  By means of the plane wave method we study spin wave dynamics in
two-dimensional bi-component magnonic crystals based on a squeezed hexagonal
lattice and consist of a permalloy thin film with cobalt inclusions. We explore
the dependence of a spin wave frequency on the external magnetic field,
especially in weak fields where the mode softening takes place. For considered
structures, the mode softening proves to be highly non-uniform on both the mode
number and the wave vector. We found this effect to be responsible for the
omnidirectional band gap opening. Moreover, we show that the enhancement of the
demagnetizing field caused by the squeezing of the structure is of crucial
importance for the non-uniform mode softening. This allows us to employ this
mechanism to design magnonic gaps with different sensitivity for the tiny
change of the external field. The effects we have found should be useful in
designing and optimization of spin wave filters highly tunable by a small
external magnetic field.
