= Solution
The independent, spatially homogeneous random-placement idealization gives a <Poisson process> of intersections along the sampling line. With line intensity $\mu$, an interval of length $s$ contains no ridge with probability $e^{-\mu s}$. The spacing $X$ thus has <exponential distribution>
$$
\boxed{f_X(s)=\mu e^{-\mu s},\qquad s\ge0,\qquad
\mathbb E[X]=\frac1\mu.}
$$
Random orientation changes the intersection intensity, which has already been represented by the measured $\mu$. Merely saying that positions are random does not prove a <Poisson process>: independence and homogeneity are additional assumptions. Finite segments, clustering or excluded widths can invalidate them.
For a <shifted lognormal distribution>, write $Z=\log(X-\theta)\sim N(m,\sigma^2)$ with $\sigma>0$. Applying the <change of variables> formula, $dZ/dX=1/(X-\theta)$, gives
$$
\boxed{
f_X(x)=
\begin{cases}
\displaystyle\frac{1}{(x-\theta)\sigma\sqrt{2\pi}}
\exp\left[-\frac{(\log(x-\theta)-m)^2}{2\sigma^2}\right],
&x>\theta,\\
0,&x\le\theta .
\end{cases}}
$$
The threshold $\theta$ is a minimum observable separation. Finite keel widths impose geometric exclusion, and the <sonar>'s footprint and ridge-identification criterion can suppress a shallow peak near a deeper peak. This <sonar ridge shadowing> or resolution effect means that $\theta$ need not be a universal physical distance between every pair of ridges; it can partly reflect how the profiles were processed.
A <lognormal distribution> is compatible with products of many positive factors: taking logarithms turns multiplicative changes into sums, which can approach a <normal distribution>. Repeated deformation, breakup, convergence and merging across scales are plausible contributors. The observed form therefore suggests correlated or multistage ridging rather than the simplest independent-intersection model. \b[It does not identify a unique ridging mechanism.] For example, directly generating $X=\theta+\exp Z$ with a <normal distribution> for $Z$ produces the same spacing law without specifying any particular mechanics. Distributional agreement must be supplemented by dynamical and spatial evidence.
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