Programming wave dispersion equation

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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 2025-10-13

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 2025-10-13

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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Answer 1

Torsten 2025-10-13

编辑：Torsten 2025-10-13

在 MATLAB Online 中打开

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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

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GIULIA 2025-10-13

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

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 2025-10-13

在 MATLAB Online 中打开

You can explicitly solve for T:

T = sqrt( 2*pi*L/g * coth(2*pi*h/L) )

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Answer 2

Sam Chak 2025-10-13

在 MATLAB Online 中打开

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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 2025-10-14

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.

GIULIA 2025-10-16

Right, thank you!

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Programming wave dispersion equation

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