understanding the meaing of k number direction in plane wave

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Hello,I am confused regarding direction in plane wave
there is X direction of the field but also i have direction of the values in "exponent."
how do i know what direction does it propogate?
how does these two directions go togother?
thanks.
 

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If this is indeed the form of a plane wave, the direction of propagation would be theta_in, which is in the y-z plane. You can check this by taking the gradient of the field -- which for a plane wave -- gives you the direction of propagation.
 
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You need to follow in what direction the phase changes. Because propagation of a planewave in a lossless media(like your case) is nothing but a phase change. Here, either of y or z changes the phase. Therefore we can say that it propagates into some direction in between y and z.
 

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