Shallow Water wave phase speed.

• I am working on a school assignment for a Tides and Water levels class and there is a question that says to make plots of the following:
• Shallow water wave phase speed (m/s) for water depths from 1 m to 4000 m. Label the axes.
• Minimum wavelength to be considered as shallow water waves for the same depth range ( 1 to 4000 meters).
• I believe I have the first part figured out but am unsure on the second. This is what I have so far:

clear;
clf;
clc;
% Constants
g = 9.81; % gravitational acceleration in m/s^2
d =1:4000 ; % water depth in meters 
 % Shallow water wave phase speed
 
c = sqrt(g*d)
%%
plot (c,d);
grid on
ylim([1, 4000]);
xlim([0,200]);
xlabel ('m/s^2')
ylabel('Depth')
title('Shallow Water Wave Phase Speed')
%%
L = g*T^2/2/pi %deep water 
L = T*sqrt(gd)%shallow water

[Ashton Faire]

Fancy seeing you here!

[Jeremiah Thomas]

And to no avail!

Answer 1

[Gabriel Ruiz-Martinez]

@Jeremiah Thomas, a solution could be this:

% Wave period (s)

T = 2:15;

% Water depths vector (m)

d = 1:4000;

% Gravitational acceleration constant (ms^-2)

g = 9.81;

% Shallow water wave phase speed $ c = sqrt{gh} $

% (ms^-1)

c = sqrt(g.*d);

% Plotting

figure;

plot(d,c,'Color','red');

ax = gca;

ax.XLim = [1 4000];

ax.YLim = [0 200];

ax.XLabel.String = 'Water depth (m)';

ax.YLabel.String = 'Wave speed (ms^-1)';

% Solving dispersion equation, using Lo aproximation **********

for i = 1 : length(T)

for j = 1 : length(d)

Lo = (g*T(i)^2)/(2*pi);

L(j) = Lo*(tanh(((2*pi)*((sqrt(d(j)/g))/T(i)))^(3/2)))^(2/3);

end

end

ld = d./L;

ldq = find(ld <= 0.04);

fprintf('Minimum wavelength to be considered as shallow waters: %5.3f m \r\n',min(L(ldq)));

Dispersion relationship for water waves

I hope this script can be useful!.

Regards.