Programming wave dispersion equation

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GIULIA
GIULIA 2025-10-13，10:43
评论： GIULIA 2025-10-16，13:41
Hi everyone,
I am quite new to MATLAB and asking for some help to programme the wave dispersion equation.
ω^2= gktanh(kh)
where
𝑔 is gravity (9.81), ℎ is water depth, and 𝑘 is the wavenumber.
I would like to programme this function using an iterative process, and use it whenever I need to solve it for finding wavelength L, wave period T (2pi/omega)
Could someone please guide me in the process to create it and help further understand the mechanisms of iteration in MATLAB?
Thanks a lot!
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GIULIA
GIULIA 2025-10-13，11:01
Yes, solve the equation for k.
And then hopefully understand the process so that I can re-arrange to solve for L and T :)
Sam Chak
Sam Chak 2025-10-13，11:57
Hi @GIULIA
If the wave number and the angular frequency ,
then
.
This is an implicit equation because the wavelength λ cannot be defined as a direct function of the other variables.
Try using the solve() command from the Symbolic Math Toolbox, as demonstrated here, before resorting to a numerical approach to solve the problem iteratively.

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Torsten
Torsten 2025-10-13，12:40
编辑：Torsten 2025-10-13，12:43
Ran in:
Code taken from
https://uk.mathworks.com/matlabcentral/fileexchange/156009-wavelength-calculation-based-on-dispersion-relationship
% Parameters
T = 10;
h = 0.2;
g = 9.81;
tolerance = 1e-6;
% Initial guess for L
L0 = T^2/g;
% Fixed point iteration to compute L
Lsol = calculateWavelength(g, h, T, tolerance, L0);
% Plot curve
f = @(L) L - g * T^2 / (2 * pi) * tanh(2 * pi * h ./ L);
L = 1:0.1:20;
hold on
plot(L,f(L))
plot(Lsol,f(Lsol),'o')
hold off
grid on
function L = calculateWavelength(g, h, T, tolerance, L0)
% Iterate until the tolerance is met
while (1)
L = g * T^2 / (2 * pi) * tanh(2 * pi * h / L0);
if (abs(L - L0) < tolerance)
break;
end
L0 = L;
end
end
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GIULIA
GIULIA 2025-10-13，16:50
Thanks Torsten!
If i'd have to re-arrange to find T (knowing k and h) would it then be:
f = @(T) 2*pi / sqrt(g*k*tanh(k*h))
tolerance = 1e-6;
function T = calculatePeriod(g, h, tolerance, T0)
% Iterate until the tolerance is met
while (1)
T = (2 * pi) / sqrt(g * k * tanh (k*h));
if (abs(T - T0) < tolerance)
break;
end
T0 = T;
end
end
My questions in that case are:
1) What would I use as initial guess for T (T0)?
2) Can I always use 1e-6 as tolerance?
Torsten
Torsten 2025-10-13，20:05
You can explicitly solve for T:
T = sqrt( 2*pi*L/g * coth(2*pi*h/L) )

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Sam Chak
Sam Chak 2025-10-13，15:44
Ran in:
An approach that directly use the solve() function.
syms lambda positive
% parameters
T = 10;
h = 0.2;
g = 9.81;
% wave dispersion equation
eq = lambda - g*T^2/(2*pi)*tanh(2*pi*h/lambda) == 0
eq = 
sol = solve(eq, lambda)
Warning: Unable to solve symbolically. Returning a numeric solution using vpasolve.
sol = 
13.988348869027691633555641776592
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Sam Chak
Sam Chak 2025-10-14，7:17
Hi @GIULIA
Can you perform algebraic operations to rearrange the lambda (λ) equation to isolate the variable T on the left-hand side? This will allow you to verify whether you arrive at the same result as @Torsten in this comment.

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