Phase velocity in oblique Angle propagation (Plane wave)

In summary, the phase velocity in the x direction is not important because it can be found in a similar manner to the z direction and is not the same as the speed of light due to the different components of the velocity.
  • #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).
question.JPG
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]
 

Attachments

  • Question.pdf
    402.7 KB · Views: 169
Last edited:
Physics news on Phys.org
  • #2
baby_1 said:
1)First: I need to know why the x velocity (on x-axis) is not important.
The "x velocity" is just as important as the "z velocity". I guess that the textbook just decided not to mention the x velocity since it can be found in a similar manner to the z velocity. Below, I denote these velocities as ##u_x## and ##u_z##.

baby_1 said:
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]
Imagine a stick moving with translational velocity ##\vec v## as shown below. The stick is the black line and ##\vec v## is perpendicular to the stick. The z and x components of ##\vec v## would be ##v_z= v \cos \theta## and ##v_x = v \sin \theta##.
1627072157553.png


The points of intersection of the stick with the z and x axes are shown in green. The speed at which the intersection with the z-axis moves along the z-axis is ##u_z = \frac{v}{\cos \theta}##. Similarly, for the speed of the intersection along the x-axis, ##u_x = \frac{v}{\sin \theta}##.

##u_z## and ##u_x## are not the z and x components of the velocity of some single point. In particular, they are not the z and x components of the velocity of some point on the stick. So, we do not expect ##v = \sqrt {u_z^2+u_x^2}##.

The textbook uses the notation ##v_{pz}## and ##v_{px}## for ##u_z## and ##u_x##. These should not be confused with the ##z## and ##x## components of the vector ##\vec v##: ##v_z## and ##v_x##.
 
  • Like
Likes Delta2, baby_1 and hutchphd
  • #3
I see the Like is nothing to thank you. I really appreciate your explanations and the time you spent to draw the shape and explain more and more.
 
  • Like
Likes Delta2
  • #4
You are welcome.
 

FAQ: Phase velocity in oblique Angle propagation (Plane wave)

1. What is phase velocity in oblique angle propagation?

Phase velocity in oblique angle propagation refers to the speed at which the phase of a plane wave travels in a medium when the wave is not propagating perpendicular to the interface between two media. It is a measure of how fast the wavefronts of the plane wave are moving in a particular direction.

2. How is phase velocity different from group velocity?

Phase velocity and group velocity are two different measures of the speed of a wave. While phase velocity refers to the speed of the wavefronts, group velocity refers to the speed at which the energy or information of the wave is transmitted. In oblique angle propagation, these two velocities may differ due to the change in direction of the wave.

3. What factors affect the phase velocity in oblique angle propagation?

The phase velocity in oblique angle propagation is affected by the refractive indices of the two media, the angle of incidence, and the frequency of the wave. The phase velocity is also dependent on the polarization of the wave and the properties of the materials through which it is propagating.

4. How does the phase velocity change with increasing angle of incidence?

As the angle of incidence increases in oblique angle propagation, the phase velocity decreases. This is because the wave is traveling a longer distance in the medium due to the change in direction, resulting in a slower speed. At a certain critical angle, the phase velocity becomes zero and the wave cannot propagate through the medium.

5. Can the phase velocity be greater than the speed of light?

No, the phase velocity cannot be greater than the speed of light. This is a fundamental principle of physics known as the universal speed limit. In certain materials, the phase velocity may appear to be greater than the speed of light, but this is due to the wave interacting with the material in a complex manner and does not violate the universal speed limit.

Back
Top