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Wave Motion in Strings
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Wave Equation Solutions
Standing Waves
Imagine a string of length
that is rigidly fixed at
and
. This implies boundary conditions of
and
.
Making use of
and Eq. (
3
), we find that
and
, so
(
4
)
Using sinusoidal sum and difference formulae,
and
, we obtain
The boundary condition
requires that
or
, so that
is restricted to values
or
.
Thus, the rigidly fixed string has normal modes of vibration given by
(
5
)
which are harmonic because each
is a integer multiple
times
. These correspond to the well known standing wave patterns on a stretched string.
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.