Energy of a Standing Wave on a String

This Demonstration shows the distribution of energy in a standing wave on a string. The mechanical energy in a differential string element varies as along the string. As the string oscillates in time, the kinetic energy and potential energy in a differential string element are periodically interchanged. The mechanical energy is conserved. Integrating the differential element of energy over one wavelength gives the total energy , which is conserved.

DETAILS

Standing waves on a string with fixed ends result from the interference of two waves with equal amplitude and frequency , traveling in opposite directions with speed . The speed of the waves depends only on the tension and linear density of the string. Standing waves exist only at frequencies that satisfy the boundary conditions where is the length of the string and is the mode (). The wavelength of a standing wave for a given mode is .
Standing waves can be represented by , where is the wave number and is the angular frequency. The displacement of the string oscillates as and the amplitude varies along the string as . The points on the string where are called nodes and the points with maximum amplitude are antinodes.
For a string element the kinetic energy is
and the potential energy is
.
You can vary the time and observe the oscillation of the standing wave and the continuous transformation of energy from kinetic to potential and back. at the nodes where the string elements do not move and the string is undisturbed. and reach maximum values (at different times) at the antinodes. The total mechanical energy in the string element , , is time independent; is conserved. The energy for wavelength λ, given by , is conserved.
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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