Record-and-replay resonance | Project S.P.H.E.R.E.

Status: proposal for public review. No experiment has been run for this page. It contains no measurements, results, expert responses or installed-hardware claims. Numbers are calculations or assumptions and are tagged as such. The SPHEREVL / Control Room figures cited are from an educational simulation, not measurements or scientific validation. Nothing here amends the current draft LOC-1 protocol.

Evidence tags used below: Sourced Calculated Assumption Open question

1. Short answer

Recording and replaying the sphere’s resonance is technically feasible. Using playback alone to change the sphere’s natural resonance frequency is not.

A passive elastic body driven by a recorded signal responds at the frequencies present in that signal. Replay re-excites the same modes; it does not retune them.
A natural frequency is set by geometry, mass distribution, elastic moduli, residual stress, temperature and how the body is supported. Moving it requires changing one of those, or adding a coupled active system.
Playback can still be useful: it can concentrate energy in chosen modes, validate a modal model, and drive the sphere under controlled conditions.
The mechanisms that can move a mode are listed in section 4. Only temperature and closed-loop feedback are both controllable and large enough to matter at 230 kHz. Feedback reshapes the measured response of the combined system; it does not alter the passive structure.

The most important practical finding

A temperature change of 1 K moves a 230 kHz mode by roughly 23 to 46 Hz (estimate, section 4). That is larger than the linewidth of a mode with Q = 10⁴ (23 Hz). Drive heating, record-to-replay drift and any position-dependent thermal change are therefore first-order effects in this experiment. Temperature must be controlled and logged at millikelvin level before any frequency shift is attributed to playback or to stage location.

This article treats the S.P.H.E.R.E. locational hypothesis as an experimental question under standard metrology. It does not claim that a location effect exists. The test protocol in section 8 and the A/B/A verification page are designed so that a null result is as informative and as publishable as a positive one.

Quick answers

Can recording and replaying a sphere's sound change its resonant frequency?

Not on its own. Replay drives the sphere at the frequencies in the recording and re-excites the same modes, because a passive elastic shell is, to first order, a linear time-invariant system. Natural frequencies move when mass, stiffness, geometry, temperature, supports or a feedback loop change.

Why is one audio recording not enough to recreate a sphere's vibration?

A microphone records air pressure at one point, while the sphere's state is a distributed field of surface displacement and velocity. Different mode shapes can give similar pressure at one point, and rebuilding surface motion from sound alone is an ill-posed inverse problem.

What can actually change a sphere's natural frequency?

Temperature, added or removed mass, geometry and wall changes, the support and transducer contacts, large drive amplitude in a nonlinear regime, permanent material changes, and a closed feedback loop that changes the combined system. Temperature is the largest practical confound, tens of hertz per kelvin at 230 kHz on our assumed coefficient.

How would a real shift be verified?

With an excitation-off A/B/A design: measure the modes at low amplitude, apply the treatment, return to the same temperature and mounting, and measure again. A claim needs the second baseline to differ from the first beyond the measurement uncertainty, with sham and reference controls and blinded analysis.

Has this been tested?

No. This is a proposal and a request for expert comment. No experiment has been run and the page reports no measurements.

2. Terms used in this article

Term	Meaning here
Natural frequency	A frequency at which the passive, undriven structure vibrates freely for a given mode. A property of the object and its boundary conditions.
Resonance peak	The measured maximum in a driven response. Equals the natural frequency only if the drive, sensors and supports do not perturb it.
Drive frequency	The frequency content of the signal applied to the sphere by an actuator.
Closed-loop response	The response of sphere plus sensor, controller and actuator operating together. It can differ from the passive sphere.
“Audio”	A convenient label. In vacuum there is no airborne sound. The recorded signal is surface velocity or displacement from a laser vibrometer or an electromagnetic or piezoelectric sensor, digitised at ultrasonic rates.

3. Why playback alone does not retune the sphere

To first order a metal shell vibrating at small amplitude is a linear, time-invariant system. Two properties follow, and both are standard results.

It cannot create frequencies that are not in the input. Replaying a recording containing frequencies f_a and f_b produces a response at f_a and f_b, weighted by the transfer function.
It does not change its own transfer function. The modal frequencies f_i depend on stiffness and mass, approximately:

f_i ∝ (1/L) · √(E_eff/ρ) ⇒ Δf/f ≈ ½(ΔE/E) − ½(Δρ/ρ) − ΔL/L

The coefficient of one half on the modulus term is the same relationship reported for longitudinal modes in nonlinear resonant ultrasound spectroscopy work, where the relative modulus change is twice the relative frequency change Sourced. It applies to small perturbations at fixed mode shape.

Near one mode, the driven amplitude relative to the peak at detuning δ from resonance follows the Lorentzian form:

A(δ)/A₀ = 1 / √(1 + (2Qδ/f₀)²)

At f₀ = 230 kHz and Q = 10⁴, a detuning of 100 Hz leaves 11% of the peak amplitude; at Q = 10⁵ it leaves 1.1% Calculated. Replay fidelity therefore depends on frequency accuracy of a few hertz, which is why the drive and the digitiser must share the rubidium reference.

Retuning a mode with a signal requires the system to be nonlinear (the modulus depends on amplitude) or time-varying (a parameter is modulated, or a feedback loop alters the effective dynamics). Those routes are covered next.

4. What can actually move a mode

Magnitudes are for a mode near 230 kHz. They are orders of magnitude for planning, not predictions for a specific sphere.

Mechanism	How it acts	Size at ~230 kHz	Reversible	Evidence
Temperature	Elastic moduli fall as temperature rises; thermal expansion contributes a smaller term.	Using −0.010 to −0.020 %/K: about 23 to 46 Hz per kelvin.	Yes	Calculated Assumption The coefficient used is the −0.015 %/°C figure cited for steel in a resonance-testing patent. Copper moduli are reported to fall near-linearly with temperature above 300 K. The copper coefficient for these shell modes must be measured.
Drive amplitude (nonlinear elasticity)	Modulus softens as strain amplitude rises, shifting the resonance down. The basis of nonlinear resonant ultrasound spectroscopy (NRUS).	Δf/f ≈ ½ΔE/E. A 10 ppm modulus softening gives about 1.2 Hz.	Mostly; slow dynamics and hysteresis are reported	Sourced (method) Open (magnitude in annealed OFE copper). Sources report pristine samples as nearly strain-independent and damaged ones as clearly nonlinear.
Mass or geometry perturbation	Changes effective mass, stiffness and symmetry. Small departures from sphericity split degenerate sphere modes roughly linearly with the distortion.	Mode-dependent. Splitting is the clearest signature. The breathing mode (l = 0) is non-degenerate and does not split; higher-order modes do (our reading).	Yes, if removable	Sourced (RUS sphericity method, Los Alamos).
Support and transducer coupling	Contact springs and added mass shift measured frequencies and damping. Sample-to-transducer coupling vibration is documented in RUS.	Depends on contact force and area. On a 0.67 g shell any contact is a large perturbation (see section 7).	Yes	Sourced
Shunted piezoelectric element	A passive circuit across a bonded piezo changes its effective stiffness between short-circuit and open-circuit values, tuning the host mode.	Up to 8 to 10% reported where the structure is piezo-dominated. A bonded patch on a 0.3 mm, 0.67 g shell is a large mass perturbation, so its net effect on frequency and Q is unknown (our judgement).	Yes	Sourced Assumption on scaling.
Closed-loop feedback	Sensor, controller and actuator add effective stiffness or damping, moving the poles of the combined system.	Design-dependent. At 230 kHz a loop delay of 100 ns adds about 8.3° of phase.	Yes	Calculated Open No copper-sphere precedent was found in the literature surveyed.
Permanent material change	Annealing, grain growth, peening or plastic strain change elastic constants and residual stress.	Measurable by RUS; not tunable in real time.	No	Sourced (RUS of heat-treated copper).

Reading the table. Playback acts through the amplitude row (small in an intact monolithic shell) and through dissipated power, which heats the shell and acts through the temperature row. If a shift appears during replay, heating must be excluded before any other explanation is considered.

5. Building the record-and-replay chain

Figure 1. Record path (top), replay path (middle) and support systems (bottom). The dashed element is the optional feedback route discussed in section 6. Chain in text: sphere → sensor → digitiser → DSP → AWG → power amplifier → actuator → sphere.

5.1 What is recorded

There is no airborne sound in ultra-high vacuum, so the recorded “audio” is the sphere’s surface motion.

Laser Doppler vibrometry. Non-contact, so it avoids mass loading. Polytec’s UHF-120 is specified to 1.2 GHz and its scanning PSV-500 line offers 3D scanning, so bandwidth at 230 kHz is not the constraint. Signal-to-noise on polished copper, optical access through vacuum windows, and heating by the measurement beam are. Polytec describes a gate function that lowers beam power to limit energy transfer to the structure Sourced.
Electromagnetic acoustic transducers (EMAT). Generate and receive through Lorentz force in conductors without couplant, so they suit a vacuum chamber and non-ferromagnetic copper. The commonly cited drawback is low transduction efficiency Sourced. Contactless EMAT-based resonance spectroscopy has been demonstrated for RUS (Ogi et al., JASA 1999). Fitting a coil and bias magnet around a 9.2 mm sphere is an open engineering question Open question.
Contact piezoelectric transducers. Efficient and standard in RUS, but they load the sample and shift the measured frequencies. Avoid them for the primary measurement.

5.2 Sampling and timebase

Nyquist requires more than twice the highest frequency of interest. For a 500 kHz band, 2.5 MS/s gives 5× margin Calculated.
At 2.5 MS/s and 32-bit samples, one channel produces 10 MB/s, or 6 GB per 10 minutes Calculated.
Frequency resolution of a single FFT segment is 1/T. Resolving 0.1 Hz needs 10 s segments Calculated.
Clock the digitiser and the arbitrary waveform generator from the same rubidium reference so record and replay are phase-coherent.

5.3 Why replaying the recorded signal is not enough

The recorded signal is the product of the actuator, sphere and sensor responses. Feeding the sensor trace straight to an actuator does not reproduce the original motion. The chain should be handled as follows.

Measure the end-to-end transfer function with a chirp or noise burst (H_act·H_sphere·H_sens).
Synthesise the drive with a regularised inverse filter. Regularisation is essential where the response is small or modes overlap.
Check amplifier and actuator linearity (total harmonic distortion) at the replay level.
Score fidelity with magnitude-squared coherence, normalised cross-correlation, and per-mode amplitude and phase error.

Mode density matters. The published baseline on the hollow copper sphere page is a 9.2 mm outer-diameter, 0.3 mm wall shell with a thin-shell breathing mode near 229 kHz, a design estimate that assumes E = 130 GPa (117 GPa would give about 218 kHz). A flat-plate-limit estimate of modal density, √3·A/(h·c_L), with A the shell surface area, h the wall and c_L ≈ 4,050 m/s, gives about 0.4 modes per kHz for that shell, a mean spacing near 2.6 kHz Calculated Assumption. The breathing mode sits at the ring frequency, where this asymptotic estimate is least reliable, so the real mode list must come from finite-element analysis and a mode survey. At Q = 10⁴ the linewidth is 23 Hz and at Q = 10³ it is 230 Hz; both are far below that spacing, so overlap is not the expected limit for the baseline shell. Thermal drift, transducer loading and amplitude and phase accuracy are. For contrast, the same estimate for a 100 mm outer-diameter, 3.0 mm wall shell (listed on that page, with a breathing mode near 21 kHz) gives about 4.5 modes per kHz at 230 kHz, a spacing near 220 Hz, where overlap and mode misidentification become real problems Calculated. These dimensions are design estimates, not confirmed specifications, and should be frozen only after finite-element analysis.

5.4 Temperature coupling between record and replay

If the sphere is 1 K warmer during replay than during recording, every mode has moved by roughly 23 to 46 Hz, more than a linewidth at Q = 10⁴. Keeping the shift under 10% of that linewidth (2.3 Hz) requires holding temperature within about 67 mK, assuming −0.015 %/K Calculated. Record and replay must therefore be thermally matched, with settle time and a logged temperature.

5.5 Why one microphone recording is not the sphere’s state

A microphone records air pressure at one point. The sphere’s state is a distributed field of displacement, velocity and strain over its surface. A single-channel recording loses or confounds several things:

Spatial information. Two different mode shapes can give similar pressure at one point.
Direction of motion. Torsional modes radiate weakly even when the structural motion is large.
Relative phase across the surface and near-field components that decay differently from far-field sound.
Room and microphone effects, including reflections and the microphone’s own frequency and phase response.
Closely spaced active modes, such as the split peaks of an imperfect sphere.

Rebuilding surface vibration from sound alone is an ill-posed inverse problem. A simulation study of a point-excited plate, which used near-field acoustic radiation modes as a basis, reported that reconstruction improved at shorter measuring distance, lower frequency and higher signal-to-noise ratio Sourced (Jiang, Lin and Liu, AIP Advances 12, 045014, 2022). A microphone array is therefore a complement to a vibrometer, not a substitute for one.

5.6 Four levels of reproduction

Level	Method	What it reproduces	Main limit
1. Replay the waveform	Drive an actuator with the recorded signal	The electrical command, not the original force	The amplifier, transducer, mounting and air path form a new transfer function
2. Inverse-filtered replay	Measure actuator-to-sensor response and drive with a regularised inverse	Motion at one sensor	Deep antiresonances and delay make exact inversion unstable
3. Multi-point or modal reproduction	Several actuators solve for commands that minimise error at several points	A spatial pattern within a bandwidth	Placement, controllability and spillover into unmodelled modes
4. Active control	Feedback or adaptive control from sensors to actuators	Suppressed, amplified or reshaped response of the combined system	Stability; the passive sphere is unchanged when the controller is off

Score every level on both the air-pressure error and the surface-velocity error. Success in one domain must not be reported as success in the other. The test design that follows from this is on the A/B/A verification protocol page; the measurement basics are in how to measure a sphere’s resonant frequency and impulse response measurement.

6. Other ways to approach the goal

The goal splits into two different questions: can we reproduce the resonant motion? and can we change the resonance? They need different tools.

Approach	Serves	Moves f₀?	Notes
Direct inverse-filtered replay	Reproduction	No	Baseline method of section 5. Fidelity bounded by noise, loading, thermal drift and, for larger shells, mode overlap.
Time-reversal re-emission	Reproduction and energy focusing	No	Record the response, reverse it in time, re-emit. Established for focusing in reverberant solids; Fink reports that in an extended elastic target it can excite the vibration modes automatically. Single-channel focusing works with broadband signals but not with a purely monochromatic phase-conjugation approach Sourced. Whether a thin spherical shell gives useful focusing gain is an open question.
Modal identification and synthesis	Reproduction with a model	No	Fit frequencies, damping and mode shapes from a 3D SLDV scan (RUS-style inversion), then synthesise drive per mode. Smaller data volume and a testable model.
Thermal tuning	Retuning	Yes	Simplest and best characterised. Tens of hertz per kelvin at 230 kHz. Slow, and it is also the dominant confound.
Active feedback (modal control)	Retuning of the combined system	Closed-loop only	Needs sub-100 ns latency (FPGA or analogue) and stability margin across neighbouring modes. Reshapes the apparent resonance and Q.
Shunted piezo patch	Retuning	Yes, weakly	Passive and simple; the patch loads the shell and lowers Q.
Controlled mass perturbation	Retuning and mode splitting	Yes	Useful as a calibration of sensitivity. Breaks symmetry and splits degenerate modes.
Amplitude sweep (NRUS)	Measurement	Slightly	Treat as a diagnostic of nonlinearity and microstructure rather than a tuning method. Risk of altering the sample at high strain.

7. Confounds and required controls

Temperature is the same order of size as the claimed location effect

The educational S.P.H.E.R.E. simulation (SPHEREVL) contains a location term of 45.2 Hz per mm of stage travel. That is a model parameter, not a measurement or validation. Against it, a 1 K change moves a 230 kHz mode by about 23 to 46 Hz, equivalent to 0.5 to 1.0 mm of stage travel in that model Calculated. Any experiment that moves the stage, or drives the sphere, without millikelvin thermometry cannot separate the two.

Drive heating. Dissipated drive power warms the shell. Log temperature through every replay and fit its contribution.
Measurement-beam heating. Keep SLDV beam power as low as the signal-to-noise allows.
Mass loading. For the 0.67 g baseline shell, an added 1 mg (adhesive, dust, a coating, a transducer tip) is 0.15% of the mass. If that mass moves with the breathing-mode amplitude, Δf/f ≈ −½Δm/m, about −170 Hz at 229 kHz, several linewidths at Q = 10⁴ Calculated Assumption (effective modal mass taken as the shell mass). Weigh the sphere before and after every handling step.
Mounting and transducer loading. RUS practitioners report that results depend on sample mounting and transducer placement. Stage motion changes support stress and contact; treat mounting as a variable and test it directly.
Mode misidentification. In a dense spectrum, track each mode by its mapped shape, not by peak order.
Electrical pickup and eddy-current interaction. Copper is a conductor; moving it in stray fields couples it to the electromagnetic environment. Faraday shielding and a field survey are required.
Environmental vibration and aging. Isolate the chamber and repeat across days.

Controls

Sham replay. Replay a phase-randomised surrogate with the same amplitude spectrum and power. If the frequency shift is the same, the effect is energy, not signal content.
Reference sphere. An identical sphere that is never driven and sits at matched temperature.
Blinded analysis. The person fitting resonance frequencies does not know which runs are real replay and which are sham.
Position-only control. Move the stage with no drive change, with temperature logged, to measure the location term separately.
Pre-registration. File hypotheses, acceptance thresholds and the analysis plan before the first replay run, and release raw data.

8. Proposed test protocol

Thresholds are set from measured repeatability in phase 0, not assumed in advance.

Phase	Activity	Exit criterion
0. Uncertainty budget	Repeat mode measurements across pump-downs and restarts. Measure the temperature coefficient of selected modes. Characterise transducer loading.	Repeatability of each tracked f_i known; detection threshold stated in hertz.
1. Characterise	Full 3D SLDV mode map, Q for each tracked mode, degeneracy splitting.	Mode list with shapes, frequencies, Q and thermal coefficients.
2. Record	Ring-down and steady-state records, temperature-stamped, Rb-locked.	Records archived with metadata and checksums.
3. Replay validation	Inverse-filtered replay; compare against the record.	Coherence and per-mode errors meet pre-registered limits.
4. Shift test	Real replay, sham replay and no-drive control; reference sphere; blinded fitting; temperature-corrected f₀.	H₀ (no shift beyond the budget) accepted or rejected.
5. Feedback trial	Only if phase 4 is null: closed-loop modal control on a few well-separated modes.	Closed-loop shift or Q change, with stability margin, reported with the loop parameters.
6. Release	Publish data, code, controls and the result whichever way it falls.	Public archive and preprint.

Prediction under standard physics: after temperature correction, passive replay produces no persistent shift in f₀ beyond the phase 0 budget. We expect H₀ to hold. A confirmed violation that survives the sham and reference controls would be a significant result and would be handed to independent groups for replication.

9. Risks and feasibility

Risk	Type	Severity	Mitigation
Expected null result for passive replay	Scientific	High likelihood	Design for a publishable null; keep feedback and thermal tuning as secondary arms.
Thermal drift larger than the effect sought	Technical	High	mK thermometry, long equilibration, reference sphere, temperature as a fitted covariate.
Low EMAT efficiency in vacuum at 230 to 500 kHz, and coil and magnet geometry on a 9.2 mm sphere with a 0.3 mm wall	Technical	Medium to high	Bench-test against laser-ultrasonic and piezo drive before committing; ask reviewers (Q4).
Contact, mount and adhesive mass dominate the frequency of a 0.67 g shell	Technical	High	Non-contact drive and receive where possible; characterise the mount directly; weigh every addition (section 7).
Thin-shell mode-density estimate is unreliable near the ring frequency; larger shells would have dense spectra	Technical	Medium	Finite-element mode list first; per-mode and model-based synthesis rather than whole-spectrum inversion.
Feedback loop latency and instability	Technical	Medium	FPGA implementation; start at lower-frequency modes of a test shell.
SLDV cost and optical access	Financial	High	The SLDV is the dominant capital item. No vendor quote has been obtained; confirm access to an existing instrument or a collaborating lab before purchase.
Geometry assumptions are unconfirmed	Technical	Medium	Run finite-element analysis and freeze dimensions afterwards.

10. Request for expert review

We are asking acousticians, resonant-ultrasound and ultrasonics specialists, vibration and structural-dynamics engineers, metrologists, and signal-processing and control engineers to critique this draft. Critical responses are the most useful. Please say where the physics is wrong, where a number is off, and what we have missed.

Questions we need answered

Missing mechanism. Is there a route by which replaying recorded signal content at sub-plastic amplitudes produces a persistent shift in the natural frequencies of a monolithic OFE copper shell that we have not listed in section 4?
Nonlinear magnitude. What strain amplitudes and fractional frequency shifts are realistic from NRUS-type softening in annealed C10100 copper near 230 kHz?
Temperature coefficient. What are measured or modelled coefficients for thin copper shell modes, and what millikelvin stability is achievable in UHV?
Transduction. For drive and receive on a 9.2 mm diameter, 0.3 mm wall copper shell at 230 to 500 kHz in vacuum, how do EMAT, pulsed-laser and piezo options compare in efficiency, loading, linearity and practical coil or magnet geometry?
Replay fidelity. Which deconvolution and regularisation methods do you recommend, for the sparse baseline spectrum (about 0.4 modes per kHz by our plate-limit estimate) and for larger shells with roughly 4.5 modes per kHz? Is the plate-limit estimate usable near the ring frequency?
Time reversal. Does single-channel or multi-channel time reversal give useful focusing on a thin spherical shell, and what gain should we expect?
Feedback. What controller architecture, latency budget and stability margin would you use for modal control near 230 kHz?
Mode identification. Is a 3D SLDV scan sufficient to identify modes uniquely, or should we also run RUS-style model inversion? How many modes validate a model?
Controls. Is a phase-randomised sham adequate, and what further controls would you require?
Shell model. Is the thin-shell treatment of a 9.2 mm OD, 0.3 mm wall sphere with a breathing mode near 229 kHz sound, given a wall-to-radius ratio near 0.065? Which finite-element benchmark would you use?
Statistics. What sample size, blinding scheme and decision thresholds would you pre-register?
Prior art. Which papers, standards or instruments have we overlooked?

Where we are asking

Acoustical Society of America technical committees: Physical Acoustics, Structural Acoustics and Vibration, and Signal Processing in Acoustics.
IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society, including the Physical Acoustics and Sensors/NDE groups of the International Ultrasonics Symposium.
The resonant-ultrasound spectroscopy community, including sessions at the American Physical Society meetings.
Time-reversal acoustics groups (for example at the Institut Langevin and the Brigham Young University Acoustics Research Group).
Application engineers at laser-vibrometer and EMAT vendors, for practical limits of their instruments.

We have not yet contacted these groups; this list is where the request is going, not a claim of endorsement.

How to respond

Copy the template, answer the questions you can, and email it to contact@spheredesci.org. Partial answers are welcome. We will log every response, publish a revision history for this page, and credit contributors by name only with their written consent.

S.P.H.E.R.E. record-and-replay review: response

Name (optional):
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Questions answered (Q1-Q12):
Q__:
  Answer:
  Confidence (low / medium / high):
  References or data:

Corrections to numbers or physics in the article:

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11. Frequency-budget calculator

Edit the inputs to test the estimates above. Every output is a first-order estimate. The temperature coefficient is an assumption until measured on the actual sphere.

–Thermal shift (heating lowers f)

–Nonlinear shift, −½ΔE/E · f₀

–Linewidth f₀/Q

–Response at the stated detuning, relative to peak

–Loop phase error at f₀

–Temperature window for a shift of 10% of linewidth

–Thermal shift expressed as stage travel in the model

12. Sources and evidence status

External sources

Nonlinear resonant ultrasound spectroscopy: arXiv 2602.21678 (Ga-induced damage in aluminium alloys), and the Politecnico di Torino study of fast and slow dynamics. The ΔE/E = 2Δf/f relation appears in US patent 11,079,505.
Resonant ultrasound spectroscopy overview: Wikipedia. Sphericity and mode splitting: Dixon, Migliori, Visscher, OSTI 869554. Measurement-science perspective: Migliori, Keithley Award abstract.
Contactless resonance spectroscopy: Ogi, Ledbetter, Kim and Hirao, J. Acoust. Soc. Am. 106(2), 660 (1999).
Transducer coupling effects: Yoneda, Earth Planets Space 54, 763 (2002). Sensitivity to mounting and placement in copper RUS: APS March Meeting 2007 abstract.
Temperature: US patent 5,965,817 (temperature compensation of resonance measurements); Chang and Himmel, The Temperature Dependence of the Elastic Constants of Cu, Ag, and Au Above Room Temperature, UCRL-16697 (1966); Ledbetter and Naimon, copper elastic properties review.
Time reversal: Fink, ASA 1997 abstract; time reversal of continuous-wave signals in elastic media, BYU.
Shunted piezoelectric tuning: J. Indian Inst. Sci., resonantly shunted piezoelectric layers; arXiv 2403.00560.
Instruments: A*STAR UHF-120 instrument overview; EMAT overview; EMAT technology note.

Internal project inputs

The 9.2 mm outer diameter, 0.3 mm wall, 0.67 g mass and 229 kHz breathing-mode estimate are the baseline on the hollow copper sphere page. They are design estimates, not confirmed specifications. The modal-density figures in section 5.3 are calculated here from that baseline.
The 45.2 Hz/mm location term is a parameter of the project’s educational metrology-lab simulation (SPHEREVL / Control Room). It is not published on this site, illustrates scale only, and is not a measurement or scientific validation.

What has not been verified

The copper temperature coefficient for these modes; the nonlinear shift of annealed C10100 copper at 230 kHz; EMAT efficiency on a 0.3 mm shell in vacuum; the feasibility of feedback at 230 kHz; any hardware cost. No experiment has been run for this article.
Calculated values were recomputed in code on 16 October 2026 and match the figures printed here.

Continue the design review

About this page. Original editorial text for Project S.P.H.E.R.E., added October 16, 2026. Sources are linked in section 12. Hypotheses are not findings.

About Project S.P.H.E.R.E.

Project S.P.H.E.R.E. is a pre-launch research project developing proposals for controlled resonance measurements. Its research pages describe hypotheses and methods, not measurement results or scientific validation.

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