You might be able to use titanium dioxide dielectric in the outer part of the cavity - I'm not sure whether its dielectric constant is high at HF and VHF. Or ferrite material. Magnetic materials can be low loss and with high relative permeability.

It will slow it down and it will compress the wave fronts in the direction it slows down in - you can't have one without the other, they are proportional.

It is the magnetic component I need to insure gets slowed down but I was thinking ferrite material would provide a path for flux and so would prevent the wave front from ever reaching the cavity walls. I remember reading in some magazine about something that slowed the propagation very much but I don't remember enough to find it again.

If the cavity is resonant then the wavefront has to reach the walls. If it doesn't it isn't resonant. Ferrite material slows the magnetic field and the electric field. Dielectric slows the magnetic field and the electric field. The wavelength is divided by the square root of the product of the permittivity and permeability, and so is the speed. If they are low loss materials then they shouldn't stop resonance. A loss tangent of 0.01 or lower (Q of 100 or more) should be reasonable.

It sounds like you do not have experience of resonant cavities so I suggest you read up dielectric or ferrite loaded microwave cavity resonators - they aren't usually used at HF, but will scale, except that ferrite and dielectric properties vary with frequency, so you might need to find other materials with the same properties at lower frequencies.

May the question be ralated to the band-gap structures (metamaterials)?

With metamaterial, or photonic crystal approach (which is not exactly the same), you can create bandgaps, where on boundaries passband dispersion branches fold either at β=0, or at the edge of Brillouin zone. So, in the latter case, you have the wave slowing effect, both in phase, and group velocity. Sometimes, it is called a "frozen wave", which have a very short wavelength, but does not go anywhere, actually. The loss is very high, so I am not sure, we can speak about any resonator.

On the opposite side, where β=0, we can enable a uniform amplitude propagation, or so-called zeroth-order resonance. The group velocity can also be high, when we widen the band-gap and increase the loss. Since, it is a fast wave, it is a radiating thing, which you may not like.

Of course, this has a meaning only in large-time scale, longer that many periods.

P.S. Slowing wave in bulk ferrites is not a good idea, unless you are making a circulator, or isolator.

On the other hand, ferrites support waves of magnetization (spin waves is a general term, or magnetostatic modes is a term for low-β spin waves). These are very-very slow and short. But normally, useful only in micron-thick ferrite film structures. However spin waves are prone to strong nonlinear effects, so you may not like these as pure resonant thing. Spin wave resonators usually resonate not in the film, but in the ring circuit, where the film is a kind of low-noise super-delay-line.

I'm looking for some propagation medium relatively cheap and easy to work with. The setup would have something like a short rod of somewhat soft magnetic material at the center of a spherical resonant cavity. The material should not have much loss and have good real permeability up to 50MHz (for example). Then the cavity size would be for a 1/4 wavelength from center to the inner walls so about 1.5 meters in air, so the cavity diameter would be 3 meters. But that is too large so I need to slow the propagation way down to 1/10 speed if possible. That would give me a cavity size of 30 cm, right? That size would be much easier to work with.

So the propagation medium (or electrical or magnetic structures) would need to do 4 things:

1. Slow the wave front down as it propagates out from the short rod

2. As a magnetic field builds around the short rod, the medium needs to cancel the magnetic field close to the rod and push that field out away from the rod and closer to the cavity walls at the same time it delays its arrival at the walls.

3. Slow the return reflection off the walls and have minimal loss as it arrive back at the short rod.

4. Relatively low losses to and from the cavity walls.

If the short rod has (for example) a relative permeability of 25 at 50MHz and if an external stimulus pulse of .04T saturates the short rod at 25*.04T= .1T peak magnetic field, then as that .1T field (peak at the rod's surface) expands out towards the walls and reflects off the walls with a reversal of the magnetic field orientation then as that reflection reaches the short rod again its strength needs to be enough to saturate the rod in the opposite orientation.

To reduce the size (radius) of the cavity by a factor of 10 you would need a relative permeability of 100, or a relative permittivity of 100, or the product of relative permeability and relative permittivity being 100. This will occupy most of a sphere 30 cm diameter, so is about 14000 cc of material, or over 14 kg of dielectric or magnetic material. I don't think this will be inexpensive. You might be able to use titanium dioxide based powder or granules, which might not cost too much.

Magnetic field does not reverse on reflection at a conducting surface - the reflected magnetic field is in the same direction as the incident magnetic field. It is the electric field that reverses on reflection from a conducting surface. Both magnetic and electric fields reverse in direction after a half-wavelength travel, which is why when the magnetic field has travelled to the outside and back it is in the opposite direction (it has taken a half cycle of time to do this).

If your cavity is resonant, the field will reverse in direction twice a cycle whatever you do.

If your cavity is spherical or cylindrical it will need to be larger than a half-wavelength across because the modes are determined by Bessel functions not sine and cos, and the wavelength for these close to the centre of curvature is larger than a wavelength in free space, or the wavelength in the dielectric or magnetic material you use. Even if your cavity is rectangular in symmetry, it can't be as small as half a wavelength across (in the dielectric or magnetic material you use), and its volume will need to be bigger than about 0.85 wavelength cube (in the dielectric or magnetic material you use).

Well those are real show stoppers. Would a cavity wall made of a dielectric reflector swap the magnetic field orientation after just 1/4 wave length?

Actually, I think you are talking about the short rod's external flux direction but I'm talking about the short rod's internal flux direction. It takes just a 1/4 wave (1/2 wave total round trip delay) to return opposite direction to the internal flux direction and trigger a swap in its internal flux direction. Look at any Lenz's law animation and see that reflected magnetic flux has swapped direction relative the flux that induced the counter EMF.

For a dielectric material, I think Barium/Strontium Titanate of 67:33 ratio and 800nm size particles might work better. Both larger and smaller size particles have lower coefficient of permittivity. But I'm not sure if I should mix a ferrite or some diamagnetic material with it.

https://www.tplinc.com/uploads/HBS-8000_Technical_Bulletin.pdf

Probably cheaper at Alibaba or IndiaMart

BSTx have nonlinear permittivity. Some people are doing parametrically modulated nonlinear circulators on those. the same reason I was not recommending ferrites

But would BST's nonlinear characteristics be a problem in this application? If it tended to slightly compress the wave front it might be a good thing.

What for?

Using ferrite (without magnetic bias ) above say 500 Mhz is not a good idea since all ferrite are rather lossy then. One can find rather high Q ferrites (when magnetically biased into saturation (thousand of single crystal YIG sphere) but again this is not very practical. At CERN a (emty) spherical cavity normal conducting (about 1 meter diameter) was used for energy storage in the early 80th on a very high order mode (mode island).Now thus is done via superconducting cavities.Anyway spherical cavities are not really good for such kind of purpose. You can find details via the CERN library (public access) ..CDS CERN Again one of the nice old stories and we have this cavity as an expo object in our museum. Whatever material you put into the cavity..it will lower the Q more or less.

OK but the frequency in question is 50MHz rather than 500MHz. The goal of the experiment is to interface with the real permeability with minimal interface with the imaginary permeability. The what for is to see if there is any arrangement of cavity design , delay medium design and magnetic material design that could ever allow reflected waves off the cavity walls to propagate back to the center mounted magnetic material after 1/2 wavelength total delay and arrive back with enough strength to flip the material's direction of magnetization thus propagating out another wave that would do the same thing again and continue the process. The magnetic material would have to have high enough relative permeability that a small external signal can make its uncompensated spins move into alignment in response and thus generate a sufficiently larger field to keep the toggling going. If the relative permeability is high enough and enough energy gets to the cavity walls and reflects back then I don't see why it couldn't work. Actually, I see a lot of things that could stop it from working but none of them are insurmountable. It just requires the right engineering.

I fully understand that perpetual motion without another energy source is not possible. The question is what in these processes would theoretically prevent the toggling result or could it theoretically work? Just saying it can't work because we have a rule against it is not sufficient. The magnetic material is not able to distinguish between an incoming signal stimulating it to saturation in 1 direction or the other and its own reflections doing the same thing. It should just be a matter of reflecting back enough energy after enough delay.

During a decrease in the core’s magnetic field strength, a portion of the magnetic field around the core will collapse back in on the core and another portion of the core’s magnetic field couples to the radiation resistance of free space and continues to radiate away from the magnetic core material. That portion that radiates away will reflect off the cavity walls. If this portion of the electromagnetic energy is large enough to saturate the magnetic core in the opposite direction then the process can continue.

Perhaps you should look at Loop-Gap resonator: It may have geometrical size way smaller than the wavelength: https://en.wikipedia.org/wiki/Loop-gap_resonator

I am familiar with the concept because of experience with magloop antenna design, which is the same idea. But I don't see how to introduce a gain element (something with greater permeability or greater permittivity) whose feedback can be delayed enough to become positive feedback.

I think adding water to a spherical cavity with a magnetic material at the center would allow the cavity size to be about 9 times smaller.

Water is highly lossy dielectric at RF frequencies, all liquids that have high dielectric constant tend to be such. Perhaps you may check these materials.

https://www.preperm.com/products/raw-materials/

https://www.preperm.com/solutions/telecom-5g/?gclid=Cj0KCQjwtsv7BRCmARIsANu-CQfD79Zjm73E8EZlxEqCtymPfnhHdeg5RIHlUUZtb9NWuVPLMgjQn8gaAnG-EALw_wcB.

"gain element (something with greater permeability or greater permittivity)" - don't understand that ..

Do you know where I could find a chart of losses per distance in water per frequency in the range of 85 to 90Mhz? I only need to cover about 22cm total distance round trip without too much loss at about 85 to 90Mhz. Never mind, I found this: https://upload.wikimedia.org/wikipedia/en/a/a1/Dielectric_loss_water.png I think I should be OK.

A magnetic material should be able to act as a gain element with the relative permeability representing the amount of gain. Please see the slides starting around the middle of this presentation for more information: https://www.vasantcorporation.com/downloads/magnetic_permeability_05-03-2017.pdf