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

[yungman]

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.
x_i=x\sin\theta_i+z\cos\theta_i\;\hbox {(9.47a)}

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?

 

[MisterX]

I think I dislike the way this was presented in this book.

Consider the incident wave like this

e^{-j\, \vec{k}_i\, \cdot \, \vec{r}}

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

\vec{k}_i \, \cdot \, \vec{r} = k_i \hat{k}_i \cdot \vec{r}

\hat{k}_i points along the direction that the incident wave travels.

\hat{k}_i = \cos\theta \hat{z} + \sin\theta \hat{x}

talking the dot product of \hat{k}_i with the position vector

\hat{k}_i \cdot \vec{r} = z\cos\theta \hat{z} + x\sin\theta \hat{x} \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} )

 

[yungman]

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