What determines the frequency of an RF cavity?

[artis]

I am reading a document describing RF cavities, in there it says that for the TM 010 mode cavities the frequency is only dependent on cavity radius but not length (which I assume is the length along the beam axis).
Also I assume the TM 010 mode is the mode in which klystron cavities and particle accelerating cavities work, where the E field points parallel to the beam axis.

If what I said so far is correct , then why does the cavity frequency depend on just the radius but not the length?

My own attempt at an answer would be that the cavity radius determines the capacitance of the cavity mostly and the capacitance directly affects the resonant frequency of an LC circuit.
Although I cannot exactly understand why having the same radius but double length wouldn't affect the frequency?My other question would be about the transformer analogy of the RF cavity.
If I have one cavity that is driven by a beam and another cavity that is simply attached via waveguide to this first cavity , then do these cavities work in phase or are they out of phase?
For example is the current/magnetic field and E field all in the same direction both for the beam driven first cavity and the waveguide coupled second cavity or not?

Or does this depend on the length and termination of the waveguide that couples the cavities? Wherein based on this one can couple another or multiple cavities all either in phase or out of phase?

 

[tech99]

The following article shows the cavity on Page 9 and shows how it resembles a short circuited transmission line having a length of about a quarter of a wavelength. We can imagine a wave travellimg back and forth in the radial direction - that is why length does not alter the frequency.
If cavities are connected together by a waveguide, that is like joining two LC circuits with a transmission line. If we assume the LC circuits are at resonance, they are resistive and the transmission line introduces a phase shift according to its length.
[sorry about the long link, due to my poor IT skills]
https://www.google.com/url?sa=t&rct...df/1111.4897&usg=AOvVaw0YJR0kgv9AG65md6EPQij_

 

[tech99]

May I also mention that the waveguide could introduce a phase reversal caused by the sense of connection to the cavities.

 

[f95toli]

The answer to your first question is that is sort of a "fundamental property" of solutions to the Helmholtz equation that for most shapes there are eigenmodes that do not depend on some of the dimensions of this shape. The same thing is true for e.g. drums so it is not in any way unique to RF cavities.
Note that there is nothing says that this will be the "dominant" mode of a given cavity; this will depend on how the cavity is excited and also on the quality factor of the different modes.

The answer to the 2nd question is "it depends"; it will depend on the details of the geometry and which modes you are working with.

 

[artis]

We can imagine a wave travellimg back and forth in the radial direction - that is why length does not alter the frequency

I am probably making a mistake here but making the cavity longer along it's beam axis would decrease the capacitance by separating the cavity walls as well as increase it's inductance , would it not?

But in general @f95toli it is possible to connect multiple cavities such that they are all in phase with one another?

 

[f95toli]

But in general @f95toli it is possible to connect multiple cavities such that they are all in phase with one another?

I can't see why not, you would just need to design is properly.

 

[tech99]

I think the resonant frequency will be decided by the length of the metallic path between the lips of the resonator, which will be approx half a wavelength. There will be slight capacitive loading between the lips and this will lower the frequency slightly.