I was wondering how we can introduce $e^{ik.x}$ terms associated with circumferencially propagating waves? In this case $\hat{e}_\theta$ is the direction of wave propagation. However, I was not able to express the exponent of exponential using a dot product because of polar coordinates. How can we introduce this dot product? Can spherical harmonics or Fourier transform help here?
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$\begingroup$ Polar coordinates don't matter. You can express your $k$ and $x$ vectors in whatever basis you want and evaluate the dot products between the basis vectors. For starters, try expressing $\hat{e}_\theta$ in the cartesian unit vector basis. $\endgroup$– smhMay 19, 2020 at 10:58
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$\begingroup$ Have you checked this answer? $\endgroup$– nicoguaro ♦May 19, 2020 at 22:28
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$\begingroup$ I don't think you can write this term in the form you want. Could you provide more detail on why you're trying to do this? $\endgroup$– LedHeadMay 23, 2020 at 13:23
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