The notion of similarity is familiar from geometry. Two triangles are said to be similar if all of their angles are equal, even if the sides of the two triangles are of different lengths. The two triangles have the same shape; the larger one is simply a scaled up version of the smaller one. This notion can be generalized to include physical phenomena. This is important when modeling physical phenomena--for instance, testing a prototype of a plane with a scale model in a wind tunnel. The design of the model is dictated by dimensional analysis.
Return to the mathematical statement of the 
-Theorem, Eq. (1.8).
We can identify the following dimensionless parameters:
and so on, such that Eq. (1.8) can be written as
The parameters 
are known as similarity parameters
Now if two physical phenomena
are similar, they will be described by the same function 
. Denote the
similarity parameters of the model and the prototype by the superscripts m
and p, respectively. Then if the two are similar, their similarity
parameters are equal:
so that
Therefore, in order to have an accurate physical model of a prototype, we must first identify all of the similarity parameters, and then insure that they are equal for the model and the prototype.
Finally, we come to estimating. In this course we will often make order of magnitude estimates, where we try to obtain an estimate to within a factor of ten (sometimes better, sometimes worse). This means that we often drop factors of two, etc., although one should exercise some caution in doing this. Estimating in this fashion is often aided by first doing some dimensional analysis. Once we know how the governed parameter (which we are trying to estimate) scales with other quantities, we can often use our own personal experience as a guide in making the estimate. More examples of this later in the course (especially elastic similarity).
