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In any real vibrating string, there are energy losses due to yielding
terminations, drag by the surrounding air, and internal friction within the
string. While losses in solids generally vary in a complicated way with
frequency, they can usually be well approximated by a small number of
odd-order terms added to the wave equation. In the simplest case, force is
directly proportional to transverse string velocity, independent of
frequency. If this proportionality constant is
, we obtain the
modified wave equation
Setting
in the wave equation to find the relationship
between temporal and spatial frequencies in the eigensolution, the wave
equation becomes
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(H.22) |