If you don't understand this animated gif, look at the next figure
Define
, and
.
Transverse standing wave:
Define
Second order differential equation with one variable: https://openstax.org/books/calculus-volume-3/pages/7-2-nonhomogeneous-linear-equations
where
is the solution to the homogeneous equation, i.e., solution to
Link to wikipedia:Fourier series?
Employ two identities:
and
To find a particular solution,
to (?) we first consider two different inhomogeneous equations:
Recall
=>
If
is proportional to
, then
, and:
=>
If
is proportional to
, then
and:
=
=>
=>
. Now use
.
By the linearity of the operator
we see that a particular solution to (?) is the sum of
In these units the speed of a
wave is
. This permits us to write an expression that does not depend on the choice of units.[1] Relating the wavenumber of the lowest order mode to string length by
:
<math></math>
wikipedia:special:permalink/1017302768
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After modifying an equation from Wikipedia:
- ↑ See David R Rowland 2011 Eur. J. Phys. 32 1475, equation 11