Group Velocity and Phase Velocity

[Joystar77]

I have no idea how to do this or where to start. Can someone please help me with this problem? Should I start with the group velocity or phase velocity? Am I suppose to come up with any type of expression such as a rational expression or algebraic expression?

Problem 4.4- Suppose n o and n e are given. In (a) you only need to find the magnitude of the group velocity. Problem #2 in HW 10 may be helpful. You can also directly use the definition of group velocity, i.e., v g = triangle k w (k), taking into account the equation of the wave normal surface.

4.4- Group Velocity and Phase Velocity

a.) Derive an expression for the group velocity of the extraordinary wave in a uniaxial crystal as a function of the polar angle 0 of the propagation vector.

b.) Derive an expression for the angle a between the phase velocity and the group velocity. This angle is also the angle between the field vectors E and D.

c.) Show that a = 0 when 0 = 0, ½ pi. Find the angle at which a is maximized and obtain an expression for a max. Calculate this angle a max for quartz with n o = 1.554, n e = 1.553.

d.) Show that for no or ne, the maximum angular separation a max occurs at 0 = 45 degrees; show that a max is proportional to [ n o – n e].

 

[HallsofIvy]

While this may be about the solution to a given differential equation, it has nothing directly to do with differential equations. And you have not given definitions for a number of things. While "group velocity" and "phase velocity" are commonly used in working with waves, it would be good to give the specific definitions you are working with. And you certainly cannot expect everyone to know what a "uniaxial crystal" is or what special properties a wave in a uniaxial crystal might have!