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.