For any given wavelength, the relation (21) then determines the frequency. Thus, from what we have considered so far, all frequencies can lead to standing waves. The demonstration in lecture, however, shows that only certain frequencies result in standing wave solutions. Therefore, it appears that only certain wavelengths are allowed in the amplitude function A(x) and, therefore, there must be additional conditions which we have not identified. These are known as boundary conditions.
What we have ignored is the motion of the very last segements of the string. The equation of motion derived from Figure 2 applies to interior segments only. Thus is because we implicitly assumed that each segment is in contact with additional segments to the left and to the right. The equations of motion for the segments at the end, or boundary, of the system will be different. These equations of motion are known as the Boundary Conditions (BC's).
There are as many different boundary conditions as there are things to which we can attach the ends of the string. To derive the boundary conditions, we proceed as with deriving any other equation of motion. We write down the laws of motion for the end of the string and express them entirely in terms of constants characterizing the system and the solution y(x,t) and its derivatives. We now consider the two most most common types of boundary condition.