One-Dimensional Forced Standing Waves

Abstract

This research-oriented report analyzes the impact of a class of time-independent forces and their time-dependent counterparts on the characteristics of solutions to the wave equation on a one-dimensional string. The solution of these equations, subject to Initial and Boundary Value (IBV) conditions, is investigated, yielding modified standing waves. Analyses are heavily Computer Algebra oriented. The popular Computer Algebra System (CAS), specifically Mathematica , has been pivotal in uncovering the numerical, algebraic, and graphical aspects of the investigation. The time-dependent force functions not only generalize the scope of the analysis but also possess 1) a character that their functionality by transitioning to the coordinate-dependent only is conducive to expected output, and 2) the absence of these forces frees the imposed IBV, conducive to the characteristics of the standard standing waves. This comprehensive report embodies an atlas of graphs for both mentioned forces. It is shown that the proposed one-dimensional time-independent forces give rise to previously unseen standing waves with a peculiar character. The report includes the essential Mathematica code for most calculations and all the graphs depicted. The report allows reproduction by individuals familiar with Mathematica . It is crafted and developed so that modification and generalization of the applied IBVs are readily possible. The Conclusion segment embodies a suggestion to expand the scope of future investigations.