Dispersion Relation for water waves

I solved my dispersion relation for water waves. Now i have confusion regarding omega that it should also be dimensionless? Because i make my parameters dimensionless. here is the following code.

% Parameter definitions
N = 3;                   %Number of terms
% Physical parameters
R_E = 47.5; % External radius (m)
R_I = 34.5; % Internal radius (m), will also compute for RI = 32
h_I = 200; % Water depth (m)
d_I = 38; % Draft (m)
g = 9.81; % Gravity (m/s^2)
% Dimensionless geometric parameters (using R_E as length scale)
RE = R_E / R_E; % R_E / R_E = 1
RI = R_I / R_E; % R_I / R_E
h = h_I / R_E; % h / R_E
d = d_I / R_E; % d / R_E
b = (h - d); % 
gamma = 1.0;             % Some constant
tau = 0.2;    % Another constant
b1 = (1 - tau) / (2 * gamma * tau^2);
X_kRE = linspace(5, 35, 1000);   % X-axis (k0*RE) 

I_given_wavenumber = 1;         % Example flag, as in your code

for c = 1:length(X_kRE)
     %if I_given_wavenumber == 1
        wk    = X_kRE(c) / RE;                  % wavenumber(k0)
        omega = sqrt(g * wk * tanh(wk * h));    % wave radian frequency
        Cg = (g*tanh(wk*h) + g*wk*h*(sech(wk*h))^2)*omega/(2*g*wk*tanh(wk*h));  %group velocity
        a = 0.93 * (1 - tau) * Cg;  % 

        fun_alpha = @(x) omega^2 + g * x * tan(x * h); % dispersion eqn
   % end
    
    for n = 1:N
        if n == 1
            k(n) = -1i * wk;
        else
            x0_n = [(2*n - 3)*pi/2/h, (2*n - 1)*pi/2/h];
            k(n) = fzero(fun_alpha, mean(x0_n));
        end
    end
end

[Javeria]

i am not understanding if i am using a dimensionless wave number i.e wk=k0RE and also RE, RI , h, d all are dimensionless then i am confused about that is my other parameters like omega which i am using in Cg is also then dimensionless and my k(n) is also then dimensionless or not to solve my problem....

[Torsten]

You have to start with a dimensionless PDE system - then you won't come into the trouble from above.

But I would strongly advice you not to do this - at least not for the first try.

[Javeria]

@Torsten what you say that i do not need to make them dimensionless?

[Torsten]

I lack the background of your question and why you think it's necessary to work with dimensionless quantities. I only see that you again try to determine zeros of a function that will most probably be used in some infinite series afterwards.

If you formulate your PDE system in nondimensional form means: all coefficients, boundary and transmission conditions are written in dimensionless quantities. Thus all analytical solutions become dimensionless. In your code from above (which I assume is a subproblem of the PDE problem), there wouldn't appear any dimensional quantities - all would have been already substituted when transforming your PDE to dimensionless form.

In my opinion, you cannot just arbitrarily non-dimensionalize some quantities during the solution process. The complete model must first be formulated in non-dimensional form. And after you've done this, you will no longer have to cope with problems as the one you posted above.

But as I wrote: don't do this in the first step. It's hard to formulate it and later on to interprete the results.

But maybe I'm comletely on the wrong track with what I think you are asking. So if it doesn't fit your needs, you will have to give some more background information to get a useful answer.