Reverse engineering formulas for directing a plane wave

yefj
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Homework Statement
plane wave
Relevant Equations
plane wave
Hello,There is a model which tunes a plane wave using certain expressions which resembles the spherical to cartesian coordinates.
There are two types of definitions:
propogation normal and electric field vector.
Why they put in propogation normal exactly the spherical to cartesian to represent propogation normal?
What is the link between propogation normal and E-field formulas?
Thanks.
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yefj said:
Why they put in propogation normal exactly the spherical to cartesian to represent propogation normal?
It looks like you have a spherical wave. The direction of propagation is radially out and a wavefront is a sphere of radius ##R##. This means that the direction of propagation at any point on the sphere is radially out, i.e. normal to the surface. A unit vector normal to the surface of a sphere is $$\mathbf{\hat n}=\sin\!\theta\cos\!\phi~\mathbf{\hat x} +\sin\!\theta\sin\!\phi~\mathbf{\hat y}+\cos\!\theta~\mathbf{\hat z}$$ which is the same as the radial unit vector ##\mathbf{\hat r}## along which the wave propagates.
 
how does E-field formulas were derived from the normal?
Thanks.
1736704647233.png
 
yefj said:
how does E-field formulas were derived from the normal?
Thanks.
View attachment 355731
The E-field vector is in the plane perpendicular to the normal. In that plane there is an infinity of directions that the E-field vector can have. Here you are given one of many. You can verify that this is a valid E-field by showing that ##~\mathbf E\cdot \mathbf{\hat n}=0.##

For a given normal you can get all the possible electric fields by varying angle ##\alpha## which rotates the electric field in the plane perpendicular to the normal. You can easily see how this works for specific choices of the angles.

For example, if I choose ##\theta=\dfrac{\pi}{2}## and ##\phi=0##, I get
(a)##~~\mathbf{\hat n}= \mathbf{\hat x}##,
(b)##~~\mathbf{\hat E}= \sin\!\alpha~\mathbf{\hat y}-\cos\!\alpha~\mathbf{\hat z}##
The above says that (a) the normal is along the ##x##-axis and (b) a unit vector in the direction of the electric field is in the ##yz##-plane and changes direction as ##\alpha## changes.
 
Hello Kuruman, I understand that E-field can have endless options.
What is the logic for choosing the expression for e-field?
Thanks.
 
yefj said:
Hello Kuruman, I understand that E-field can have endless options.
What is the logic for choosing the expression for e-field?
Thanks.
If you understand that the E-field has infinitely many options and that it must be perpendicular to the direction of propagation, there is no logic for choosing a particular direction as opposed to another. Pick a direction and move on with your life. That's what the person that made the simulation that you posted most probably did understanding that it is necessary to make a choice before drawing the surface, the normal and the E-field.
 
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