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The first four terms of the Fourier series of a square wave,
, and .
Transverse standing wave:
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:
To find a particular solution, to (?) we first consider two different inhomogeneous equations:
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. Relating the wavenumber of the lowest order mode to string length by :
After modifying an equation from Wikipedia:
- ↑ See David R Rowland 2011 Eur. J. Phys. 32 1475, equation 11