Visualization of Wave Phenomena by an Array of Coupled Oscillators

Mechanical engineering curricula typically include courses in classical mechanics and continuum mechanics. Classical mechanics is governe d by the Newtonian axioms, which lead to ordinary differential equations as the equations f motion. The mathematical description of continuum mechanics, on the other hand, is based on partial differential equations, describing the conservation laws and the constitutive relation s. The underlying theories of ordinary and partially differential equations are usually covere d in different mathematics courses, and a typical approach to a first discussion of partial d ifferential equations in engineering mathematics is the heuristic derivation of the tran sversal wave equation of a vibrating string. Another approach, the continuum limit of the loaded string, leads to the one-dimensional longitudinal wave equation. Both approaches start from the ordinary differentia l equations of Newtonian mechanics and lead to partial differential equations of continuum mechanics. The advantage of the continuum limit of a chain of masses connected by s prings is that it is easily comprehensible for students and, in addition, many materials respo nd t small perturbations just as if they were a system of coupled oscillators. This harmonic scillator response to perturbations leads in a continuum model to the appearance of wave phen om a. For the visualization of such wave phenomena, a com puter program that simulates a twodimensional spring-mass system has been developed w ithin an undergraduate student project. The model consists of a rectangular lattice of regu larly spaced point masses connected to each other and to the rigid boundary by a network of mas sless springs. The force on each mass is computed due to its spring connections with its nei ghbors, along with external forces such as gravity. Energy dissipation can be added to the mod el n demand by viscoelastic damping. The motion of each particle is governed by Newton’s second law, which requires the solution of a system of coupled ordinary differential equati ons. This is done in the C# programming language via a variety of implemented numerical int egration schemes. The computer program allows the visualization of the motion of the point masses, which can be initiated by the displacement of an arbitrary number of masses via m ouse drag. The motion of a single mass connected by springs to the adjacent walls closely resembles the behavior of the well-known harmonic oscillator. The introduction of additional masses brings particle interaction into effect, which leads to en ergy transfer in the system. When the number of involved mass points is increased, the moving pa rticles increasingly appear to the observer as a continuous system. This is due to a scale chan ge that links the microstructure of coupled particles with the macroscopic behavior of a contin uous material, which includes wave behavior like interference effects. The dynamic vis ual output of the program can increase and enhance understanding of various wave phenomena and is therefore well suited as both a teaching aid and an analysis tool.