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The aim of this article is to describe an improved mathematical model for the basic problem of evaluating the void fraction of a slab material by using the responses of a collimated detector from a typical neutron source-detector system. In contrast to previous work, the model reported here: (i) is able to treat explicitly and exactly a parallel neutron beam with normal incidence; (ii) allows for an arbitrary order of anisotropy of the neutron scattering phase function and (iii) provides us with a void fraction evaluation method that is free from computer overflow exceptions even for angular quadrature sets of high order and arbitrarily thick slabs. We perform a numerical experiment in order to illustrate the positive features of the mathematical model and the void fraction evaluation method reported here, and we conclude this article with closing remarks and directions for future work.

A shallow water, nonlinear spectral wave transformation model is developed for conditions of a mild sloping bottom (μ = ∇h/kh ≪ 1) and small amplitude effects (ε = ζ/h ≪ 1). Nonlinearities and combined shoaling and refraction effects act on the same time and length scales. The evolution equation of the wave action is prescribed by the wave Boltzmann equation, whereby resonant collinear triad interactions transfer energy among Fourier components. Combined shoaling and refraction effects are taken into account through the geometrical optics approximation. A numerical solution of the three‐wave collision integral is presented, and the steady state wave Boltzmann equation is integrated using a piece wise ray method. The model is tested using the high‐resolution frequency‐directional wave spectra of Freilich, Guza, and Elgar (1990) that show nonlinear transfers of energy between both harmonic and nonharmonic frequencies. A digitized version of the measured frequency‐directional spectrum at 10‐m depth is evolved 246 m shoreward over a bathymetry of straight and parallel bottom contours to 4‐m depth. The model predicts the prominent spectral features in the measured wave field. The model results are in general superior to estimates using linear, finite depth wave theory, and they compare well with the observations in the region of the spectrum dominated by nonlinear effects.