Standing waves on a hanging rope

Abstract

A simple model is presented for waves traveling on a vertical rope of uniform density. The waves travel with an upward acceleration of g2. The standing wave frequencies and nodal positions for the modes of oscillation are calculated. The standing wave frequencies form a series of odd harmonics. The results are compared to those derived by Satterly.

On the Counterpropagation of Waves

“…Also, it has been observed that both approaches are equivalent when the same reference phase of the resultant wave is considered. However, for full validation, the analytical expressions obtained for the phase velocity through both approaches must be confronted with the results obtained from the direct numerical simulation of equation 9with equation (10) and equation (11) or equation (12). In this way, in the next section, a direct comparison between the analytical and numerical results for the resultant wave and its phase velocity will occur.…”

“…Reference 11 goes a little bit further and obtain the resultant wave in the medium but still just perform an analysis of the wave amplitude. Many books treat this problem still in a more restrict context, in the sense that one of the waves is a result of reflection of the other, since this configuration permeates many mechanical and electromagnetic systems, such as waves in strings of musical instruments, [12] waves in energy and communication lines, [13] to name a few examples. In this situation, dimensionless quantities like standing wave ratio (SWR) and the reflection coefficient acquire importance, which are associated with the amplitude of the individual waves.…”