A Study on Steady State Traveling Waves in Strings and Rods


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Virginia Tech


The main focus of this present work is to study how mechanical steady state traveling waves can be generated and propagated through one dimensional media by applying forces. By steady state traveling waves we refer to propagating mechanical waves in a finite medium that never exhibit reflections at the boundaries and continuously move from one end of the structure to the other.

Mechanical waves can be classified as traveling, standing and hybrid waves that are the results of the interplay of excitation forces, applied force locations, and the boundary conditions of the structure. Traveling waves carry energy through a defined medium while standing waves keep energy at certain areas that are associated with the modes of excitation. To understand the interaction of systems that exhibit traveling waves with their surrounding media (i.e., swimming flagella, manta ray locomotion), it is crucial to first understand the wave propagation and what is desired in these structural systems.

The parameters that affect the generation and propagation of waves should be welldefined to control and manipulate the desired system’s response. One-dimensional string and rod equations are studied with various boundary conditions to generate steady-state traveling waves in a string and longitudinal traveling waves in a rod. Two excitation forces are applied to a string and a rod near the boundaries to understand the generation and propagation of traveling and standing waves at various frequencies. The work examines the quality of the wave propagation in a string, and in a rod. A cost function approach is applied to identify the quality of such waves. Furthermore, steady-state square traveling waves are generated in a string and in-plane in a rod, both theoretically and experimentally. To the authors’ knowledge this is the first time this has been attempted in the literature.

Determining the quality of traveling waves and understanding the parameters on the wave propagation of a string and rod can lead to further understand and leverage various engineering disciplines such as mechanical actuation mechanisms, propulsion of flagella, and the basilar membrane in the ear’s cochlea.



Traveling Waves, Vibrations, Two-force excitation, Cost Function, Strings, Bars, Rods