String vibration/Nonlinear

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The first four terms of the Fourier series of a square wave, .

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:

NOW[edit | edit source]

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 :


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Other identities[edit | edit source]

wikipedia:special:permalink/1017302768

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After modifying an equation from Wikipedia:

  1. See David R Rowland 2011 Eur. J. Phys. 32 1475, equation 11