Is there a particularly good function form for this curve?

Question

This curve shape seems to appear in various natural phenomena:

Do you recognize it? Do you know a specific function form that could match it or approximate it closely?

Answer 1

FIRST PART : Search for a model function.

In the empirical approach, the curve given by L_R_T is used (without the scattered points) : copy on Figure 1 below, curve drawn in red.

On Figure 2, instead of $x$ the abscissas are $\ln(x)$. The curve tends to become sinusoidal. But it is not symmetrical relatively to the horizontal axe. This draw to think that a damping factor should be taken into account.

On Figure 3, with a damping function very roughly adjusted by trial and error, the shape of the curve becomes closer from a sinusoid (dashed curve).

This leads to think that a good candidat might be on the form $$y(x)\simeq x^{\alpha}\left(b\:\sin\left(\omega \ln(x)\right)+c\:\cos\left(\omega \ln(x)\right)\right) $$

They are four adjustable parameters $\omega,\alpha,b,c$ in the proposed formula.

This is the same as $$y(x)\simeq x^{\alpha}\rho\:\sin\left(\omega \ln(x)+\varphi\right) \quad \begin{cases} \rho=\sqrt{b^2+c^2} \\ \tan(\varphi)=\frac{c}{b}\end{cases}$$

The dashed curves are drawn with $\omega=\frac{\pi}{2}$ , $\alpha=-0.25$ , $\rho=5.6$ , $\varphi=-0.5$ Of course, this is only a rough preliminary result.

SECOND PART : Method to compute approximate values of the parameters.

Generally, this requires a non-linear regression method. For example of the kind of Levenberg–Marquardt algorithm. They are iterative processes, starting from guessed values of parameters.

A non-conventional approach (not iterative, no initial guess) is described in the paper : https://fr.scribd.com/doc/14674814/Regressions-et-equations-integrales

In case of damped sinusoidal function the method of calculus is given page 66. The so-called "short way" is sufficient for the next result.

The data used comes from the figure given by L_R_T. Only the scattered points are used which coordinates where picked by graphical scanning.

One cannot expect an accurate result because they are only few points, not well distributed and with a big scatter.

The computed values are shown below (symbols defined p.66 in the paper referenced above. Note that in the paper $x$ must be replaced by $\ln(x)$ to be consistent with the actual case).

The computed curve is drawn in blue on the next figure.

The full computation process (page 67 of the referenced paper) leads to :