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VinnyCee
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Homework Statement
Given...
[tex]v\left(z,\,t\right)\,=\,5\,e^{-\alpha\,z}\,sin\left(4\pi\,\times\,10^9\,t\,-\,20\pi\,z\right)[/tex]
where z is distance (m), find...
(a) Frequency
(b) Wavelength
(c) Phase Velocity
(d) At z = 2m, the amplitude is 1 [V], Find the attenuation constant ([itex]\alpha[/itex]).
Homework Equations
[tex]f\,=\,\frac{1}{T}[/tex]
[tex]y\left(x,\,t\right)\,=\,A\,cos\left(\frac{2\pi\,t}{T}\,-\,\frac{2\pi\,x}{\lambda}\,+\,\phi_0\right)[/tex]
[tex]u_p\,=\,f\,\lambda[/tex]
The Attempt at a Solution
(a)Using the first term ([itex]\frac{2\pi\,t}{T}[/itex]) in the argument to the cosine in the general form above...
[tex]\frac{2\pi}{T}\,=\,4\pi\,\times\,10^9\,\,\longrightarrow\,\,T\,=\,\frac{2\pi}{4\pi\,\times\,10^9}\,=\,0.5\,\times\,10^{-9}[/tex]
[tex]f\,=\,\frac{1}{T}\,=\,\frac{1}{0.5\,\times\,10^{-9}}\,=\,2\,\times\,10^9\,=\,2\,Ghz[/tex](b)
Using the second term ([itex]-\,\frac{2\pi\,x}{\lambda}[/itex]) in the argument to the cosine in the general form above...
[tex]\frac{2\pi}{\lambda}\,=\,20\pi\,\,\longrightarrow\,\,\lambda\,=\,\frac{2\pi}{20\pi}\,=\,\frac{1}{10}\,=\,0.1\,m[/tex](c)
[tex]u_p\,=\,f\,\lambda\,=\,\left(2\,\times\,10^9\right)\,(0.1)\,=\,200,000,000\,\frac{m}{s}[/tex](d)
[tex]1\,=\,5\,e^{-2\,\alpha}\,sin\left(4\pi\,\times\,10^9\,t\,-\,40\pi\right)[/tex]
[tex]5\,e^{-2\alpha}\,=\,1\,\,\longrightarrow\,\,-2\alpha\,=\,ln\left(\frac{1}{5}\right)\,\,\longrightarrow\,\,\alpha\,=\,0.8047[/tex]
Right?
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