Energy of a Standing Wave on a String

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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.
Contributed by: Anna Petrova-Mayor (August 2022)
Open content licensed under CC BY-NC-SA
Snapshots
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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