The microwave is reflected at the wall of the cavity to form the standing wave in a closed cavity. If the plane boundary surfaces are at z=0 and z=d, the boundary conditions can be satisfied at each surface only if
\(d=p \frac{\lambda_g}{d}\,\,\,\, p=1,2,3,…\)
where \(lambda_g\) is the waveguide length, p is the mode number in the z-axial direction. However, if the cavity boundary is open, there is no clear walls at the end of the cavity. The microwave energy is stored mainly in the cavity and a little part of the energy goes into the tapper section. So the cavity length is slightly smaller than a half of the waveguide wavelength. It means,
\(k=\frac{2 \pi f_0}{c}=\sqrt{k_z^2+(\frac{\chi_{mn}}{R_0})^2}\)
\(\lambda_g=\frac{2 \pi}{k_z}>L_2\)
where, k is the wave number in vacuum, \(f_0\) is the frequency, \(k_z\) is the wave number in the z-axial direction, c is the speed of light, \(\chi_{mn}\) is the n-th root of the equation \(J_m'(x)=0\).
The main mode is found by compare the waveguide wavelength for all the TE modes and the length of the uniform section in the high power cavity.