APPENDIX 1
What is deep water?
When the depth don't interfere the waves, the water is referred as deep water'. That is, the depth of the water does not affect the regular motion of the waves. The criteria of deep water is:
d / L = d / Lo ≥ 0.5
where d is the depth, L is the wavelength at this depth and L o is the deep water wavelength. At deep water L = L o . The interpretation of this formula is, the water is still ( i.e. the water molecules don't move ) under depth d when:
d 3≥ 0.5 * Lo
To determine Lo ,
Lo = 1.56 * T2
can be, where T is the wave. We are assuming no energy loss or gain during these calculations.
How to find wave energy
At deep water, the potential energy of a wave is equal to its kinetic energy. The wave energy is equal to the sum total of the kinetic energy and the potential energy. This can be calculated by:
wave energy = 1/16 * r * g * H2 + 1/16 * r * g * H2
= 1/8 * r * g * H2
Rate of energy transmission
The rate of energy transmission can be calculated by :
Rate of energy transmission = E * Cg
where Cg = n * C = velocity of energy propagation, C is the wave celerity and n is a variable. Also, C can be calculated by
C = L / T
where L is the wavelength and T is constant at every depth. Given T and d, C may be worked out.
the variable n = 1/2 * ( 1 + 2kd/ sinh2kd )
for deep water : n → 0.5
for shallow water : n → 1.0
transition : 0.5 < n < 1.0
Conservation of energy
Obeying the conservation of energy rule ;
Eo * Cgo * bo = E1 * Cg1 * b1
where b is the width of the section. From the equation above,
(H1 / Ho )2 = Cgo / Cg1 * bo / b1
is derived.
As the water becomes shallower and the wave height increases. The front of the wave moves slower than the back, therefore the water is pushed up ( i.e. wave height is increased ). No change in b is assumed.
k s = H i / Ho = v Cgo / v Cg 1
(k s )2 = no /n 1 * Co /C 1
Co = 1.56 * T
(k s )2 = no /n 1 * 1/tanhkd
Run up
Run up (Ru) is the height above still water level that is used to calculate the discharge (see fig app.1).
 Ru can be calculated by:
Ru = ? * H
? = tan a / (H / Lo )1/2
where H is the wave height
and L o is the deep water wavelength. The discharge can be calculated by evaluating the green area in the figure.
Calculating discharge from Ru
From the following equation, the green area in fig app.1 can be calculated. The angle z is the top angle of the green triangle.
Q = ((Ru h)2 / 2T)*{[(1 / tan (a z)] ( 1 / tan z)}
APPENDIX 3
The results obtained from the experiments are in the tables below:
data no |
h (m) |
tan a |
tanq |
H (m) |
T (s) |
t1 (s) |
t2 (s) |
t3 (s) |
t4 (s) |
t5 (s) |
tav (s) |
P(m) |
V (m3) |
Q1(m3/s) |
611 |
-0,04 |
0,260 |
0,050 |
0,052 |
1,1 |
6,6 |
6,4 |
7,2 |
|
|
6,7 |
|
4,080E-02 |
6,068E-03 |
612 |
-0,04 |
0,260 |
0,050 |
0,050 |
1,0 |
6,0 |
6,1 |
7,2 |
|
|
6,4 |
|
4,080E-02 |
6,358E-03 |
613 |
-0,04 |
0,260 |
0,050 |
0,051 |
0,7 |
7,4 |
10,4 |
7,7 |
|
|
8,5 |
|
4,080E-02 |
4,817E-03 |
614 |
-0,04 |
0,260 |
0,050 |
0,043 |
1,0 |
6,0 |
5,8 |
7,3 |
|
|
6,4 |
|
4,080E-02 |
6,402E-03 |
615 |
-0,04 |
0,260 |
0,050 |
0,065 |
0,8 |
7,5 |
7,4 |
7,8 |
|
|
7,6 |
|
4,080E-02 |
5,390E-03 |
616 |
-0,04 |
0,260 |
0,050 |
0,069 |
0,7 |
7,7 |
7,8 |
8,3 |
5,4 |
|
7,3 |
|
4,080E-02 |
5,612E-03 |
617 |
-0,02 |
0,260 |
0,050 |
0,051 |
0,8 |
5,2 |
4,8 |
6,4 |
7,7 |
|
6,0 |
|
2,040E-02 |
3,393E-03 |
618 |
-0,02 |
0,260 |
0,050 |
0,055 |
0,7 |
9,0 |
7,9 |
9,7 |
|
|
8,9 |
|
2,040E-02 |
2,302E-03 |
619 |
-0,02 |
0,260 |
0,050 |
0,061 |
0,7 |
8,9 |
9,5 |
9,8 |
|
|
9,4 |
|
2,040E-02 |
2,171E-03 |
623 |
-0,02 |
0,260 |
0,050 |
0,039 |
1,6 |
8,1 |
6,4 |
7,5 |
7,6 |
8,8 |
7,7 |
|
2,040E-02 |
2,656E-03 |
624 |
-0,02 |
0,260 |
0,050 |
0,055 |
1,0 |
5,6 |
5,8 |
5,7 |
5,6 |
|
5,7 |
|
2,040E-02 |
3,592E-03 |
626 |
-0,02 |
0,260 |
0,050 |
0,062 |
1,0 |
5,8 |
3,6 |
6,0 |
5,2 |
|
5,2 |
|
2,040E-02 |
3,953E-03 |
627 |
-0,04 |
0,260 |
0,060 |
0,024 |
1,3 |
10,5 |
9,3 |
9,5 |
|
|
9,8 |
|
3,520E-02 |
3,605E-03 |
628 |
-0,04 |
0,260 |
0,060 |
0,022 |
2,5 |
5,7 |
6,9 |
4,9 |
|
|
5,8 |
|
3,520E-02 |
6,052E-03 |
629 |
-0,04 |
0,260 |
0,060 |
0,014 |
0,8 |
6,4 |
8,5 |
10,6 |
|
|
8,5 |
|
3,520E-02 |
4,131E-03 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
630 |
-0,02 |
0,260 |
0,060 |
0,037 |
1,1 |
3,7 |
5,7 |
4,7 |
5,6 |
5,2 |
5,0 |
|
1,760E-02 |
3,536E-03 |
631 |
-0,02 |
0,260 |
0,060 |
0,036 |
1,1 |
4,3 |
4,6 |
5,1 |
5,4 |
4,5 |
4,8 |
|
1,760E-02 |
3,673E-03 |
632 |
-0,02 |
0,260 |
0,060 |
0,035 |
1,0 |
4,8 |
4,6 |
4,7 |
4,8 |
4,7 |
4,7 |
|
1,760E-02 |
3,734E-03 |
633 |
-0,02 |
0,260 |
0,060 |
0,035 |
1,3 |
6,2 |
7,1 |
7,3 |
5,5 |
8,2 |
6,9 |
|
1,760E-02 |
2,563E-03 |
data no |
Ru (m) |
Q (m2/s) |
Q/(gH3)1/2 |
(Ru-h)/H |
log(Q/(gH3)1/2) |
Q(m2/s)(130) |
Q(m2/s)(100) |
Q(m2/s)(120) |
Q(m2/s)(9,50) |
611 |
0,078 |
1,190E-02 |
0,32054373 |
2,268408484 |
-0,49411271 |
6,83E-02 |
1,46E-02 |
3,74E-02 |
1,13E-02 |
612 |
0,073 |
1,247E-02 |
0,35847121 |
2,261187995 |
-0,445545719 |
5,29E-02 |
1,13E-02 |
2,90E-02 |
8,73E-03 |
613 |
0,048 |
9,445E-03 |
0,25966744 |
1,722366818 |
-0,585582509 |
5,31E-03 |
1,13E-03 |
2,91E-03 |
8,75E-04 |
614 |
0,066 |
1,255E-02 |
0,45459126 |
2,486204402 |
-0,342378916 |
3,46E-02 |
7,38E-03 |
1,90E-02 |
5,71E-03 |
615 |
0,064 |
1,057E-02 |
0,20170455 |
1,581948546 |
-0,695284311 |
3,61E-02 |
7,69E-03 |
1,98E-02 |
5,95E-03 |
616 |
0,061 |
1,100E-02 |
0,19542572 |
1,4657431 |
-0,709018274 |
2,98E-02 |
6,36E-03 |
1,63E-02 |
4,92E-03 |
617 |
0,061 |
6,653E-03 |
0,18720861 |
1,611996997 |
-0,727674189 |
1,02E-01 |
2,17E-02 |
5,58E-02 |
1,68E-02 |
618 |
0,056 |
4,513E-03 |
0,1129644 |
1,392497831 |
-0,947058394 |
8,79E-02 |
1,87E-02 |
4,81E-02 |
1,45E-02 |
619 |
0,054 |
4,257E-03 |
0,0906351 |
1,214205316 |
-1,042703592 |
8,51E-02 |
1,82E-02 |
4,66E-02 |
1,40E-02 |
623 |
0,104 |
5,208E-03 |
0,21346721 |
3,145979458 |
-0,670668832 |
2,17E-01 |
4,63E-02 |
1,19E-01 |
3,58E-02 |
624 |
0,080 |
7,042E-03 |
0,17270483 |
1,803192812 |
-0,762695504 |
1,71E-01 |
3,65E-02 |
9,37E-02 |
2,82E-02 |
626 |
0,080 |
7,752E-03 |
0,1594751 |
1,605680203 |
-0,797307124 |
1,82E-01 |
3,88E-02 |
9,96E-02 |
3,00E-02 |
627 |
0,067 |
8,194E-03 |
0,72564274 |
4,542190142 |
-0,139277144 |
2,68E-02 |
5,71E-03 |
1,47E-02 |
4,42E-03 |
628 |
0,120 |
1,375E-02 |
1,3878088 |
7,41255612 |
0,142329638 |
1,27E-01 |
2,70E-02 |
6,94E-02 |
2,09E-02 |
629 |
0,032 |
9,390E-03 |
1,8421849 |
5,188407864 |
0,265333219 |
4,03E-03 |
8,58E-04 |
2,21E-03 |
6,64E-04 |
|
|
|
|
|
|
|
|
|
|
630 |
0,070 |
8,035E-03 |
0,36065294 |
2,427625099 |
-0,442910525 |
1,11E-01 |
2,36E-02 |
6,08E-02 |
1,83E-02 |
631 |
0,069 |
8,347E-03 |
0,39036996 |
2,460106468 |
-0,408523607 |
1,06E-01 |
2,26E-02 |
5,80E-02 |
1,75E-02 |
632 |
0,063 |
8,485E-03 |
0,41395707 |
2,362297052 |
-0,383044691 |
8,82E-02 |
1,88E-02 |
4,83E-02 |
1,45E-02 |
633 |
0,078 |
5,826E-03 |
0,28421113 |
2,798289281 |
-0,546358912 |
1,31E-01 |
2,79E-02 |
7,16E-02 |
2,16E-02 |
data no |
h (m) |
tan a |
tanq |
H (m) |
T (s) |
t1 (s) |
t2 (s) |
t3 (s) |
t4 (s) |
t5 (s) |
tav (s) |
P(m) |
V (m3) |
Q1(m3/s) |
634 |
-0,02 |
0,260 |
0,060 |
0,037 |
1,2 |
4,6 |
6,1 |
5,9 |
5,4 |
|
5,5 |
|
1,760E-02 |
3,196E-03 |
635 |
-0,02 |
0,260 |
0,060 |
0,036 |
1,4 |
5,2 |
4,8 |
4,8 |
4,6 |
5,4 |
5,0 |
|
1,760E-02 |
3,554E-03 |
636 |
-0,04 |
0,260 |
0,060 |
0,037 |
1,3 |
4,7 |
7,9 |
7,3 |
7,3 |
6,4 |
6,7 |
|
1,760E-02 |
2,620E-03 |
637 |
-0,04 |
0,260 |
0,060 |
0,037 |
1,2 |
3,0 |
5,5 |
5,4 |
4,4 |
5,5 |
4,8 |
|
1,760E-02 |
3,704E-03 |
638 |
-0,04 |
0,260 |
0,060 |
0,037 |
1,3 |
6,4 |
5,9 |
5,7 |
5,4 |
4,1 |
5,5 |
|
1,760E-02 |
3,188E-03 |
|
|
|
|
|
|
|
|
|
|
|
t b (s) |
|
|
|
641 |
0,01 |
0,260 |
0,050 |
0,030 |
1,5 |
|
|
|
|
|
24,1 |
0,077 |
2,851E-02 |
1,140E-03 |
642 |
0,01 |
0,260 |
0,050 |
0,061 |
1,1 |
|
|
|
|
|
19,4 |
0,139 |
5,146E-02 |
2,573E-03 |
643 |
0,01 |
0,260 |
0,050 |
0,070 |
1,0 |
|
|
|
|
|
14,4 |
0,111 |
4,110E-02 |
2,740E-03 |
644 |
0,01 |
0,260 |
0,050 |
0,037 |
1,3 |
|
|
|
|
|
24,2 |
0,080 |
2,962E-02 |
1,185E-03 |
645 |
0,01 |
0,260 |
0,050 |
0,057 |
1,1 |
|
|
|
|
|
19,0 |
0,110 |
4,073E-02 |
2,036E-03 |
646 |
0,01 |
0,260 |
0,050 |
0,069 |
0,8 |
|
|
|
|
|
13,9 |
0,075 |
2,777E-02 |
1,851E-03 |
647 |
0,03 |
0,260 |
0,050 |
0,050 |
1,2 |
|
|
|
|
|
24,1 |
0,062 |
2,296E-02 |
9,182E-04 |
648 |
0,03 |
0,260 |
0,050 |
0,063 |
0,9 |
|
|
|
|
|
19,4 |
0,061 |
2,258E-02 |
1,129E-03 |
649 |
0,03 |
0,260 |
0,050 |
0,078 |
0,9 |
|
|
|
|
|
14,1 |
0,047 |
1,740E-02 |
1,160E-03 |
650 |
0,03 |
0,260 |
0,050 |
0,052 |
1,2 |
|
|
|
|
|
23,7 |
0,095 |
3,517E-02 |
1,407E-03 |
651 |
0,03 |
0,260 |
0,050 |
0,075 |
1,0 |
|
|
|
|
|
19,2 |
0,115 |
4,258E-02 |
2,129E-03 |
652 |
0,03 |
0,260 |
0,050 |
0,071 |
0,8 |
|
|
|
|
|
14,0 |
0,052 |
1,925E-02 |
1,284E-03 |
653 |
0,03 |
0,260 |
0,060 |
0,045 |
1,2 |
|
|
|
|
|
24,4 |
0,098 |
3,628E-02 |
1,487E-03 |
data no |
Ru (m) |
Q (m2/s) |
Q/(gH3)1/2 |
(Ru-h)/H |
log(Q/(gH3)1/2 ) |
Q(m2/s)(130) |
Q(m2/s)(100) |
Q(m2/s)(120) |
Q(m2/s)(9,50) |
634 |
0,078 |
7,263E-03 |
0,32597918 |
2,648939402 |
-0,486810138 |
1,35E-01 |
2,87E-02 |
7,37E-02 |
2,22E-02 |
635 |
0,086 |
8,078E-03 |
0,37775704 |
2,949946001 |
-0,422787434 |
1,56E-01 |
3,33E-02 |
8,57E-02 |
2,58E-02 |
636 |
0,082 |
5,954E-03 |
0,26724179 |
3,285777006 |
-0,573095632 |
6,61E-02 |
1,41E-02 |
3,62E-02 |
1,09E-02 |
637 |
0,078 |
8,418E-03 |
0,3778052 |
3,17934341 |
-0,422732065 |
5,69E-02 |
1,21E-02 |
3,12E-02 |
9,39E-03 |
638 |
0,084 |
7,246E-03 |
0,325241 |
3,358422159 |
-0,48779471 |
7,25E-02 |
1,55E-02 |
3,97E-02 |
1,20E-02 |
|
|
|
|
|
|
|
|
|
|
641 |
0,085 |
2,236E-03 |
0,13766485 |
2,491422286 |
-0,861176929 |
2,97E-01 |
6,34E-02 |
1,63E-01 |
4,90E-02 |
642 |
0,086 |
5,045E-03 |
0,10606337 |
1,246082103 |
-0,974434577 |
4,32E-01 |
9,21E-02 |
2,37E-01 |
7,13E-02 |
643 |
0,082 |
5,372E-03 |
0,09335821 |
1,03235604 |
-1,029847492 |
4,41E-01 |
9,41E-02 |
2,42E-01 |
7,28E-02 |
644 |
0,080 |
2,323E-03 |
0,10567836 |
1,907951318 |
-0,976013917 |
3,14E-01 |
6,70E-02 |
1,72E-01 |
5,19E-02 |
645 |
0,082 |
3,993E-03 |
0,0927707 |
1,25686487 |
-1,032589178 |
4,02E-01 |
8,56E-02 |
2,20E-01 |
6,62E-02 |
646 |
0,069 |
3,630E-03 |
0,06444909 |
0,862626766 |
-1,190783225 |
3,85E-01 |
8,21E-02 |
2,11E-01 |
6,36E-02 |
647 |
0,084 |
1,800E-03 |
0,05131658 |
1,079626127 |
-1,28974233 |
5,62E-01 |
1,20E-01 |
3,08E-01 |
9,27E-02 |
648 |
0,075 |
2,214E-03 |
0,04511539 |
0,720807293 |
-1,345675315 |
5,97E-01 |
1,27E-01 |
3,27E-01 |
9,85E-02 |
649 |
0,081 |
2,275E-03 |
0,0330942 |
0,653741755 |
-1,480248113 |
6,92E-01 |
1,47E-01 |
3,79E-01 |
1,14E-01 |
650 |
0,088 |
2,759E-03 |
0,07470765 |
1,109871018 |
-1,126634904 |
5,83E-01 |
1,24E-01 |
3,19E-01 |
9,61E-02 |
651 |
0,085 |
4,174E-03 |
0,06491999 |
0,73321363 |
-1,187621581 |
6,91E-01 |
1,47E-01 |
3,79E-01 |
1,14E-01 |
652 |
0,067 |
2,517E-03 |
0,0422972 |
0,523700333 |
-1,3736884 |
6,09E-01 |
1,30E-01 |
3,34E-01 |
1,00E-01 |
653 |
0,085 |
3,380E-03 |
0,11473403 |
1,23868773 |
-0,940307733 |
5,33E-01 |
1,14E-01 |
2,92E-01 |
8,80E-02 |
654 |
0,081 |
4,363E-03 |
0,10255189 |
0,90304251 |
-0,989056352 |
5,90E-01 |
1,26E-01 |
3,23E-01 |
9,74E-02 |
655 |
0,078 |
4,032E-03 |
0,0627068 |
0,646591018 |
-1,202685327 |
6,66E-01 |
1,42E-01 |
3,65E-01 |
1,10E-01 |
data no |
h (m) |
tan a |
tanq |
H (m) |
T (s) |
t1 (s) |
t2(s) |
t3(s) |
t4(s) |
t5(s) |
tav (s) |
P(m) |
V (m3) |
Q1(m3/s) |
656 |
0,03 |
0,260 |
0,060 |
0,036 |
1,2 |
|
|
|
|
|
24,3 |
0,071 |
2,629E-02 |
1,082E-03 |
657 |
0,03 |
0,260 |
0,060 |
0,064 |
1,0 |
|
|
|
|
|
19,2 |
0,083 |
3,073E-02 |
1,601E-03 |
658 |
0,03 |
0,260 |
0,060 |
0,063 |
0,8 |
|
|
|
|
|
14,6 |
0,048 |
1,777E-02 |
1,217E-03 |
659 |
0,01 |
0,260 |
0,060 |
0,037 |
1,2 |
|
|
|
|
|
24,9 |
0,108 |
3,999E-02 |
1,606E-03 |
660 |
0,01 |
0,260 |
0,060 |
0,063 |
1,0 |
|
|
|
|
|
18,7 |
0,107 |
3,962E-02 |
2,118E-03 |
661 |
0,01 |
0,260 |
0,060 |
0,065 |
0,9 |
|
|
|
|
|
13,0 |
0,069 |
2,555E-02 |
1,965E-03 |
662 |
0,01 |
0,260 |
0,060 |
0,041 |
1,3 |
|
|
|
|
|
24,9 |
0,111 |
4,110E-02 |
1,650E-03 |
663 |
0,01 |
0,260 |
0,060 |
0,063 |
1,1 |
|
|
|
|
|
18,3 |
0,165 |
6,109E-02 |
3,338E-03 |
664 |
0,01 |
0,260 |
0,060 |
0,069 |
1,0 |
|
|
|
|
|
14,2 |
0,107 |
3,962E-02 |
2,790E-03 |
data no |
Ru (m) |
Q (m2/s) |
Q/(gH3)1/2 |
(Ru-h)/H |
log(Q/(gH3)1/2 ) |
Q(m2/s)(130) |
Q(m2/s)(100) |
Q(m2/s)(120) |
Q(m2/s)(9,50) |
656 |
0,076 |
2,459E-03 |
0,11628468 |
1,274966639 |
-0,93447748 |
4,52E-01 |
9,64E-02 |
2,48E-01 |
7,46E-02 |
657 |
0,079 |
3,638E-03 |
0,07148234 |
0,765302583 |
-1,145801226 |
6,18E-01 |
1,32E-01 |
3,39E-01 |
1,02E-01 |
658 |
0,068 |
2,766E-03 |
0,05567307 |
0,595667816 |
-1,254354831 |
5,74E-01 |
1,22E-01 |
3,15E-01 |
9,48E-02 |
659 |
0,078 |
3,650E-03 |
0,16400995 |
1,833694229 |
-0,785129799 |
3,09E-01 |
6,59E-02 |
1,69E-01 |
5,10E-02 |
660 |
0,080 |
4,815E-03 |
0,09798616 |
1,121508339 |
-1,008835265 |
4,12E-01 |
8,79E-02 |
2,26E-01 |
6,80E-02 |
661 |
0,071 |
4,466E-03 |
0,08660973 |
0,945189472 |
-1,062433335 |
3,82E-01 |
8,15E-02 |
2,09E-01 |
6,30E-02 |
662 |
0,088 |
3,751E-03 |
0,14518274 |
1,898656057 |
-0,838084999 |
3,56E-01 |
7,59E-02 |
1,95E-01 |
5,88E-02 |
663 |
0,088 |
7,587E-03 |
0,15330115 |
1,238335627 |
-0,814454589 |
4,44E-01 |
9,47E-02 |
2,43E-01 |
7,33E-02 |
664 |
0,087 |
6,341E-03 |
0,11131266 |
1,119383575 |
-0,953455434 |
4,63E-01 |
9,87E-02 |
2,54E-01 |
7,64E-02 |
For the results obtained from the experiments, please see the tables above.
Q/(gH3)1/2 is the non-dimensionalized discharge. This is useful to evaluate the discharge at any given wave height, wave period and height h. The logarithm of this value is used to obtain regular shaped graphs which makes it much easier to get the values from a graph. However, the graph drawn from the non-dimensionalized discharge (Q/(gH3)1/2 ) gives only a scatter of points which makes the graphs unpredictable.
Also, (Ru h)/H is the non-dimensionalized version of Ru. This value is non-dimensionalized to get the Ru at every given wave height and height h.
The graphs obtained from the table above are below:
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