# Hybrid field mode in electromagnetic cylindrical wave guide

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.

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).