The paper deals with a non-linear problem of long water waves approaching a sloping beach. In order to describe the phenomenon we apply the Lagrange’s system of material variables. With these variables it is much easier to solve boundary conditions, especially conditions on a shoreline. The formulation is based on the fundamental assumption for long waves propagating in shallow water of constant depth that vertical material lines of fluid particles remain vertical during entire motion of the fluid. The analysis is confined to one – dimensional case of unsteady water motion within a ’triangular’ body of fluid. The partial differential equations of fluid motion, obtained by means of a variational procedure, are then substituted by a system of equations resulting from a perturbation scheme with the second order expansion with respect to a small parameter. In this way the original problem has been reduced to a system of linear partial differential equations with variable coefficients. The latter equations are, in turn, substituted by a system of difference equations, which are then integrated in a discrete time space by means of the Wilson-µ method. The procedure developed in this paper may be a convenient tool in analysing non-breaking waves propagating in coastal zones of seas. Moreover, the model can also deliver useful results for cases when breaking of waves near a shoreline may be expected.

Six enigmatic fossils from the Famennian (Devonian) Cleveland Shale in Ohio, U.S.A., are interpreted here as

arthrodiran (Placodermi) egg cases. Recognition as egg cases is confirmed based on the observation of layered

collagen fibers. The presence of a tuberculated bone fragment preserved within one case confirms a vertebrate

source. The nature of the tubercles and the unique morphology of the egg cases supports the interpretation of

an arthrodiran source. Reports of Devonian egg cases are limited to either assumed chondrichthyan producers

or a putative ‘egg sac’ with a morphology atypical for any vertebrate. The Cleveland Shale egg cases thus

represent the first record for a non-chondrichthyan producer. Among placoderms, behaviors of a pelagic life

style with obligate nesting sites, reef fishes with live birth, and estuarine and fluvial nurseries, along with eggcase

oviparity testifies to the diversity of reproductive strategies. As with modern fishes these strategies may

be ecologically driven and the derived and variable reproductive biology of extant chondrichthyans is actually

a primitive condition among gnathostomes. One consequence of the diversity of reproductive strategies (dependent

on the topology of relationships) is the independent origin of internal fertilization within placoderms,

possibly suggesting external fertilization as the primitive gnathostome reproductive mode.