Inverting dispersion relation with Mathematica

[Takeshin]

Hi guys,

I'm currently working on a project related to Faraday instability, and I of course came across the dispersion relation for capillary-gravity waves, i.e.

ω² = tanh(kh)(gk + σk³/ρ) .

Now I would need to numerically solve this relation for the wavenumber k as a function of depth h (in fact, this is the wavelength I'm interested in, but this is given freely if I know the wavenumber). My first reflex was to rewrite this relation in terms of h as a function of k

h = (1/k) arctanh [ω² (gk + σk³/ρ)^(-1) ]

and then try to invert it with Mathematica (which I'm not really familiar with, but it seemed the easiest choice to me). I tried using Findroot, with something like this :

h[x_] := (1/x) ArcTanh[((70 \[Pi])^2)/ (9.81x +
(0.0206 x³/950) )]
hmax = 0.01;
hmin = 0.001;
dh = 0.0001;
Sol = Table[{{b -> i}, FindRoot[i == h[x], {x, 0.001}]} //
Flatten, {i, hmin, hmax, dh}];

but this seems to stumble upon some convergence error (here I replaced ω, ρ and σ with numerical values related to my experimental setup), and I found nothing that seemed really helpful in the doc. Does anyone have an idea on what I should try to make it work ?

Thanks in advance for you replies.

 

[NateD]

</code>I think you should try using Mathematica's NDSolve function instead of FindRoot. NDSolve is a numerical solver that can solve differential equations, and since your dispersion relation is an equation relating the wavenumber k to the depth h, it might be useful here. The way you would use it is by making a function f[k, h] that is equal to 0 when the dispersion relation is satisfied. You can then plug this function into NDSolve and it will solve for the relationship between k and h. For example, let's say you have the following dispersion relation: ω² = tanh(kh)(gk + σk³/ρ) You would then make a function f[k, h] that is equal to 0 when this equation holds true: f[k_, h_] := (Tanh[k h] (g k + σ k^3/ρ) - ω^2)Then, you can plug this function into NDSolve to solve for the relationship between k and h: NDSolve[{f[k, h] == 0, hmin <= h <= hmax}, k, {h, hmin, hmax}]This should give you a numerical solution for the relationship between k and h that you can use in your project. Hope this helps!