The magnetic field along the propagation direction inside a rectangular waveguide with the cross section shown in the figure is $${H_Z} = 3\,\,\cos \,\,(2.094\,\, \times \,\,{10^2}x)\,\,\,\cos \,(2.618\,\, \times \,\,{10^2}y)$$
$$\cos \,\,(6.283\,\, \times \,\,{10^{10}}t\, - \beta \,z)$$
The phase velocity $${V_p}$$ of the wave inside the waveguide satisfies

A rectangular waveguide of internal dimensions (a = 4 cm and b = 3 cm ) is to be operated in $$T{E_{11}}$$ mode. The minimum operating frequency is

The $$\mathop E\limits^ \to $$ field in a rectangular waveguide of inner dimensions $$a\,\, \times \,\,b$$ is given by $$\mathop E\limits^ \to = {{\omega \,\mu } \over {{h^2}}}\,\left( {{\pi \over a}} \right)\,{H_0}\,\sin \,\left( {{{2\,\pi \,x} \over a}} \right)\,\,\sin \,(\omega \,t - \,\beta \,z)\hat y$$,

where $${H_0}$$ is a constant, a and b are the dimensions along the x-axis and the y-axis respectively. The mode of propagation in the waveguide is

An air-filled rectangular waveguide has inner dimensions of $$3\,cm\,\, \times \,\,2\,\,cm\,$$. The wave impedance of the $$T{E_{20}}$$ mode of propagation in the waveguide at a frequency of 30 GHz is (free space impedance $$\,{\eta _0} = \,377\,\,\Omega $$)