Energy of a Traveling Wave on a String

This Demonstration shows how mechanical energy is transported by a traveling harmonic wave on a string. For a differential string element , the kinetic and potential energies and are always equal and they vary as the disturbance travels through the medium. The total mechanical energy in one wavelength, , is conserved.

SNAPSHOTS

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DETAILS

The top plot shows a view of the wave along the string. A transverse wave traveling in the direction is described by , where is the amplitude (maximum displacement from equilibrium), is the wave number and is the angular frequency. The wave travels with speed , which depends on the tension and the linear density . For a given frequency of oscillation , the wavelength is . Vary the tension or the linear density of the string to change the propagation speed and consequently the wavelength. Vary the time to see how the disturbance travels down the string, as each point oscillates up and down in time.
The bottom plot shows a view of the energy density along the string. The kinetic and potential energies in the differential string element are always equal but vary by location and with time as .
The string element at the position of zero disturbance has maximum instantaneous speed , therefore maximum kinetic energy . At the same places the string is stretched the most (largest slope ), thus the most elastic potential energy is stored there. At the crests of the wave, , the string element is instantaneously at rest and in equilibrium . The mechanical energy of a segment is . Each string element is a non-isolated system and its mechanical energy is not conserved. changes as the disturbances passes. Integrating over one wavelength gives the energy transported by the wave as it travels along the string: , which is a conserved quantity.
Reference
[1] L. M. Burko, "Energy in One-Dimensional Linear Waves in a String," European Journal of Physics, 31(5), 2010 L71–L77. doi:10.1088/0143-0807/31/5/L01.
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