(Note I uploaded a picture of a dog because its the only picture that successfully uploaded and I needed one.)
Holy crap I did it! This was far and away the hardest project I've ever done.
I decided to type up my results to tell you all about this because a few of you helped me get here, but also to just put what I learned out there as another data point for anyone who is going down the same road as I am.
This took 3 months of work, sometimes for 10+ hours a day. A big thank you to Harold Kapp his help was key to making this work. The vast majority of that time was understanding how the heck all of this work, and then machining the actual horn.
But slowing down, what and why? A sonicator is a device that can both extract substances from plants or animal cells by literally exploding them open with a shockwave from cavitation. They can also emulsify two non miscible liquids, like oil and water. This is a permanent mixture that requires no additional chemicals, this is the freaky looking stuff in the cup.
That is a mixture of mineral oil and water, I'd like to note that both of those are clear liquids. The craziest part about this whole device is how it works, It sends vibrations through the horn, which is the aluminum piece shown in the picture. This device amplifies those vibrations until they reach the tip where the amplitude is greatest. That tip then moves back and fourth 28,000 times a second. Or in the case of my particular device 26,500 times a second, this is really important because the device must be in resonance in order to work. If you don't get this right you will get transverse waves within the horn, this is where the horn literally wiggles back and fourth like a wet noodle.
Of course in real life this is visually imperceptible, however I think you can hear it. While the frequency of the driver is 28 kHz which is out side of human hearing, when the horn has a transverse wave due to it being out of resonance you can hear a fairly loud audible tone. I think this is due to the wave.
Now the ideal vibration mode is the longitudinal wave. This is where the tip acts like a jack hammer as you can see from my simulation picture. This is what happens if the horn is resonating correctly. To get into the city where the ball park is you calculate the wavelength of sound within the material the horn is made out of. For me this was 6061 aluminum which has a speed of sound AROUND 5000 m/s. Apparently its really hard to know how fast waves travel in metal. Some sources said as low as 4000 m/s some as high as 6300 m/s. I learned all of the necessary calculations and considerations from this paper:
It has been my bible, they state that in a stepped horn, which is what I've designed here, the ratio of the major and minor diameter will affect the resonant frequency of the horn. This all boils down to the fact that it's practically impossible to calculate the dimensions of a horn that will resonate. So I made everything longer than it needed to be, but this is a path to insanity. If you cut the horn too long you can start to get into the node or anti-node of the next wave which will throw everything off. This is why parametric FEM is your best friend, you can vary lengths on your horn until it resonates in simulation. The key thing here is if you remove material from the front face of the major diameter section you can reduce the resonant frequency (every time I made a horn it resonated too high) cutting material off the very end of the horn raises the resonant frequency.
The techniques of how to do this and how to measure this are detailed here:
But I came to realize I would sink dozens of dollars (aluminum is surprisingly cheap) and thousands of hours into machining a horn that resonated at output frequency of my driver board. So this is where Harold helped me to figure out a way to vary the output frequency of the board. Which is detialed here:
The TL;DR is there is a toroid on the board that is used to trigger two transistors, on that toroid is what I think is called a sense coil. It has much fewer turns than the other coils. Disconnecting this wire and powering it with an H-bridge circuit allows you to control the output frequency of the ultrasonic driver board.
The board to do this consists of a 555 timer circuit with variable frequency output connected to an H-bridge. I initially planned to use an arduino to control this, with the hopes of incorporating resonance tracking with ease, however the arduino simply isn't fast enough. So a 555 circuit worked a treat. In addition to this I needed to create an active low version of the 555 output. Connecting the active high and active low signals to opposite sides of the H-bridge created the desired, albeit crude, AC I wanted. However the H-bridge needs to be a little beefy as in my case I has 12V @ 1A running through. I also used a 20 W 10 Ω resistor in series with the coil to not blow up the whole thing. And the resistor still gets hot. Lastly of note I used the pride of my electronics collection a 10 turn 2 kΩ potentiometer to vary the output frequency. This is only for fine tuning as a simple pot is soldered to the board to get the frequency into the right range.
I then buttoned it all up inside of an enclosure and bam, $2500 tool for <$100.
Thank you all so much for your willingness to help