Square Pulse Train Fourier Series help?

[asdf12312]

Square Pulse Train Fourier Series help??

Homework Statement

problem+directions below:

Homework Equations

ω=2\pif
β=\frac{2\pi}{\lambda}

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 t_d (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\pif*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?

 

[asdf12312]

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 u_p=f*λ and given f, then the wavelength λ=300m. So now my Fourier series equation looks like this:

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

can someone confirm this is right so far?