# Expression of $\omega$ in fundamental mode of EM cavity resonator

In Physics (Part 2), D. Halliday - R. Resnick, on page 957 it is written that, "We state without proof that the angular frequency of oscillation for the electromagnetic cavity of Fig. 38-8 is, in the fundamental mode shown in that figure, $$\omega_1 = \frac {1.19c}{a}$$ in which $$a$$ is the cavity radius and $$c$$ is the speed of electromagnetic radiations in free space." I couldn't find any derivation on the net or in the texts that I have. I would be obliged if anyone could provide me with an outline of the derivation. Note that the cavity is cylindrical. Moreover, the factor $$1.19$$ appears to be near $$\frac {π}{\sqrt 7}$$. This observation was inspired by the corresponding acoustic cavity analogue of the expression i.e. $$\omega_1 = \frac {πv}{l}$$

According to Stratton: Electromagnetic Theory, pp 561-562, where it is derived in gory details, the spherical resonator's lowest frequency mode has the the resonant wavelength $$\lambda_{11}'= \frac{2\pi a}{2.75} =2.28a$$ where $$a$$ is the radius of the sphere. The "2.75" is the first root of the function $$(xj_1(x))'$$, and $$j_n(x)$$ are the spherical Bessel functions, see pp 405-406.