String Theory

The vibrating string has been employed by nearly every human culture to create musical instruments. Although the musical application has attracted the attention of mathematical and scientific analysts since the time of Pythagoras (570 BC–495 BC), we will study the string primarily because its vibrations are easy to visualize and string vibrations introduce concepts and techniques that will recur throughout our study of the vibration and the acoustics of continua. In this chapter, we will develop continuous mathematical functions of position and time that describe the shape of the entire string. The amplitude of such functions will describe the transverse displacement from equilibrium, y(x, t), at all positions along the string. The importance of boundary conditions at the ends of strings will be emphasized, and techniques to accommodate both ideal and “imperfect” boundary conditions will be introduced. Solutions that result in all parts of the string oscillating at the same frequency which satisfy the boundary conditions are called normal modes, and the calculation of those normal mode frequencies will be a focus of this chapter.