How is the constant pi/L deduced in Fourier series?

[henry wang]

How is pi/L part deduced in (n*pi*x)/L?

 

[mathman]

Essentially it has to do with the interval that the series is defined for. The length of the interval is 2L while \pi is the normalization (based on the fact that sines and cosines have period 2\pi).

 

[strauser]

How is pi/L part deduced in (n*pi*x)/L?

We're requiring that the fundamental frequency is periodic in space, with spatial period $2L$. So we have for the fundamental frequency: $$\cos \omega x = \cos(\omega(x+2L)) = \cos(\omega x + 2L \omega) \Rightarrow 2L \omega = 2\pi \Rightarrow \omega = \frac{\pi}{L}$$

 

[mfig]

It is simply a change of variables to allow the series to work on $[-L,L]$ instead of the restricted domain of the simple series given by $[-\pi,\pi]$. Have a look at the transformation from equations (6-9) to the more general equations (13-16) via the substitution given in equation (11).

 

[henry wang]

Thank you guys, I understand now.