Exercise on Linear and Nonlinear Waves
Problem 1
%Wave Data
g=9.81;h=1.3;T=1.5;H=0.26;tetha=0;
%Calculation of Wave Lenght (L) & Wave Number (K)
Lo=1.56*T^2;
L=Lo*(tanh((2*pi()*h/Lo)^(3/4)))^(2/3);
k=2*pi/L;
w=2*pi/T;
eta=H/2*cos(tetha);
high=h+eta;
%Calculation of U using Linear Wave Theory
for i = 0:100
u1(i+1)=eta*g*T*cosh(k*(i*high/100))/(L*cosh(k*h));
end
%Calculation U using Chakrabarti method
for i = 0:100
u2(i+1)=eta*g*T*cosh(k*(i*high/100))/(L*cosh(k*(h+eta)))*cos(tetha);
end
%Calculation U using Wheeler stretching method
for i = 0:100
u3(i+1)=eta*g*T*cosh(k*(i*high/100)*(h/(h+eta)))/(L*cosh(k*h))*cos(te
tha);
end
%Data read manually from R15BU(z).gif
u=[0.1,0.11,0.13,0.2,0.3,0.34,0.39,0.43,0.44,0.54];
z2=[-1.1,-0.99,-0.75,-0.49,-0.25,-0.19,-0.04,0.006,0.05,0.1];
%Plotting result into graph
z=linspace(-h,eta,101);
plot(u1,z,':r',u2,z,'bo',u3,z,'g--',u,z2,'bs',0,z);
grid on
xlabel('Horizontal particle velocity (U) in m/s')
ylabel('Water Depth (Z) in m')
title('Horizontal particle vecolity profile')
h = legend('LWT','Chakrabarti','Wheller stretching','R1BU(z).gif',4);
set(h,'Interpreter','none')
Page 1 of 6
Horizontal particle vecolity profile
0.2
0
-0.2
Water Depth (Z) in m
-0.4
-0.6
-0.8
-1
LWT
Chakrabarti
Wheller stretching
R1BU(z).gif
-1.2
-1.4
0
0.1
0.2
0.3
0.4
0.5
Horizontal particle velocity (U) in m/s
0.6
0.7
0.8
Problem 2
%Problem 2 Wave Data
clc
g=9.81;h=6;T=10;H=1.2;tetha=0;
%Calculation of Wave Lenght (L), Wave Speed (C), Crest (Sc) & Trough
(St)
%from seabed
Lo=1.56*T^2;
L=Lo*(tanh((2*pi()*h/Lo)^(3/4)))^(2/3)
k=2*pi/L;
w=2*pi/T;
eta=H/2*cos(tetha);
C=L/T
Sc=h+eta
St=h-eta
Result Using Linear Wave Theory:
L
C
Sc
St
=
=
=
=
74.7429
7.4743
6.6000
5.4000
m
m/s
m
m
Page 2 of 6
Result using Dean Stream function theory (Dalrymple Java Applet):
From the graphics, we can read that the Length of the wave is (L) =
75.33m.
Therefore C=L/T=75.33/10=7.53 m/s.
Page 3 of 6
Sc = h+eta crest  Sc = 6 + 0.78 = 6.78m
St = h-eta trough  St = 6 – 0.41 = 5.59m
Result using Fourier:
Through Solution.res file we obtain wave length and celerity: (check file
Instructions.pdf, page 13 inside the Fourier folder, to check how to make the variables
dimensional again).
L=76.74m; C=L/T=7.67m/s
Page 4 of 6
File Surface.res shows the no dimensional values for the surface elevation, going from
trough to crest to trough:
Trough: St= 0.9303*d= 5.58m
Crest: Sc=1.1303*d= 6.7818
Summary of the results:
Items
LWT
L
C
Sc
St
74.7429 m
7.4743 m/s
6.6 m
5.4 m
Dean Stream
function theory
75.33 m
7.53 m/s
6.78 m
5.59 m
Problem 3
Page 5 of 6
Fourier
76.74 m
7.67 m/s
6.78 m
5.58 m
a. Wave length
Distance of antinode = L/2 = 2 m  L = 4 m
b. Incident wave height
( H max  H min )
HI 
2
(0.14  0.1)
HI 
 0.12m
2
c. Reflection coefficient
( H max  H min )
H
KR  R 
H I ( H max  H min )
(0.14  0.1)
KR 
 0.167
(0.14  0.1)
d. The depth of water is 0.2 m; calculate
 Horizontal component of orbital motion at the bottom under the antinode
 Hi
 Hr
1
1
Uc 

T sinh  kd 
T sinh  kd 
L
L
Shallow water  C   gd  T 
T
gd

Vertical component of orbital motion there at a depth of 0.1m
Wc ,max 

 H i sinh(k ( z  d ))  H r sinh(k ( z  d ))
T
sinh(kd )

T
sinh(kd )
The difference between mean water level and still water level
Measure from figure take an average MWL-SWL
Page 6 of 6