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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. $$\omega_1 = \frac {πv}{l}$$

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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.

That "1.19" in your formula is probably a typo if the subject of Halliday-Resnick (I do not have the book) is indeed a spherical cavity and should be more like "1.14".

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