The first step in the general analysis is to identify the degrees of freedom, the minimal set of variables needed to define the configuration of the system at a given instant in time. Because the string consists of a large collection of particles or segments, we must be able to specify the location of each such segment. For simplicity, we here consider only up and down motions of the string, thus the degrees of freedom must specify the x and y coordinates of each segment.
Simplifying approximation -- We shall make one single simplifying approximation in our analysis of the string. We shall assume that the amplitude of the waves on the string is small. One could begin without this approximation, carry the analysis in full and then take the limit of small amplitudes. It turns out, however, that in practice the small amplitude approximation is quite accurate even for relatively large vibrations and that the full analysis becomes quite complicated. Without this approximation, we would learn little for much additional effort. Thus, we shall assume from this point onward we shall assume that the amplitude of the waves we study is small.
This approximation has a very important implication of which we shall make much use. If the amplitude of the waves on the string is small, then the length of string between the walls is always very nearly L. Thus, there is very little pulling of the string in and out of the hole where we apply the tension and the x position of each segment changes negligibly. As the x positions of the segments do not change, they do not need to be specified as degrees of freedom. Moreover, we can use the x position as a way to identify or label the segments.
To specify the state of the string we need only specify the y-location of each segment at horizontal location x between the walls. Mathematically, this is the same a giving a function y(x). This specifies the state of the string because to determine what the string looks like, one would simply plot the given function y(x).
Given y(x) as a way to specify the degrees of freedom, a solution
for the string should specify a function y(x) for each value of time
t. Mathematically, this is the same as a function of the form
y(x,t) because, for any given instant in time 
, we can get the
state of the string by plotting 
versus x. Note that
because a solution y(x,t) is a function of two variables, we shall
now be working primarily with partial derivatives.