ ^
,0 fw o r- r-
N e 0> ii m
w-ir
J3
-4 a> >»■ o r^
<m ■*■ r^ o (\i
tt\ ^^ O 0C' <o
(\| (Nj ^< _< — 1
00 -4 iTI
in * m ff i/i
<\j in ao o c^
rvi o oo r- it
<r\ rf <\j rs, im
in ^- >o *o 0^
u in j h ^
<~i o* ^ o -t
■4" ft m r«i f<i
o r«- ^ o ■*
c • • • • •
<f P CT^ g? CNi
rg O <e «M in
(7- oo -o m <o
<* •*■ •♦ <* •*
c
o
uuo
Ol\j INJ (%> (M
X
•-0-
uupg ao
3— oo
f> oo
— ,^ ^ ^, r\
0<N (Ni (\J l\| (NJ
2
l— <M >0 O <f •*
a
aim <o »^ ^- >»■
ac> -4- ao m a*
>*" (7s ~* nO in
—i —i (M
0<M fM rg pg Pg
•— »*■ SCOW
Q. • • • • •
Luc- in in n p
Oin oc «< p-i m
PO ro j- r>4 o
Opg f\> pg rg rg
>-fM f\, »o -J- -4?
a. • • • • •
luO o in r>- <f
Oin o o 7 N
r**\ ?** (*\ ro wk
oro
o
c
o
-^
U.O
c
r^
o
c-
iO
o
o
o
o
X<-.
c
o
,
r^
j
c
n
O
OO
O
O
o
O
XO
o
o
o
<~>
xo
o
<->
o
c;
<
Li-<-'
*~>
—^
UJO
O
o
o
■3
QIS.,
<->
\—'
o
<_>
ac-
l_-
o
o
c*
arir. f. O o o
ulct o o o o
X-h o _) o o
x 3> c. o a <r
m o o o o
Off o o o o
x^ o o o o
X ff o o o o
<
u. -■> T- • o •..'
LUO o o o o
OCO o o o o
CO o o c o
U_gD M ff O (-■
X<_) —i IM O O
x~- ff ff o r>
r*°* O f^ O O
oo pg ff o o
Xini — i og o o
Iff ff ff o o
u.g3 o o ~> c
Lum o o o o
ac-^ o o o o
ao o c o o
o
orff o o o o
LLP- f» O
I-* —t «_1
Iff ff o
pp P" pg o O
O-* P- — I P- o
!<»• pg O -< O
Iff ff ff ff o
a.m pg X •"> ^
oj-^ pm pg o o
ac** pg ff o o
"DC '"• ff O r3
(_)••♦••
U- •
ii
-o
— i
■c
— <
-0
t—
VPvi
-^
ff
X
•c
Ci
</i
X
m
p-
p-
p-
O
LU
gD
r-
X
ff
o
><_'
ac
^-4
>o
o
**u
LL
>»■
X
pg
in
ff
□
•
•
•
•
•
o
X
>!■
^H
p-
r<^
O
ac
O
ppi
go
X
^-*
>in
LU
x^
pg
o
X
P»
xgr
ao
pg
pg
pg
—4
—*
rg X vC in p^
• • • « ■
Pvl ^ if f4 4^
>-p- P- in gr pg
ff ff ff ff ff
o c- x ff o
pg pg gD O m
ff pp> ff so og
o gr -c ff pg
xpn -h ff p- go
ppi pp pg pg pg
INJ LO
\t\
in p- m -^ gD
>x — • "• o x
l/l
o
— »
«_4
-*
o
i/>
LU
p-
X
ff
O
tM
LU
ac
•— i
t^t
ac
o
>j
u.
<c
ppi
^3
gr
X
LL.
o
o
X
p-
rg
<»■
O
ac
in
pg
K
O
pp>
ac
LU
X-J
o
X
p-
in
Mi
00
gr
-l-
<r\
t*>
pn
m
I
X
■D
o
z
z
O P- x pg <0
>x pp in gp gf
-* og pg pg pg
P- X ff O "■ *
43 pg n o n
• • • • •
x go ff go ff
x g3 pg ff pp»
xff to f» m gf
SAMPLE COMPUTER OUTPUT
\
I
1
wn
GOV
083
004
005
00C
00?
806
K-SCALE - 1.00E+04 UNITS- IMOi.
1 -SCALE - 1.00E + Q4 LKTVINCH.
f-PRELL WAUE REFRACTION PROGRAM MONTER EY BAY
TIDE .LUL 0 ANGLE = 0 PERIOD = 18
APPENDIX 3
SUMMARY OF EQUATIONS
Wave Speed (C)
zir K tc /
Ray Curvature (FK)
FK : £[>W?|§--^A|^| .
Coefficient of Refraction (Kr)
The value of ray separation (p ) is calculated by solving the
second order, non-linear differential equation:
where
p- -«•** t& - «*~ ft t If
The above equation is solved by the finite difference method. This
results in an equation for the Beta value at the n+1 point in terms
of the Beta values at the two previous points. The equation to be
solved is then:
^ " Z+PP
where
D is the incremented distance
p,q are as defined above
& I /» 2 are tne Beta values of the two previous points. The
The coefficient of refraction is calculated by the relation:
59
Direct Shoaling Coefficient (K£) :
Hs- 3.zstf - ,z, 8\5o(%-) + Z 8. 8M ( -^y -?r,usi(0
Wave Height Reduction Factor Due to Bottom Friction (Kf) :
I
where
«r =
f H.
0f 3$* l/^h
Z1fC*
60
INITIAL DISTRIBUTION LIST
No. Copies
1. Defense Documentation Center 20
Cameron Station
Alexandria, Virginia 22314
2. Library 2
Naval Postgraduate School
Monterey, California 93940
3. Professor Glenn H. Jung 3
Department of Meteorology and Oceanography
Naval Postgraduate School
Monterey, California 93940
4. LT Charles A. Farrell 3
Department of Meteorology and Oceanography
Naval Postgraduate School
Monterey, California 93940
5. Department of Meteorology and Oceanography 3
Naval Postgraduate School
Monterey, California 93940
6. Office of the Naval Weather Service 1
Naval Station (Washington Navy Yard Annex)
Washington, D. C. 20390
7. Officer in Charge 1
Naval Weather Research Facility
Naval Air Station, Building R-48
Norfolk, Virginia 23511
8. Commanding Officer 1
U. S. Fleet Weather Central
Box 12, COMNAVMARIANAS
FPO San Francisco, California 96601
9. Commanding Officer 1
U. S. Fleet Weather Central
Box 10
FPO San Francisco, California 96610
10. Commanding Officer 1
U. S. Fleet Weather Central
FPO New York, New York 09540
11. Officer in Charge 1
Fleet Weather Facility
Naval Air Station
San Diego, California 92135
61
No. Copies
12. Officer in Charge 1
Fleet Numerical Weather Facility
Naval Postgraduate School
Monterey, California 93940
13. Director, Naval Research Laboratory 1
Attn: Tech. Services Info. Officer
Washington, D. C. 20390
14. Superintendent 1
Naval Academy
Annapolis, Maryland 21402
15. Oceanographer of the Navy 1
The Madison Building
732 N. Washington Street
Alexandria, Virginia 22314
16. Naval Oceanographic Office 1
Attn: Library
Washington, D. C. 20390
17. Director, Coast & Geodetic Survey 1
Department of Commerce
Attn: Office of Oceanography
Washington, D. C. 20235
18. Office of Naval Research
Department of the Navy
Washington, D. C. 20360
Attn: Special Projects (Code 418) 1
Attn: Geophysics Branch (Code 416) 1
Attn: Director, Surface and Amphibious 1
Programs (Code 463)
Attn: Senior Marine and Amphibious 1
Warfare Officer (Code 407M)
19. Mission Bay Research Foundation 1
7730 Herschel Avenue
La Jolla, California 92038
20. LT David E. Stouppe, USN 1
USS INDEPENDENCE (CVA-62)
FPO, New York, New York 09501
62
No, Copies
21. LCDR Gale M. Griswold 3
Fleet Numerical Weather Facility
Naval Postgraduate School
Monterey, California 93940
22. Professor J. A. Putnam 1
University of California
Berkeley, California
23. Professor C. L. Bretschneider 1
Chairman of Ocean Engineering
University of Hawaii
Honolulu, Hawaii 96822
24. Coastal Engineering Research Center 1
Corps of Engineers, U. S. Army
5201 Little Falls Road, N. W.
Washington, D. C. 20016
25 . Chairman 1
Department of Meteorology and Oceanography
New York University
University Heights, Bronx
New York, New York
26. Chairman 1
Department of Meteorology and Oceanography
University of Hawaii
Honolulu, Hawaii
27. Chairman 1
Department of Oceanography
Oregon State University
Corvallis, Oregon 97331
28. Chairman 1
Department of Oceanography
Texas A&M University
College Station, Texas 77843
29. Director, Biological Laboratory 1
Bureau of Commercial Fisheries
U. S. Fish and Wildlife Service
450-B Jordan Hall
Stanford, California
30. Geophysical Institute 1
Tokyo University
Bunkyo-ku
Tokyo, Japan
63
31. Director
Pacific Oceanographic Group
Nanaimo, British Columbia
Canada
32. Ocean Research Institute
University of Tokyo
Tokyo, Japan
33. Secretary
Canadian Committee on Oceanography
Ottawa, Canada
34 . Chairman
Department of Oceanography
University of Rhode Island
Kingston, Rhode Island
64
UNCLASSIFIED
Security Classification
DOCUMENT CONTROL DATA - R&D
(Security claaaltlcatlon ot tltta, body ot abatract and Indexing annotation muat ba entered whan tha orarall report la claaaitted)
1. ORIGINATING ACTIVITY (Corporate author)
Naval Postgraduate School
Monterey, California
2a. REPORT SECURITY CLASSIFICATION
UNCLASSIFIED
2b. GROUP
3. REPORT TITLE
TIDE-LEVEL AND BOTTOM-FRICTION EFFECTS ON WAVE REFRACTION AS DETERMINED BY
NUMERICAL WAVE REFRACTION PROCEDURES
4. DESCRIPTIVE NOTES (Type ot report and Inctualva dmtaa)
Thesis
5 AUTHORS (Laet nana, tint name, Initial)
FARRELL, Charles Augustus, Jr,
«. REPORT DATE
June 1967
7« TOTAL NO. OF PASES
62
7b. NO. OP PEPS
• «. CONTRACT OR GRANT NO.
b. PROJECT NO.
9«. ORIOINATOR'S REPORT NUMBERfSj
d.
oJLJJlJUJL
»s. OTHER REPORT NOf5J (A ny other numbere that may be aaaltnad
thla report)
fen
10. AVAILABILITY/LIMITATION NOTICES
"— ™I"""L ■■ j » "' ■ »j-- '■' t ----- rrr—
11. SUPPLEMENTARY NOTES
12. SPONSORING MILITARY ACTIVITY
Naval Postgraduate School
Monterey, California
19. ABSTRACT
Numerical wave refraction programs permit a detailed study of the trans-
formation of wave energy as waves move from deep water to shallow water. By
eliminating the subjectivity that is present with hand drawn diagrams the effect
of small variations in the initial assumptions and wave conditions can be in-
vestigated. The effects on wave refraction of tide level changes and bottom
friction are investigated here. It is demonstrated that a uniform increase in
water level as would be caused by tidal fluctuations can cause a significant
change in the wave refraction pattern for a given nearshore region. A computing
procedure is developed to permit numerical refraction programs to account for
bottom friction. The reduction in wave height caused by bottom friction depends
primarily on bottom slope; this relation is shown in tabular and graphical form.
DD
FORM
1 JAN 84
1473
65
UNCLASSIFIED
Security Classification
UNCLASSIFIED
Security Classification
key wo RDS
Wave Refraction
Numerical Analysis
Bottom Friction
Tide level Changes
DD ,F°1\.1473 «"«>
5/N 0101-807-6821
1
66
UNCLASSIFIED
Security Classification
A- 3 1 409
thesF2293
Tide-level and bottom-friction effects o
3 2768 002 13380 3
DUDLEY KNOX LIBRARY
.:. ... ■ • j
iffli
sHKPHIt:
wbF
niB&t
iw-Jjl
tauD
" • ' ' ' - \
PS
reraH
',;■*'■■
KB
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11