The structure of the numerical range and root zones of a class of operator functions, arising from one or two parameter polynomial operator pencils of waveguide type is studied. We construct a general model of such kind of operator pencils. In frame of this model theorems on distribution of roots and eigenvalues in some parts of root zones are proved. It is shown that, in general the numerical range and root zones are not connected but some connected parts of root zones are determined. It is proved that root zones, under some natural additional conditions which are satisfied for most of waveguide type multi-parameter spectral problems, are non-separated, i.e., they overlap.