Speedmeter: Principles and Applications

• Speedmeter is a measurement scheme that converts various observables (e.g., time of flight, pressure difference, image displacement) into accurate speed estimates.
• Classical implementations, from echo acoustics to smartphone-based Pitot tubes and inertial sensors, demonstrate high precision through calibrated observable-to-speed inversion.
• In gravitational-wave interferometry, speedmeters utilize advanced optical techniques to reduce quantum noise and enhance low-frequency sensitivity.

Speedmeter denotes an instrument or measurement scheme that estimates speed from an observable linked to motion. In the cited research, the term spans classical devices that infer speed from time of flight, pressure difference, angular rate, image displacement, or magnetic-field harmonics, and also a specialized interferometric usage in which the optical output is proportional to a velocity-like observable rather than instantaneous position (Berg et al., 2011, Shah et al., 2018, Freise et al., 2019). Across these settings, the defining operation is the conversion of a measured quantity into a speed estimate, but the relevant transfer function, calibration procedure, and dominant noise sources differ markedly.

1. Conceptual basis

At the most elementary level, a speedmeter implements $v = d/t$. In the echo-based acoustic experiment, the measured quantity is a round-trip propagation time $\Delta t$ over a known path $d_{rt}$, giving $v = d_{rt}/\Delta t = 2L/\Delta t$ for a planar reflector at distance $L$ (Berg et al., 2011). In a Pitot-tube implementation, the measured quantity is a pressure difference, and Bernoulli’s equation yields

$$v = \sqrt{\frac{2\left(P_{\text{stagnation}} - P_{\text{static}}\right)}{\rho}}, \qquad \rho = \frac{p}{R_* T},$$

with $p = P_{\text{static}}$ for dry air (Dorsel et al., 2022). In a wheel-mounted inertial sensor, the primary observable is angular velocity, so the relevant kinematic relation is

$v(t) = R\,\omega(t),$

with $R$ the effective wheel radius (Shah et al., 2018). In camera-based systems, the observable is image displacement over time, which must be mapped from pixels to physical distance by calibration (Ikon, 2017). In magnetic inductive sensing, the observable is the harmonic spectrum of a motor’s magnetic field, from which the rotation frequency and then RPM are inferred (Hoskeri et al., 13 Mar 2025).

In interferometric gravitational-wave detection, the same term acquires a more restrictive meaning. A position meter measures a displacement $x(t)$ or differential arm length, whereas a speedmeter measures a quantity approximating $\Delta t$0 for a delay $\Delta t$1, so that the readout is proportional to velocity in the low-frequency limit (Freise et al., 2019, Onglao et al., 7 Apr 2026). This suggests a common structural feature across otherwise disparate devices: a speedmeter is defined less by its hardware than by the observable-to-speed map that it realizes.

2. Classical implementations

The acoustic “speedmeter for sound” is an explicit time-of-flight system. A balloon is popped next to a microphone at a measured distance from a large flat wall, the waveform is digitized, and the time between the pop and the echo is read from the recording (Berg et al., 2011). In the reported experiment, the microphone/balloon pair was $\Delta t$2 from the wall, so the round-trip distance was $\Delta t$3. With a sampling rate of $\Delta t$4, three trials gave speeds $\Delta t$5, $\Delta t$6, and $\Delta t$7, with a mean $\Delta t$8. The authors state that the method is accurate to better than $\Delta t$9 (Berg et al., 2011).

The smartphone Pitot-tube speedometer is pressure-based rather than time-based. A smartphone barometer measures ambient static pressure and then stagnation pressure when exposed to airflow; phyphox computes air density from the ideal gas law and converts pressure difference to speed through Bernoulli’s equation (Dorsel et al., 2022). In the wind-speed configuration, the paper reports meaningful values up to about $d_{rt}$0 for the wind machine. In the vehicle-speed configuration, simultaneous static and stagnation pressure measurements are obtained from a smartphone barometer and an external Texas Instruments SensorTag CC2650 BLE pressure sensor. The resulting Pitot-derived speed is described as noisier than GPS but as fitting very well overall (Dorsel et al., 2022).

The following representative cases illustrate the range of meanings attached to the term.

Domain	Measured quantity	Reported characteristic
Echo acoustics	Round-trip acoustic time	$d_{rt}$1 mean at $d_{rt}$2
Smartphone Pitot tube	$d_{rt}$3	Meaningful values up to about $d_{rt}$4
Wheel-mounted IMU	Gyroscope angular velocity	Localization error about $d_{rt}$5 of total path length
Smartphone accelerometer DNN	$d_{rt}$6 acceleration window	MAE $d_{rt}$7
Magnetic inductive sensing	Harmonic magnetic spectrum	RMAE $d_{rt}$8 below $d_{rt}$9

These examples show that classical speedmeters need not share a sensor modality. What they share is a calibrated inversion from an observable to speed, plus an error model that specifies how distance, pressure, orientation, timing, or spectral uncertainty enters the estimate (Berg et al., 2011, Dorsel et al., 2022, Shah et al., 2018, Or, 2024, Hoskeri et al., 13 Mar 2025).

3. Wheel, camera, inertial, and magnetic speedmeters

A non-intrusive inertial speed sensor for ground vehicles mounts an IMU directly on the wheel rim and uses gyroscope output as the primary speed observable (Shah et al., 2018). The measurement model is

$v = d_{rt}/\Delta t = 2L/\Delta t$0

with $v = d_{rt}/\Delta t = 2L/\Delta t$1 the true angular velocity, $v = d_{rt}/\Delta t = 2L/\Delta t$2 the bias, and $v = d_{rt}/\Delta t = 2L/\Delta t$3 zero-mean noise. The paper develops both a Kalman-based estimator and an RLS alternative to recover denoised angular velocity and then wheel speed via $v = d_{rt}/\Delta t = 2L/\Delta t$4. The hardware stack includes a Pololu MinIMU-9, Arduino Nano, nRF24L01 wireless module, and an 18650 Li-ion cell, with ROS and GTSAM used downstream for odometry and factor-graph localization. In the reported Husky robot experiment, the resulting localization error is about $v = d_{rt}/\Delta t = 2L/\Delta t$5 of total path length (Shah et al., 2018).

A camera-based smartphone speedmeter treats the vehicle as an image object. The Android implementation uses OpenCV 2.4.11 and an MB-LBP cascade classifier to detect a vehicle and track the position of the bounding rectangle across frames (Ikon, 2017). Pixel displacement is computed from the rectangle coordinates, divided by the measured elapsed time, and then converted to physical speed using a calibration derived from a measured vehicle length in the image. In the reported test geometry, a $v = d_{rt}/\Delta t = 2L/\Delta t$6 car occupied $v = d_{rt}/\Delta t = 2L/\Delta t$7 pixels, implying $v = d_{rt}/\Delta t = 2L/\Delta t$8. The application computes multiple speed estimates over 20 detections and takes the median to suppress outliers. The paper concludes that the measured speeds are proportional to the ground truth speeds (Ikon, 2017).

CarSpeedNet replaces explicit geometry with supervised learning. The model uses only three-axis smartphone accelerometer data, with no CAN bus, no wheel sensors, and no gyroscope (Or, 2024). The dataset comprises $v = d_{rt}/\Delta t = 2L/\Delta t$9 hours across $L$0 distinct sessions, with accelerometer data sampled at $L$1, low-pass filtered, and downsampled to $L$2. The best configuration uses a $L$3 window, corresponding to an $L$4 input, and combines Bi-LSTM, LSTM, Conv1D, and dense layers. Ground truth speed is taken from $L$5 GPS. The best reported performance is RMSE $L$6 and MAE $L$7, which the abstract summarizes as a precision of less than $L$8 (Or, 2024).

MagTach is a handheld magnetic-inductive rotation speedmeter for electric machines (Hoskeri et al., 13 Mar 2025). The sensed field is modeled as

$L$9

and the induced voltage follows Faraday’s law: $$v = \sqrt{\frac{2\left(P_{\text{stagnation}} - P_{\text{static}}\right)}{\rho}}, \qquad \rho = \frac{p}{R_* T},$$0 Unlike conventional magnetic tachometers that search mainly for the fundamental, MagTach exploits the full harmonic structure using a custom inductive sensor array, adaptive noise cancellation, delay-and-sum spatial filtering, Welch PSD estimation, a Pyramid Power Spectrum Parsing Network, and fuzzy-logic frequency inference (Hoskeri et al., 13 Mar 2025). The prototype uses four inductive sensors, MAX9814 amplifiers with $$v = \sqrt{\frac{2\left(P_{\text{stagnation}} - P_{\text{static}}\right)}{\rho}}, \qquad \rho = \frac{p}{R_* T},$$1 gain, an SGTL5000 ADC at $$v = \sqrt{\frac{2\left(P_{\text{stagnation}} - P_{\text{static}}\right)}{\rho}}, \qquad \rho = \frac{p}{R_* T},$$2 and 16-bit resolution, a Teensy 4 microcontroller, and a $$v = \sqrt{\frac{2\left(P_{\text{stagnation}} - P_{\text{static}}\right)}{\rho}}, \qquad \rho = \frac{p}{R_* T},$$3 lithium battery. The paper reports RMAE $$v = \sqrt{\frac{2\left(P_{\text{stagnation}} - P_{\text{static}}\right)}{\rho}}, \qquad \rho = \frac{p}{R_* T},$$4, $$v = \sqrt{\frac{2\left(P_{\text{stagnation}} - P_{\text{static}}\right)}{\rho}}, \qquad \rho = \frac{p}{R_* T},$$5, and $$v = \sqrt{\frac{2\left(P_{\text{stagnation}} - P_{\text{static}}\right)}{\rho}}, \qquad \rho = \frac{p}{R_* T},$$6 at $$v = \sqrt{\frac{2\left(P_{\text{stagnation}} - P_{\text{static}}\right)}{\rho}}, \qquad \rho = \frac{p}{R_* T},$$7, $$v = \sqrt{\frac{2\left(P_{\text{stagnation}} - P_{\text{static}}\right)}{\rho}}, \qquad \rho = \frac{p}{R_* T},$$8, and $$v = \sqrt{\frac{2\left(P_{\text{stagnation}} - P_{\text{static}}\right)}{\rho}}, \qquad \rho = \frac{p}{R_* T},$$9, respectively; range up to $p = P_{\text{static}}$0; maximum RPM $p = P_{\text{static}}$1; and operation through walls and thin metals, though not through a $p = P_{\text{static}}$2 stainless-steel pot without major degradation (Hoskeri et al., 13 Mar 2025).

Taken together, these systems show three distinct modern trajectories in speedmeter design: explicit model-based inversion, learned inference from raw sensors, and hybrid sensing-plus-ML pipelines. A plausible implication is that contemporary speedmeters increasingly trade direct interpretability for portability, non-intrusiveness, or sensing range.

4. Speedmeter in gravitational-wave interferometry

In gravitational-wave detection, a speedmeter is a quantum-non-demolition measurement scheme whose output is proportional to the speed of the end test masses instead of their differential position (Freise et al., 2019). Standard Michelson interferometers are position meters: their main observable is $p = P_{\text{static}}$3. By contrast, a speedmeter implements an effective finite difference,

$p = P_{\text{static}}$4

so the low-frequency response is velocity-like (Freise et al., 2019). The motivation is quantum-noise engineering: radiation-pressure noise is the dominant low-frequency back-action in position meters, and speedmeter topologies are designed so that this back-action is partially canceled or strongly suppressed (Onglao et al., 7 Apr 2026).

The Sagnac topology is the canonical realization. In a speed-meter Sagnac, clockwise and counterclockwise beams traverse the interferometer in opposite orders and sample the test masses at different times; in the low-frequency limit the output obeys

$p = P_{\text{static}}$5

which is proportional to velocity $p = P_{\text{static}}$6 (Onglao et al., 7 Apr 2026). A related realization is the sloshing speedmeter, where a Michelson is coupled to an auxiliary sloshing cavity that returns a delayed copy of the signal field so that the outgoing light encodes a difference of mirror positions at two times (Freise et al., 2019).

The EPR speedmeter introduces two independent optical readouts with different bandwidths $p = P_{\text{static}}$7 and circulating powers $p = P_{\text{static}}$8, constrained by

$p = P_{\text{static}}$9

When the two outputs are combined at a beamsplitter, the common output is proportional to position and the differential output to velocity (Knyazev et al., 2017). The paper argues that this scheme is attractive because it requires only minor changes to a standard signal-recycled Michelson interferometer and allows switching between position-meter and speed-meter modes without hardware changes (Knyazev et al., 2017).

Recent tabletop demonstrations have focused on validating the velocity response itself. A triangular-cavity EPR speedmeter based on two orthogonal polarisation modes with different effective bandwidths shows a differential response that vanishes at DC and increases linearly with signal frequency up to the bandwidth of the slower mode (Kranzhoff et al., 15 Aug 2025). A polarization-circulation speed meter, implemented first in a single-cavity configuration with all mirrors fixed, uses a quarter-wave plate, a polarization beam splitter, and a polarization circulation mirror to make the field interact twice; its measured transfer function confirms that the device measures the speed of mirror motion (Nishino et al., 13 Nov 2025). A hybrid readout cavity with a movable test mass has now shown simultaneous speedmeter and position-meter response in two output ports, constituting the first experimental observation of speedmeter behavior in an optomechanical system with a movable test mass (Korobko et al., 18 Sep 2025).

At the detector-concept level, CHRONOS extends the speedmeter idea into the sub-Hz regime. It combines cryogenic operation, a cross torsion-bar configuration, and a triangular Sagnac interferometer with speed-meter readout, targeting $v(t) = R\,\omega(t),$0 to $v(t) = R\,\omega(t),$1, $v(t) = R\,\omega(t),$2 around $v(t) = R\,\omega(t),$3, and $v(t) = R\,\omega(t),$4 at $v(t) = R\,\omega(t),$5 (Onglao et al., 7 Apr 2026). In that design, the speedmeter is not an auxiliary feature but the mechanism that keeps quantum back-action from overwhelming the low-frequency band.

5. Non-idealities, asymmetries, and enabling optics

The idealized advantage of a speedmeter is highly sensitive to losses and asymmetries. A generalized analytical treatment of the Sagnac speed meter with arm cavities shows that losses, asymmetric beamsplitters, and unequal arm-cavity input mirror reflectivities materially affect the quantum-noise-limited sensitivity (Danilishin et al., 2014). This indicates that the speedmeter is not merely a topological label; it is a topological response that must survive realistic optical imperfections.

Beamsplitter imbalance is a particularly consequential example. In an asymmetric Sagnac, imbalance increases laser-noise coupling to the detection port and can reduce the low-frequency superiority of the speed meter. Balanced homodyne readout with a co-moving local oscillator or a BSAR local oscillator partly compensates this excess noise (Zhang et al., 2018). For a beam splitter imbalance of $v(t) = R\,\omega(t),$6 in the Glasgow proof-of-concept experiment, the required relative intensity noise at $v(t) = R\,\omega(t),$7 decreases from $v(t) = R\,\omega(t),$8 to $v(t) = R\,\omega(t),$9, shifting the requirement from a not practical achievable value to one routinely achieved with moderate effort (Zhang et al., 2018).

Polarisation-based speedmeters move the critical tolerances into polarisation optics. In a 10 m prototype polarisation Sagnac speedmeter, the PBS was characterized with an extinction ratio better than $R$0 in transmission and $R$1 in reflection at $R$2 angle of incidence, the QWP had birefringence $R$3, and cavity mirror birefringence lay between $R$4 and $R$5 radians (Spencer et al., 2021). The paper shows that cavity birefringence and QWP imperfections can be canceled by careful adjustment of the QWP angle, leaving PBS extinction ratio as the leading limitation in terms of polarisation effects (Spencer et al., 2021).

All-polarisation beamsplitters are therefore an enabling technology. Coatings from Laseroptik GmbH and Optoman were investigated on a best-effort basis for equal 50/50 power splitting and equal overall phase shift for orthogonal linear polarisations (Kranzhoff et al., 30 Jul 2025). The Laseroptik coating uses 22 alternating layers of $R$6 and $R$7, reaches equal power reflectivity of $R$8 at $R$9, and zero phase shift between both polarisations at $x(t)$0. The Optoman coating uses 5 $x(t)$1 and $x(t)$2 layers, giving power reflectivities of $x(t)$3 for s-polarisation and $x(t)$4 for p-polarisation with a differential phase shift around $x(t)$5 that is largely independent of angle (Kranzhoff et al., 30 Jul 2025). The cited measurements do not yet coincide perfectly with the ideal speedmeter condition, but they identify the required optical specifications with unusual clarity.

A common misconception is that “speedmeter” in interferometry simply means a different data-analysis convention applied to a Michelson output. The cited analyses show instead that the speedmeter advantage is contingent on the optical plant itself: delays, symmetry, homodyne strategy, and polarisation control are part of the observable.

6. Applications, pedagogy, and outlook

Classical speedmeters continue to function as compact laboratories for measurement science. The echo-based sound-speed experiment is suitable for high-school and undergraduate physics labs, illustrates time-of-flight measurement, and can be used either to determine the speed of sound or, if the speed of sound is known, to determine unknown distances (Berg et al., 2011). The phyphox Pitot-tube experiment similarly ties Bernoulli’s principle to real-time sensing and extends naturally from fan-generated airflow to vehicle speed measurement with a BLE pressure sensor (Dorsel et al., 2022).

Vehicle and mobile applications cover several distinct use cases. The wheel-mounted inertial speed sensor addresses low-speed autonomy and odometry without modifying the drivetrain (Shah et al., 2018). The smartphone-camera application acts as a mobile speed camera that stores image, time, and location together with the estimated speed (Ikon, 2017). CarSpeedNet targets app-based and telematics-style inference from commodity smartphone accelerometers alone (Or, 2024). MagTach addresses industrial and diagnostic settings—factory motor calibration, engine-health monitoring, and appliance fault detection—by eliminating line-of-sight and direct mechanical access (Hoskeri et al., 13 Mar 2025). These cases suggest that the contemporary speedmeter is increasingly a software-defined instrument built around a calibrated or learned sensing front end.

The interferometric branch has a different application horizon. Speedmeters are being developed to mitigate quantum radiation-pressure noise in future gravitational-wave detectors (Knyazev et al., 2017). CHRONOS pushes this into the $x(t)$6 to $x(t)$7 band and, beyond gravitational-wave astronomy, evaluates prompt gravity-gradient detection for earthquake early warning; at $x(t)$8, the predicted lead time over conventional seismometers is $x(t)$9 to $\Delta t$00 seconds (Onglao et al., 7 Apr 2026). In that context, the speedmeter is not an ancillary sensor but a candidate foundational observable for low-frequency precision measurement.

A plausible synthesis of the cited literature is that the term now denotes two mature but still converging traditions. One tradition concerns practical instruments for acoustic, vehicular, fluid-dynamic, and electromechanical speed estimation. The other concerns quantum-limited interferometers in which a velocity-like optical observable is engineered to reshape the measurement back-action itself. The unifying theme is not hardware similarity but the deliberate construction of an observable whose inversion yields speed.