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Please check what's in the Ulaby book regarding reflection.

  1. May 24, 2013 #1
    Attached is a scanned of the page in question. This is regarding to Perpendicularly polarized plane wave. in equation (9.47a) at the lower left corner it is the distance ##x_i## to the origin.
    [tex]x_i=x\sin\theta_i+z\cos\theta_i\;\hbox {(9.47a)}[/tex]

    That is not a distance. distance of ##|\vec x_i|=\sqrt{x^2 \sin^2\theta_i+z^2\cos^2\theta_i}##, not ##x_i=x\sin\theta_i+z\cos\theta_i##.

    Actually ##\hat x_i=\hat x\sin\theta_i+\hat z\cos\theta_i\;\hbox { and }\vec x_i=\hat x|x_i|\sin\theta_i+\hat z|x_i|\cos\theta_i##

    Am I missing something because it's Memorial Day this weekend?!!! Did I read the book wrong?
     

    Attached Files:

    Last edited: May 24, 2013
  2. jcsd
  3. May 25, 2013 #2
    I think I dislike the way this was presented in this book.

    Consider the incident wave like this

    [itex]e^{-j\, \vec{k}_i\, \cdot \, \vec{r}}[/itex]

    now if we factor out the magnitude of the wave vector so that [itex]\vec{k}_i = k_i \hat{k}_i[/itex]

    [itex]\vec{k}_i \, \cdot \, \vec{r} = k_i \hat{k}_i \cdot \vec{r}[/itex]

    [itex] \hat{k}_i[/itex] points along the direction that the incident wave travels.

    [itex] \hat{k}_i = \cos\theta \hat{z} + \sin\theta \hat{x}[/itex]

    talking the dot product of [itex] \hat{k}_i[/itex] with the position vector

    [itex] \hat{k}_i \cdot \vec{r} = z\cos\theta \hat{z} + x\sin\theta \hat{x}[/itex]


    [itex]\vec{k}_i \cdot \vec{r} = k_i \hat{k}_i \cdot \vec{r} = k_i(z\cos\theta \hat{z} + x\sin\theta \hat{x} )[/itex]
     
  4. May 25, 2013 #3
    Yes I figured this out today. None of the books present this well at all. I had to read Cheng's, Griffiths, and Ulaby and work on the vector calculus to figure this and interpreted it out, it's like what you have.

    I am surprised good book like Griffiths does not do a good job in this particular section.

    Notice the nomenclature of the E field is deceiving too. They all use ##\vec E_I(\vec r,t)##. But in fact ##\vec r=\hat xx+\hat yy+\hat zz## is not the direction of the propagation. ##\vec r## is only used to provide the c,y and z terms by the dot product. If it is according to Ulaby that I scanned, it should be ##\vec E_I(\vec x_i,t)## as ##\vec x_i## is the direction of propagation of the ##\vec E_I##.

    Thanks
     
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