Hyperbolic Equations: Waves and Vibrations Problems
This chapter considers wave phenomena and vibrations of continuous media; problems that are described by hyperbolic equations. Such equations admit discontinuous solutions in space and time. The discontinuities may be present in the initial conditions or may develop in time for nonlinear problems or if the speed of wave propagation is a function of space. The chapter covers problems of transverse, longitudinal, and torsional vibrations of one-dimensional structures such as strings, beams, and bars. It solves the displacement field, and hence the acceleration is represented by a second-order time derivative. For strings and bars the space derivative is second order, whereas for Euler-Bernoulli's beam equation the space derivative is fourth order. The chapter discusses the time response of the aforementioned structures.