Square Pulse Train Fourier Series help?

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Square Pulse Train Fourier Series help??

Homework Statement


problem+directions below:
33jjyn8.jpg



Homework Equations


ω=2[itex]\pi[/itex]f
β=[itex]\frac{2\pi}{\lambda}[/itex]

The Attempt at a Solution


Since the problem asks to make all time-dependent sinusoidal functions deal with x-direction, i don't think i need to worry about the sin function because it is dependent on td (duty cycle) and not time. I am aware of the cos format for x and t dependent but this is how i would plan to change it (for the equation with both sin and cos):
cos(n*2[itex]\pi[/itex]f*t) -> cos[n(ωt-βx)]

basically I just factored out the n, and I don't think I need to include any phase shift. I just changed the cos part, the rest of the equation stays the same.

I can worry about using Fourier series with MATLAB later, I just wanted to know if the equation for the fourier series I had was right. also, I don't know how to find the value of wavelength for β, since it is not given anywhere. would i need to use the equation for phase velocity, if it is related to the f given somehow?
 
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Answers and Replies

  • #2
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OK, so thankfully our professor went over how to find the wavelength in a lecture. According to him in a non-dispersive medium, the phase velocity is simply c=speed of light=3x108. So if I use the equation up=f*λ and given f, then the wavelength λ=300m. So now my fourier series equation looks like this:

1+[itex]\frac{20}{\pi} \sum{\frac{1}{n}sin(\frac{n\pi}{10})cos[n(ωt-βx)]}[/itex] where ω=2[itex]\pi[/itex]*106, β=[itex]\frac{2\pi}{300} = \frac{\pi}{150} [/itex], and the fourier series goes from n=1 to a finite number N=1000 (not infinity).

can someone confirm this is right so far?
 

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