The purpose of this investigation was to examine the relationship between tension and velocity in two systems, and to explore the dispersion relationship in each case.
So, what is dispersion anyway? When wave velocity is dependent on frequency for a given medium, that medium is said to be dispersive. - (Marion & Thornton). In other words, if the velocity of waves in a system changes with frequency (while tension remains constant), the material through which the waves move is dispersive - there is dispersion in the system.
Specifically, in the beaded string system, we expect at a certain frequency to see the 1/wavelength of oscillations to roll over sinusoidally. Velocity should decrease as the frequency of oscillations increases, forcing the system beyond a certain mode of oscillation. Typically, that final mode has each bead oscillating asymmetrically with its neighbors. An w vs k graph that is linear, or does not roll over at some point, indicates a non-dispersive system.
In each system, it was necessary to produce standing waves
in the medium. Frequency at
resonance was read from the voltage source of the driver. Wavelength was determined either by directly measuring node
separation or by noting the total length of the medium and the number of nodes
on it at a given frequency. Wavelength
multiplied by frequency gives the velocity of a wave.
Wave velocities were calculated for varying tensions and frequencies of
oscillation and results were analyzed using Microsoft Excel.
Follow this link to an interactive Physlet demonstration of dispersion on a beaded string designed by W. Christian at Davidson College. For more information about Physlets, visit Dr Christian's website.