Mechanical and Electromagnetic Vibrations and Waves

Chapter 5

Mechanical Waves

In this chapter we study mechanical waves in solids and fluids. Any disturbance displaces the molecules or atoms of the medium and manifests itself as local displacements or deformations (called strains) and internal forces (called stress). If the disturbance depends on time, it propagates as an elastic wave in solids, sound wave in fluids or a surface wave in liquids. We study the elastic waves in the case of simple systems: strings, rods and membranes. The study of sound, also called acoustics, includes physical acoustics (which we introduce in this chapter), physiological acoustics, and architectural acoustics. Sound waves include audible sounds as well as infrasounds and ultrasounds, which are inaudible. We also briefly consider surface waves.

5.1. Transverse waves on a taut string

Consider a line of masses m joined by an ideally massless string taut to a tension F (Figure 5.1a). Let Ox be the axis of equilibrium of the string, and d the distance between consecutive masses. Due to the string tension, a displacement u_n of the mass (n) in the transverse y-direction propagates to the mass (n + 1) producing a displacement u_n+1 and so on. Each mass (n) is subject to two forces of equal magnitude F exerted by the string toward the masses (n –1) and (n + 1). Their resultant is generally not in the transverse direction Oy. However, if the oscillations are small, the longitudinal component of this resultant is of the second order in u_n, u_n−1 or u_n+1 and the ...