Nature of approximation

to thine own self be true

The smallness of a term in an equation is a hint that the term might be omitted, but we must be careful because a quantity that is negligible in one equation, may not be in another, perhaps because it appears there multiplied by a large factor. This is because it represents a different process in the new context. Micawber commented that even a small excess of expenditure over income eventually led to disaster.

Mathematicians like to expand in powers of a small number ∊ say, and keep only the lowest orders. This often presents difficulties in physical interpretation because the terms in ∊2 often represent very different physics to those in ∊. For example a wave of small amplitude ∊ advects wave properties at a rate proportional to ∊2, and often it is the transfer we are really interested in, not just the existence of the wave. Thus we find the shape of the motion pattern, represented by the correlation between different properties, interesting, as well as the amplitude.

The longer-term evolution of the amplifying waves found in unstable linearised systems, depends on the non-linear terms, because it is only these that prevent the unlimited increase in amplitude. In contrast, the amplitude of forced waves is often a more incidental property. It might be argued that this is untrue of breaking waves where non-linearity is an essential ingredient.