2.1.1. Free Vibrations of a String

Let us examine a string with length L and mass per unit length ? bound at both ends and subjected to a tension T ^[1]. We will ignore the interaction with the bridge and the important resulting attenuation; these will be the subject of Study problem 2.6.5. We will also ignore other sources of attenuation, for which you may refer to [VAL 93]. There are three possible types of vibrations: transverse, longitudinal and torsional vibrations. We will only be studying the first kind, and we will assume that the vibration occurs in an xOy plane. In this plane, the string's ends are located at points (0,0) and (0, L), and the string's position at a time t is given by the equation y = u( x, t). The boundary conditions (at the ends of the string, where it is bound) impose that u(0, t) = u( L, t) = 0 for any t.

In order to obtain the equation that governs the string's movement, we have to consider, for a given time t, the forces applied to a small portion of the string located between the x-coordinates x and x + dx (see Figure 2.1). The angle between the string and the Ox axis is denoted by ?( x, t). At point x + dx, the...