# Calculating displacement amplitude of ultrasonic power transducer

I am not a physicist, but my current project drives me to some physics-related computations, hence seeking help.

I have some ultrasonic transducers, 5938D-25LBPZT, for which very limited information is available.

In order to define the parameters for an ultrasonic booster and amplification horn for this device, I need to compute the displacement amplitude of its radiating surface. The procedure I am following, and the base data (or where data is unavailable, my assumptions) are thus:

Operating power: $$50\ \mathrm W$$ RMS (the transducer is rated for $$60\ \mathrm W$$)
Emitting horn material: stainless steel SS316 (best-guess, unfortunately, as SS316 is typical in other such transducers)
Emitting face diameter: $$5.865\ \mathrm{cm}$$ (measured)
Frequency at resonance: $$f=24\,989\ \mathrm{Hz}$$ (measured)
Best-case power transmission: $$90\ \%$$ (in forward direction, according to various sources)

Constants:

Density of SS316: $$\rho=7.8\ \mathrm{g/cm^3}$$
Velocity of longitudinal sound propagation: $$C=5.8\ \mathrm{mm/μs}$$ (from this reference, approx)
Should the axial velocity of sound be considered instead for this calculation?

Calculations so far:

Face area: $$27.016\ \mathrm{cm^2}$$ (ignoring area lost to bolt-hole)
Sound intensity at face: $$I = 90\ \% \times \text{power}/\text{area} = 1.666\ \mathrm{W/cm^2}$$
Displacement amplitude $$A$$:
$$A=\sqrt{\frac{2I}{(\rho C)\omega^2}}$$ where $$\omega = 2 \times \pi \times f$$
Thus, $$A = 172.8296514\ \mathrm{nm}$$

My questions are:

1. I am not sure I have managed to convert everything to consistent SI units, so I need help confirming that I've got that part right
2. If anyone has experience in such power ultrasonic piezo transducers, a confirmation that my results are more or less within the realms of plausibility would really help.
3. Is there some significantly incorrect assumption or constant value in the foregoing?
4. Should I be using some online resource that trivially calculates all this from the rated power and other inputs, instead of struggling with it manually?

Update: Adding diagrams to show the excitation direction and materials involved.

This is the final arrangement of the Ultrasonic Stack. The booster and horn in the diagram above will both be SS316. In order to compute dimensions for these two parts for resonance, I need the displacement amplitude of the front face of the steel "radiation head" in the transducer (first diagram here), which is to be bolted to the booster. The bolt-hole in the radiation head can be seen in the photograph above.

Another diagram, for a similar "Ultrasonic Stack", with the Langevin transducer itself split up. This diagram has a cylindrical radiation head, as opposed to the cone section shape in the transducers I need to compute for.

(Last two diagrams are from articles on ScienceDirect.com)

2. The value you have computed is reasonable. There are a number of (pay-wall blocked) academic articles discussing transducer amplitudes. These amplitudes range between minimum detectable values of a bit over 1 angstrom (0.1 nm) to rupture inducing amplitudes of ~5 micron (5000 nm). Of course, these values depend heavily on the driving frequency, power, and geometry of the transducer. However, you should have confidence in the values you have computed. I would add that, since the uncertainty in your material density and sound speed is somewhat large, so too will be the uncertainty in your amplitude. To more accurately reflect this uncertainty (using only significant figures, rather than a rigorously computed uncertainty), your amplitude is about $1.7\times 10^{-5}$cm.