- #1

baby_1

- 159

- 15

- Homework Statement
- Maxwell equation

- Relevant Equations
- V=xt

Hello,

Regarding the wave oblique angle propagation and based on Balanis "Advanced engineering Electromagnetic" book on page 136 ( it has been attached) I need to know why the phase velocity in x direction is not important to keep in step with a constant phase plane( Just equation 4-23).

I derived the electric filed:

[tex]\vec{E}=E_{x}(z)e^{-j\beta (xSin(\theta )+zCos(\theta ))}[/tex]

For calculating the phase velocity we need to obtain the time-domain form:

[tex]\vec{E(t)}=Re\{E_{x}(z)e^{-j\beta (xSin(\theta )+zCos(\theta ))}e^{jwt}\}=E_{x}(z)Cos(\omega t-\beta (xSin(\theta )+zCos(\theta ))[/tex]

Calculating the Z- phase velocity (X=0):

[tex]\frac{d}{dt}(\omega t-\beta zCos(\theta ))=0=>\omega -\beta \frac{dz}{dt}Cos(\theta )=0=>v_{z}=\frac{\omega}{\beta Cos(\theta )}[/tex]

which is the same as the book. However,

1)First: I need to know why the x velocity (on x-axis) is not important.

x phase velocity:(z=0)

[tex]\frac{d}{dt}(\omega t-\beta xSin(\theta ))=0=>\omega -\beta \frac{dx}{dt}Sin(\theta )=0=>v_{x}=\frac{\omega}{\beta Sin(\theta )}[/tex]

2)Second: Why the resultant phase velocity is not the same as the light velocity([itex]v_{c}=\frac{\omega}{\beta}[/itex])

[tex]\sqrt{v_{z}^2+v_{x}^2}=\sqrt{({\frac{\omega}{\beta Cos(\theta )})}^2+({\frac{\omega}{\beta Sin(\theta )})}^2}=(\frac{\omega}{\beta Sin(2*\theta )})[/tex]

Regarding the wave oblique angle propagation and based on Balanis "Advanced engineering Electromagnetic" book on page 136 ( it has been attached) I need to know why the phase velocity in x direction is not important to keep in step with a constant phase plane( Just equation 4-23).

[tex]\vec{E}=E_{x}(z)e^{-j\beta (xSin(\theta )+zCos(\theta ))}[/tex]

For calculating the phase velocity we need to obtain the time-domain form:

[tex]\vec{E(t)}=Re\{E_{x}(z)e^{-j\beta (xSin(\theta )+zCos(\theta ))}e^{jwt}\}=E_{x}(z)Cos(\omega t-\beta (xSin(\theta )+zCos(\theta ))[/tex]

Calculating the Z- phase velocity (X=0):

[tex]\frac{d}{dt}(\omega t-\beta zCos(\theta ))=0=>\omega -\beta \frac{dz}{dt}Cos(\theta )=0=>v_{z}=\frac{\omega}{\beta Cos(\theta )}[/tex]

which is the same as the book. However,

1)First: I need to know why the x velocity (on x-axis) is not important.

x phase velocity:(z=0)

[tex]\frac{d}{dt}(\omega t-\beta xSin(\theta ))=0=>\omega -\beta \frac{dx}{dt}Sin(\theta )=0=>v_{x}=\frac{\omega}{\beta Sin(\theta )}[/tex]

2)Second: Why the resultant phase velocity is not the same as the light velocity([itex]v_{c}=\frac{\omega}{\beta}[/itex])

[tex]\sqrt{v_{z}^2+v_{x}^2}=\sqrt{({\frac{\omega}{\beta Cos(\theta )})}^2+({\frac{\omega}{\beta Sin(\theta )})}^2}=(\frac{\omega}{\beta Sin(2*\theta )})[/tex]

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