Physics/A/String vibration/Nonlinear

Define

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

, 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>

Other identities

edit

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