Numerical study of linear and nonlinear string vibrations by means of physical discretization

To solve partial differential equations numerically, discretization of the continuous model is required and may be achieved either mathematically or physically. This paper illustrates how physical discretization of a continuous string may be accomplished by employing discrete model theory which has as its essential substance Newtonian mechanics.Typical examples of wave motion in discretized ‘linear’ and ‘non-linear’ strings are discussed. They include the transverse vibrations of a string after having been subjected to a given initial displacement, reflection and superposition of wave pulses in the string, and resonance of the string when coupled to a harmonic vibrator. The equations that arise after application of discrete model theory to these problems, describe the subsequent motion of the string, and are solved numerically by computer. In all cases the results obtained for the discrete linear string agree remarkably well with those for the corresponding continuous physical string. The stability of the solutions obtained by discretization are also investigated.