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If we have a boundary-value problem that consists of a circular cylindrical waveguide (axis along $\hat{z}$, say), with some arbitrary boundary condition (so not necessarily the usual conductive wall) applied to the guide's wall at $r=a$, with $a$ being the radius, we notice that the general field representation assumes a hybrid mode [i.e. a combination of transverse electric (TE) and transverse magnetic (TM) modes]. The TE mode is governed by the $H_{z}$ component, while the TM mode is governed by the $E_{z}$ component. Following conventional treatments of the Helmholtz equation in cylindrical coordinates, which gives a Bessel equation and periodic angular dependence, we generally get the following field form for the longitudinal component (suppressing the propagation factor $e^{-i\omega t +i\beta z}$):

$$ u=u(r,\phi)=u_{0}J_{n}(k_{t}r)\cos(n \phi + \phi_{0}), $$ where $u$ is either $E_{z}$ (for the TM case) or $H_{z}$ (for the TE case), $n$ is integer and $k_{t}$ is the transverse wavenumber (also called cut-off wavenumber). This is the most general expression mathematically for this problem.

My question now is: why when the hybrid mode (TE+TM) is considered, most books (e.g. Stratton, Slater or Mahmoud) proceed by writing one of these functions now with a cosine, while the other one with a sine? Namely, in the general case, they write

$$ E_{z}=E_{0}J_{n}(k_{t}r)\cos(n\phi), $$ $$ H_{z}=H_{0}J_{n}(k_{t}r)\sin(n\phi). $$

As if the authors know a priori that choosing the option for the angular functions (i.e. choosing $\phi_{0}$ for TM or TE) in a different way will not eventually work out physically $-$ but they don't spell it out!

Any hint is appreciated.

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Seeing "$\cos(n \phi)$" and "$\sin(n\phi)$", one might be tempted to form the combination $\cos(n \phi) + \mathrm{i} \sin(n \phi)$, which one can arrange through $E_z + \mathrm{i} H_z$ (ignoring a few constants).

One perhaps recognizes this as similar to the Riemann-Silberstein vector (still at the level of ignoring a few constants), in terms of which Maxwell's equations without sources (no charges, no currents) can be written. The Riemann-Silberstein vector is a bridge to the Faraday tensor.

These connections are discussed further here: https://physics.stackexchange.com/a/80039/50480

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  • $\begingroup$ Thanks for this insightful connection, which gives insight from a geometric point of view and links to tensor formalism. However, I think there is a rather simpler and more "practical" or physical reason behind the traditional use of sine and cosine in this question, even without considering tensors or spacetime formalism. Just classical 3D-space vector analysis arguments were used in all of the cited references -- which begs the question as to what practical/physical rationale was for them to use these choices. $\endgroup$
    – user135626
    Dec 20, 2020 at 7:29

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