Analytical solutions of a linear internal-tide generation problem are presented for a stratification that involves both an interface (thermocline) and a lower layer of constant stratification. It is shown that an internal tidal beam scatters strongly at a moderately developed thermocline but much less at a thermocline that is either very weak or very strong. This scattering goes along with a local wave activity at the thermocline as well as with a downward leaking from the thermocline. It is argued that these processes offer a clue to explaining observations in the Bay of Biscay of 'local generation' of internal soliton packets. This is substantiated by numerical results from a nonlinear nonhydrostatic model, which is derived in this paper. Also, the effects of Coriolis dispersion are discussed. It appears that the observations from solitary waves in the Bay of Biscay can be contrasted with those in the Sulu Sea, to which a different stratification regime applies. An interpretation is given in terms of the modal structure.