A waveguide becomes a resonant cavity upon placing conducting walls at the two ends. Show that a resonance occurs when the length \(L\) is an integral number \(n\) of guide halfwavelengths \(\lambda_{e} / 2 ;\) specifically, \(\left(\frac{\omega}{c}\right)^{2}=\left(\frac{l \pi}{a}\right)^{2}+\left(\frac{m \pi}{b}\right)^{2}+\left(\frac{n \pi}{L}\right)^{2} \quad\) rectangular parallelepiped \(\left(\frac{\omega}{c}\right)^{2}=\left(\frac{u_{l m}}{a}\right)^{2}+\left(\frac{n \pi}{L}\right)^{2} \quad\) right circular cylinder. Cavity modes, requiring three integral indices, are named \(\mathrm{TE}_{l m n}\) or \(\mathrm{TM}_{l m n} . \mathrm{Make}\) a mode chart for cylindrical cavities by plotting loci of resonances on a graph of \((d / L)^{2}\) against \((f d)^{2}\), where \(d \equiv 2 a, f \equiv \omega / 2 \pi . \dagger\)

We are given the resonance condition as: $\left(\frac{\omega}{c}\right)^{2}=\left(\frac{l \pi}{a}\right)^{2}+\left(\frac{m \pi}{b}\right)^{2}+\left(\frac{n \pi}{L}\right)^{2}$. Resonance occurs when the length \(L\) is equal to an integral number \(n\) of guide halfwavelengths, i.e., \(L = n\cdot\frac{\lambda_{e}}{2}\). Rearrange this equation to get \(\lambda_{e} = \frac{2L}{n}\). The relation between the wavelength, wave speed, and frequency of the wave can be expressed as \(\lambda_e = \frac{c}{f}\). Replace \(\lambda_e\) with \(\frac{c}{f}\) to get \(\frac{2L}{n} = \frac{c}{f}\). Now, rearrange the equation to isolate \(\omega\) and square it to get: $\left(\frac{\omega}{c}\right)^{2} = \left(\frac{2 \pi f}{c}\right)^{2} = \left(\frac{l \pi}{a}\right)^{2}+\left(\frac{m \pi}{b}\right)^{2}+\left(\frac{n \pi}{L}\right)^{2}$.

For a right circular cylinder, the formula is given as: \(\left(\frac{\omega}{c}\right)^{2}=\left(\frac{u_{lm}}{a}\right)^{2}+\left(\frac{n \pi}{L}\right)^{2}\) The formula remains the same as a rectangular waveguide for the longitudinal component. Now, to find the dependence of the resonance condition on \(f\), we can rearrange the given equation to get: \(\left(\frac{\omega}{2 \pi}\right)^{2} = \left(\frac{u_{lm}^2}{(2a)^2}\right)c^2 + \left(\frac{n^2 \pi^2}{L^2}\right)c^2\)

To plot the mode chart for a cylindrical cavity, we will graph the resonance condition in terms of \((d / L)^{2}\) against \((f d)^{2}\). Here, \(d = 2a\) and \(f = \frac{\omega}{2 \pi}\). First, find the dependency of \((fd)^2\) on \(\left(\frac{u_{lm}}{a}\right)^{2}+\left(\frac{n \pi}{L}\right)^{2}\). \((fd)^2 = \left(\frac{2 \pi \frac{\omega}{2 \pi}}{2a}\right)^2 = \frac{\omega^2}{(2a)^2}\) And for \((d/L)^2\) on \(\left(\frac{n \pi}{L}\right)^{2}\), we have: \((d/L)^2 = \left(\frac{2a}{L}\right)^2 = \frac{(2a)^2}{L^2}\). Now, substitute these into the resonance condition: \(\left(\frac{\omega}{c}\right)^{2} = \frac{\omega^2}{(2a)^2} = \frac{u_{lm}^2}{(2a)^2} + \frac{n^2 \pi^2}{L^2}\) Rearrange it into the form: \((fd)^2 = \frac{u_{lm}^2}{(2a)^2} + \left(\frac{d}{L}\right)^2\) Now, we can plot \((d/L)^{2}\) against \((f d)^{2}\) for given values of \(l, m, n\) and \(u_{lm}\) calculated for a specific mode of TE or TM. The graph will display the resonance condition surface for cylindrical waveguides for each mode, affecting the propagation of electromagnetic waves inside it.