Mid-Air Audio & Duty Cycle Acoustic Lab — Ultrasonic Heterodyning & Electronic Coil Whine Simulator
Interactive simulation laboratory investigating mid-air parametric acoustic self-demodulation (original ultrasonic carrier vs demodulated audible audio in free space) and electronic PWM duty cycle acoustic emanations (coil whine, magnetostriction, and piezoelectric MLCC singing). Explore real-time Web Audio synthesis, 3D acoustic field propagation, duty cycle modulation mathematics, and psychoacoustic disambiguation.
1. Mid-Air Parametric Non-Linear Demodulation (Westervelt & Berktay)
Parametric acoustic arrays launch intense ultrasonic carrier waves (40 kHz to 100 kHz) into the air. Due to the non-linear compressibility of air (parameter β ≈ 1.2), wave crests travel faster than troughs, causing the air column itself to self-demodulate the audio envelope into an audible, highly directional sound beam.
2. Electronic PWM Duty Cycle Acoustic Emanations & Coil Whine
Switched-Mode Power Supplies (SMPS) and voltage regulators rapidly pulse inductors and capacitors at switching frequencies (10 kHz–100 kHz). When dynamic workloads alter the pulse-width modulation duty cycle (D = T_on / T), magnetostriction in ferrite inductor cores and piezoelectric strain in MLCC ceramic capacitors generate loud acoustic squeal, turning power supplies into unintentional loudspeakers.
Interactive Simulation Presets
- Mid-Air Parametric Ultrasound Demodulation (Non-Linear Acoustics): A 40 kHz ultrasonic beam modulated with human speech. As the beam travels through the air column, the non-linear elasticity of air demodulates the envelope into an audible voice in mid-air.
- SMPS Inductor Coil Whine (Magnetostriction) (Power Electronics): A Switched-Mode Power Supply (SMPS) buck converter running at 16 kHz. As the duty cycle shifts with dynamic CPU load, the magnetic flux expansion in the ferrite inductor emits high-pitched acoustic squeal.
- Ceramic MLCC Capacitor Piezoelectric Singing (Component Physics): Multi-layer Ceramic Capacitors (MLCC) with BaTiO3 dielectric act as piezoelectric transducers. High-frequency voltage pulses at 22 kHz create physical mechanical deformation that vibrates the PCB.
- Repetitive High-Pitch Pulse Train (Temporal Loudness Summation) (Psychoacoustics): Repetitive 14 kHz pulsed tone bursts at 80 Hz PRF. The human auditory cortex integrates energy over a ~200 ms temporal window; increasing the duty cycle of the pulse train dramatically increases perceived loudness.
- Auditory Pareidolia & Phonemic Restoration Filter (Cognitive Neuroscience): Demonstrates how the human brain fills in missing speech signals when listening to repetitive high-frequency noise or duty cycle switching transients, synthesizing phantom words or speech.
Mathematical & Physics Foundations
- 1. Berktay's Non-Linear Acoustic Demodulation in Air:
p_demod(t, r) ≈ (β · P₀² · S) / (16 · π · ρ₀ · c₀⁴ · α · r) · (∂² / ∂t²) [ E²(t) ]
When an intense ultrasonic beam (e.g. 40 kHz) travels through the air, high-pressure crests travel slightly faster than low-pressure troughs due to adiabatic heating and non-linear bulk modulus. This non-linear distortion self-demodulates the envelope E(t), generating audible baseband sound directly in the mid-air column along the beam path. - 2. PWM Duty Cycle Fourier Series & Harmonic Energy:
v(t) = V_in · D + 2 · V_in · Σ [ sinc(n · D) · cos(2π · n · f_sw · t) ]
Every electronic PWM switching stage creates infinite harmonic frequencies whose amplitudes are directly shaped by the duty cycle D. When D changes dynamically (duty cycle modulation), harmonic sidebands appear in the audible acoustic spectrum. - 3. Magnetostrictive Force & Acoustic Power Scaling in Inductors:
F_mag(t) = (B² · A_core) / (2 · μ₀) ∝ (L · I_peak · D)²
Ferrite and iron cores change their physical dimensions when magnetized (magnetostriction). Because magnetic force scales with B², the acoustic vibration power is proportional to the square of current and duty cycle. Higher duty cycles deliver more magnetic energy per cycle, making the acoustic whine louder. - 4. Psychoacoustic Temporal Loudness Summation:
L_perceived = k · [ ∫₀^T (p(t))² · dt ]^0.35 = k · (I_avg · T_int)^0.35
The human auditory system does not measure instantaneous peak sound; it integrates acoustic energy over a 200 ms temporal window. A repetitive high-pitch tone with a high duty cycle (longer 'on' duration) pumps significantly more integrated acoustic energy into the cochlea, resulting in a dramatic increase in perceived loudness.
4-Step Diagnostic Disambiguation Protocol
- Step 1: Spatial Proximity & Inverse-Square Localization Test [Spatial Test]
Move your head or walk 1–2 meters in different directions while listening to the high pitch.
Physics: Physical point sources (like electronic power supplies, chargers, monitors) drop in volume by 6 dB per doubling of distance (1/r² law). In contrast, a parametric ultrasonic beam is collimated (pencil beam 3°–10°), cutting in and out abruptly across a ~10 cm boundary. Internal tinnitus does not change volume with head movement. - Step 2: Power Supply & Circuit Breaker Isolation Test [Electrical Test]
Systematically unplug laptop chargers, LED dimmer switches, monitor power bricks, and USB hub power supplies.
Physics: Switched-mode power supplies (SMPS) operate at 15 kHz–100 kHz. When under light load or standby, pulse-skipping modes drop the switching frequency into the audible 8 kHz–18 kHz range. Unplugging the device instantly cuts magnetostrictive and piezoelectric acoustic vibration. - Step 3: Smartphone Microphone Spectrum Recording (FFT Verification) [Objective Proof]
Use a spectrum analyzer app (e.g. Phyphox, Spectroid, Audacity) on a smartphone or external USB microphone.
Physics: Standard smartphone MEMS microphones capture frequencies from 20 Hz up to 20 kHz (at 44.1 kHz or 48 kHz sample rates). If an external physical sound or coil whine is present, the FFT waterfall will display a sharp, persistent vertical line at the exact frequency (e.g. 15.625 kHz CRT/SMPS line, 18.2 kHz inverter harmonic). - Step 4: Phonemic Restoration & Auditory Pareidolia Test [Cognitive Check]
Record the ambient sound and play it back in slow motion (0.5x speed) or inverted phase.
Physics: The human auditory cortex contains high-level pattern-matching neural networks that constantly attempt to parse ambiguous, repetitive high-frequency noise (e.g., fan motor clicks, SMPS pulse trains, water pipe hiss) into familiar linguistic phonemes. Slowing down the recording reveals whether the sound contains actual modulated speech formants or simply white/colored noise.
Python Simulation Code
# berktay_demodulator.py — Simulates mid-air non-linear acoustic self-demodulation
import numpy as np
import scipy.signal as signal
def simulate_midair_demodulation(carrier_freq=40000, sample_rate=192000, duration=1.0):
"""
Computes Berktay's non-linear acoustic self-demodulation of an
ultrasonic beam traveling through ambient air.
"""
t = np.linspace(0, duration, int(sample_rate * duration), endpoint=False)
# 1. Baseband audio message: two spoken vowel formants (500 Hz & 1500 Hz)
message = 0.6 * np.sin(2 * np.pi * 500 * t) + 0.4 * np.sin(2 * np.pi * 1500 * t)
# 2. Square-root envelope modulation for low-distortion demodulation
m = 0.8 # modulation index
envelope = np.sqrt(np.maximum(0, 1.0 + m * message))
# 3. Modulate onto 40 kHz ultrasonic carrier
ultrasonic_tx = envelope * np.sin(2 * np.pi * carrier_freq * t)
# 4. Air non-linear propagation (Berktay's equation: second derivative of envelope squared)
# p_demod(t) ∝ d^2/dt^2 [ E^2(t) ]
envelope_sq = envelope ** 2
d_env_sq = np.gradient(envelope_sq, t)
demodulated_audio = np.gradient(d_env_sq, t)
# 5. Low-pass filter to extract audible band (< 4 kHz)
b, a = signal.butter(4, 4000 / (sample_rate / 2), btype='low')
audible_speech = signal.filtfilt(b, a, demodulated_audio)
# Normalize
audible_speech /= np.max(np.abs(audible_speech) + 1e-9)
print(f"[+] Mid-Air Non-Linear Demodulation Computed:")
print(f" Carrier: {carrier_freq} Hz | Sample Rate: {sample_rate} Hz")
print(f" Demodulated Audio Peak SNR: {20 * np.log10(np.std(audible_speech)):.2f} dB")
return t, ultrasonic_tx, audible_speech
if __name__ == '__main__':
t, tx, rx = simulate_midair_demodulation()
# pwm_duty_cycle_whine.py — Simulates PWM duty cycle acoustic magnetostriction
import numpy as np
import scipy.fft as fft
def analyze_duty_cycle_harmonics(f_sw=16000, duty_cycle=0.65, sample_rate=192000):
"""
Computes the acoustic emanation spectrum produced by an inductor
under variable PWM duty cycle modulation.
"""
t = np.linspace(0, 0.05, int(sample_rate * 0.05), endpoint=False)
# 1. Generate PWM square wave with duty cycle D
period_samples = int(sample_rate / f_sw)
high_samples = int(period_samples * duty_cycle)
pwm_single_period = np.zeros(period_samples)
pwm_single_period[:high_samples] = 1.0
repeats = int(len(t) / period_samples) + 1
pwm_wave = np.tile(pwm_single_period, repeats)[:len(t)]
# 2. Inductor current ripple: i_L(t) = (1/L) * integral(V * dt)
# Magnetostrictive mechanical stress: F(t) ∝ i_L(t)^2
current_ripple = np.cumsum(pwm_wave - duty_cycle) / sample_rate
acoustic_pressure = np.gradient(np.gradient(current_ripple ** 2))
# 3. FFT spectrum
spectrum = np.abs(fft.rfft(acoustic_pressure))
freqs = fft.rfftfreq(len(acoustic_pressure), 1.0 / sample_rate)
peak_idx = np.argmax(spectrum[1:]) + 1
print(f"[+] PWM Duty Cycle Acoustic Analysis:")
print(f" Switching Freq: {f_sw} Hz | Duty Cycle: {duty_cycle * 100:.1f}%")
print(f" Dominant Acoustic Peak: {freqs[peak_idx]:.1f} Hz (Magnitude: {spectrum[peak_idx]:.2f})")
return freqs, spectrum
if __name__ == '__main__':
analyze_duty_cycle_harmonics()
Frequently Asked Questions
How can high-pitched ultrasonic sounds create audible audio in mid-air without a physical speaker?
This phenomenon is governed by Westervelt's (1963) and Berktay's (1965) non-linear acoustic theory. Normal sound propagation in air is linear at low amplitudes. However, when high-intensity ultrasound (e.g., 40 kHz at >120 dB SPL) is launched into the air, the non-linear compressibility parameter of air (β ≈ 1.2) causes the crests of the ultrasonic wave to travel slightly faster than the troughs. This self-demodulates the audio envelope, creating a virtual acoustic source directly in the mid-air column. Listeners inside the narrow beam hear clear audible speech or tones, while bystanders off-axis hear complete silence.
What is the difference between the original mid-air ultrasonic audio and the demodulated audio?
The 'original' mid-air audio consists of an intense ultrasonic carrier (typically 40,000 Hz to 100,000 Hz) that is inaudible to human ears because the hair cells in the cochlea cannot vibrate at that frequency. The 'demodulated' audio is the baseband difference frequency signal generated in the air itself via non-linear wave interaction (second derivative of the envelope squared). The demodulated audio is in the normal human hearing range (20 Hz to 20,000 Hz).
Why do electronic duty cycles produce loud high-pitched squeals (coil whine)?
Switched-Mode Power Supplies (SMPS) in laptop chargers, monitors, and graphics cards regulate voltage by rapidly pulsing switches at frequencies from 10 kHz to 500 kHz. The ratio of on-time to total period is the 'duty cycle' (D = Ton / T). Inside the power supply, inductor coils experience magnetostriction (physical expansion/contraction under magnetic flux), and ceramic MLCC capacitors experience piezoelectric deformation. When the duty cycle is high or modulates with CPU/GPU workloads, the component physically vibrates and radiates acoustic sound pressure.
Can electronic power supplies accidentally broadcast verbal language or speech?
Yes, as demonstrated in peer-reviewed side-channel security research (Choi et al., ACM CCS 2020 'TEMPEST Comeback'; Topcuoglu et al., NDSS 2024 'Periscope'). When an electronic device plays audio or runs software that changes power consumption with speech rhythms, the switching regulator's duty cycle dynamically mirrors the audio envelope. This creates an unintentional acoustic loudspeaker out of the power supply's inductors and capacitors, which can be measured acoustically or intercepted electromagnetically.
Why do repetitive high-pitched clicks or squeals sometimes sound like voices or words (Auditory Pareidolia)?
The human auditory cortex is an active predictive pattern engine. Under the 'Phonemic Restoration Effect' (Warren 1970) and auditory pareidolia, when the ear receives continuous, ambiguous, or repetitive high-frequency noise (such as an SMPS pulse train, fan bearing click, or water pipe resonance), the brain attempts to match these rhythmic transients against known phoneme models in memory. This can generate the vivid perception of spoken words or whispering even though the physical sound is purely random electronic switching noise.
How can I conclusively test if a high pitch is an external physical sound, electronic whine, or tinnitus?
Follow the 4-step diagnostic protocol: 1) Spatial test: move your head — physical sound drops with distance (1/r² law) or sharp beam edges, while tinnitus is completely constant. 2) Power test: switch off chargers and circuit breakers to see if electronic coil whine vanishes. 3) Spectrum test: record with a phone microphone and check the FFT waterfall for a sharp vertical spike (e.g. 15.6 kHz). 4) Pareidolia test: record the sound and play it at 0.5x speed — real speech retains formant structure, while pareidolic noise remains random clicks and hiss.
Primary Literature & Citations
- Westervelt, P. J. (1963). Parametric Acoustic Array. Journal of the Acoustical Society of America (JASA), 35(4):535–537. DOI: 10.1121/1.1918525
Finding: Derived the fundamental non-linear wave equation proving that collimated high-frequency ultrasonic beams self-demodulate in air to produce directional low-frequency sound. - Berktay, H. O. (1965). Possible exploitation of non-linear acoustics in underwater transmitting applications. Journal of Sound and Vibration, 2(4):435–461. DOI: 10.1016/0022-460X(65)90122-7
Finding: Established Berktay's Far-Field Solution showing the demodulated acoustic pressure is directly proportional to the second time-derivative of the envelope squared. - Yoneyama, M., Fujimoto, J., Kawamo, Y., Sasabe, S. (1983). The Audio Spotlight: An approach to voice and music source with large nonlinearity. Journal of the Acoustical Society of America (JASA), 73(5):1532–1536. DOI: 10.1121/1.389414
Finding: Pioneered the physical creation of parametric acoustic arrays operating in airborne environments for voice and musical reproduction. - Pompei, F. J. (2002). The Sound of Light: Large-Scale Directional Audio using Ultrasound. Audio Engineering Society (AES) Convention 112 / MIT PhD Thesis, Paper 5539. DOI: 10.1121/1.427357
Finding: Built the commercial Audio Spotlight parametric array, achieving 3-degree acoustic beamwidth in air using 40 kHz piezoelectric transducers. - Gan, W.-S., Yang, J., Kamakura, T. (2012). A review of parametric acoustic array in air. Applied Acoustics, 73(12):1211–1219. DOI: 10.1016/j.apacoust.2012.04.001
Finding: Comprehensive survey of non-linear parametric acoustic arrays in air, covering pre-processing modulation algorithms, Khokhlov-Zabolotskaya-Kuznetsov (KZK) numerical models, and directivity patterns. - Roy, N., Hassanieh, H., Roy Choudhury, R. (2017). BackDoor: Making Microphones Hear Inaudible Sounds. ACM MobiSys 2017 (Best Paper), pp. 2–14. DOI: 10.1145/3081333.3081366
Finding: Discovered that non-linear characteristics of front-end operational amplifiers in commodity MEMS microphones record inaudible ultrasonic tones as audible baseband signals. - Zhang, G. et al. (2017). DolphinAttack: Inaudible Voice Commands. ACM Conference on Computer and Communications Security (CCS 2017 - Best Paper), pp. 103–117. DOI: 10.1145/3133956.3134052
Finding: Demonstrated that ultrasonic AM carriers (25–40 kHz) demodulate into baseband speech commands due to non-linearities in smartphone MEMS microphones. - Genkin, D., Shamir, A., Tromer, E. (2014). RSA Key Extraction via Low-Bandwidth Acoustic Cryptanalysis. CRYPTO 2014 / Communications of the ACM 2016, 59(6):70–79. DOI: 10.1007/978-3-662-44371-2_25
Finding: Extracted 4096-bit RSA keys from laptop computers within one hour by recording high-frequency acoustic emanations (10–150 kHz) caused by SMPS CPU power load vibrations. - Genkin, D., Pattani, M., Schuster, R., Tromer, E. (2019). Synesthesia: Detecting on-screen content via acoustic emissions of monitors and power converters. IEEE Symposium on Security and Privacy (S&P 2019), pp. 83–96. DOI: 10.1109/SP.2019.00042
Finding: Reconstructed displayed text and web pages by measuring acoustic emissions from switching power converters and inductors inside LCD monitors and power bricks. - Choi, H. et al. (2020). TEMPEST Comeback: Realistic Audio Eavesdropping on Mixed-Signal SoCs. ACM Conference on Computer and Communications Security (CCS 2020), pp. 341–355. DOI: 10.1145/3372297.3417241
Finding: Showed that power supply switching regulator duty cycles unintentionally modulate with audio processing, creating both acoustic coil whine and electromagnetic leakage. - Topcuoglu, U. et al. (2024). Periscope: Eavesdropping on Black-Box Mobile Devices via Audio Amplifiers' EMR. Network and Distributed System Security (NDSS 2024), NDSS 2024. DOI: 10.14722/ndss.2024.24108
Finding: Demonstrated eavesdropping on audio output through electromagnetic and switching ripple emanations from Class-D audio amplifiers and power management ICs. - Murata Manufacturing Technical Division (2022). Causes and Countermeasures for Acoustic Noise in Ceramic Capacitors. Murata Technical Engineering Report, Tech Note 0020. Source Link
Finding: Detailed the inverse piezoelectric and electrostrictive effects in barium titanate (BaTiO3) MLCCs causing physical micro-strain and acoustic radiation via PCB resonance. - Texas Instruments Power Application Group (2021). Understanding and Mitigating Audible Acoustic Noise in Switch-Mode Power Supplies. TI Application Report, SNVA823. Source Link
Finding: Formulated inductor core magnetostrictive stress F ∝ (I_peak · D)^2 and MLCC piezoelectric singing in SMPS buck/boost converters; prescribed spread-spectrum PWM and acoustic potting solutions. - Warren, R. M. (1970). Perceptual Restoration of Missing Speech Sounds. Science, 167(3917):392–393. DOI: 10.1126/science.167.3917.392
Finding: Discovered the Phonemic Restoration Effect: when speech sounds are masked or replaced by repetitive high-frequency noise, the brain synthesizes the missing linguistic content automatically. - Fastl, H. & Zwicker, E. (2007). Psychoacoustics: Facts and Models (3rd Edition). Springer Series in Information Sciences, Vol. 22. DOI: 10.1007/978-3-540-68888-4
Finding: Comprehensive experimental reference on human temporal loudness summation (200 ms integration window), critical bandwidths (Bark scale), and non-linear cochlear response. - Florentine, M., Buus, S., Poulsen, T. (1996). Temporal integration of loudness as a function of level. Journal of the Acoustical Society of America (JASA), 99(3):1633–1644. DOI: 10.1121/1.415236
Finding: Showed that perceived loudness of repetitive high-frequency tone bursts grows non-linearly with pulse repetition rate due to neural energy summation in the central auditory system.
Related
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