We use a simple effective model, obtained through the application of
high-frequency homogenisation, to tackle the fundamental question of how the
choice of gradient function affects the performance of a graded metamaterial.
This approach provides a unified framework for comparing gradient functions
efficiently and in a way that allows us to draw conclusions that apply to a
range of different wave regimes. We consider the specific problem of
single-frequency localisation, for which the appropriate effective model is a
one-dimensional Schrodinger equation. Our analytic results both corroborate
those of existing studies (which use either expensive full-field wave
simulations or black-box numerical optimisation algorithms) and extend them to
other metamaterial regimes. Based on our analysis, we are able to propose a
design strategy for optimising monotonically graded metamaterials and offer an
explanation for the lack of a universal optimal gradient function