Abstract
We apply Rice's multidimensional formulas, in a mathematically rigorous way, to several problems which appear in random sea modeling. As a first example, the probability density function of the velocity of the specular points is obtained in one or two dimensions as well as the expectation of the number of specular points in two dimensions. We also consider, based on a multidimensional Rice formula, a curvilinear integral with respect to the level curve. It follows that its expected value allows defining the Palm distribution of the angle of the normal of the curve that defines the waves crest. Finally, we give a new proof of a general multidimensional Rice formula, valid for all levels, for a stationary and smooth enough random fields X:ℝd→ℝj(d>j).
Highlights
In 1944, Rice 1 proposed the model ζt cn cos σnt εn, to describe the noise in an electrical current
The present work is aimed at studying functionals of random field level sets in order to understand certain phenomena occurring in random sea modeling such as the movement of the luminous points which appear over any water surface. These points are called specular points and originate when the light is reflected in agreement to Snell’s Law from different zones which act as small mirrors
They can be modeled as level sets of certain derivatives of the original random field ζ
Summary
In 1944, Rice 1 proposed the model ζt cn cos σnt εn , 1.1 n to describe the noise in an electrical current. The present work is aimed at studying functionals of random field level sets in order to understand certain phenomena occurring in random sea modeling such as the movement of the luminous points which appear over any water surface These points are called specular points and originate when the light is reflected in agreement to Snell’s Law from different zones which act as small mirrors. These velocities are computed both for Gaussian and non-Gaussian random fields, formalizing and generalizing, the deep and inspired work of Longuet-Higgins. In what follows λd and σd−m will denote, respectively, the Lebesgue measure in the space Rd and the Hausdorff measure defined in the subspaces of dimension d − m, trivially by definition λd σd
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
