Subharmonic Frequency Research Articles

CMUT for ultrafast passive cavitation detection during ultrasound-induced blood–brain barrier disruption: proof of concept study

Objective.Cavitation dose monitoring plays a key role in ultrasound drug delivery to the brain. The use of capacitive micromachined ultrasonic transducer (CMUT) technology has a great potential for passive cavitation detection (PCD).Approach.Here, a circular (diameter 7 mm) CMUT centered at 5 MHz was designed to be inserted into a therapeutic transducer (1.5 MHz) used for ultrasound-induced blood-brain barrier (BBB) disruption on mice. CMUT-based real-time cavitation detection was performed during the ultrasound procedure (50μl intravenous injection of SonoVue microbubbles, frequency 1.5 MHz, PNP 480 kPa, duty Cycle 10%, PRF 10 Hz, duration 60 s). BBB disruption were confirmed by contrast-enhanced 7T-MRI.Main results.The CMUT device has a fractional bandwidth of 140%, almost twice a conventional piezocomposite PCD transducer. As expected, the CMUT device was able to detect the occurrence of harmonic, subharmonic and ultraharmonic frequencies as well as the increase of broadband signal indicating inertial cavitation in a wide frequency range (from 0.75 to 6 MHz). Signal-to-noise ratio was high enough (>40 dB) to perform ultrafast monitoring and follow the subtle intrapulse variations of frequency components at a rate of 10 kHz.Significance. This firstin vivoproof of concept demonstrates the interest of CMUT for PCD and encourages us to develop devices for PCD in larger animals by integrating an amplifier directly to the CMUT front-end to considerably increase the signal-to-noise ratio.

A numerical study on the relationship between local defect resonance and sub-harmonic resonance for closed cracks

Abstract The sub-harmonic resonance in closed cracks is caused by resonant properties of the system and the contact of crack surfaces. Local defect resonance is considered a potential resonant phenomenon near cracks. This study compares local resonance and sub-harmonic resonance in a surface-breaking crack using numerical simulations. The authors previously simulated nonlinear sub-harmonic resonance using the harmonic balance-boundary element method. The investigation of local defect resonance utilizes both the boundary element method and the Sakurai-Sugiura method. The numerical results indicate a strong relation between the vibration fields of sub-harmonic frequency components and local defect resonance.

THD improvement of piezoelectric MEMS speakers by dual cantilever units with well-designed resonant frequencies

In this study, a piezoelectric MEMS (Micro-Electro-Mechanical-System) speaker with thoughtfully selected resonant frequencies of dual cantilever units is designed and implemented. Based on the lead zirconium titanate (PZT) thin film with superior piezoelectric coefficient, the cantilever diaphragm is designed through the modal analysis. In the 1.5 mm by 1.5 mm cantilever array, a high-frequency actuation unit with resonance of 13.1 kHz and a low-frequency actuation unit with resonance of 6.35 kHz constitute the proposed piezoelectric MEMS speaker. The boost of sound pressure level (SPL) due to the resonant mode of low-frequency actuation unit reduces the total harmonic distortion (THD) around the subharmonic frequency of high-frequency actuation unit. Meanwhile, the asymmetric triangle diaphragm reduces the maximum stress in the structure when operating at the resonant frequency, thereby mitigating the THD spikes resulting from the nonlinear structural behavior. In pressure-field measurements, the proposed design can reach SPL ≥ 70.0 dB from 2.7 kHz to 15.0 kHz with only 0.179 Vrms input. At the same time, THD remains below 3.0 % from 3.2 kHz to 20 kHz. The outstanding performance under larger input signal and bias voltage is also verified. Actuated by 0.354 Vrms and 2 VDC bias, the maximum SPL achieves 111.9 dB and the SPL is higher than 80.0 dB from 3.3 kHz to 14.1 kHz. Furthermore, THD is lower than 3.0 % from 1.0 kHz to 8.9 kHz. A promising solution of piezoelectric MEMS speaker for in-ear applications is demonstrated in this study.

Decay-protected superconducting qubit with fast control enabled by integrated on-chip filters

Achieving fast gates and long coherence times for superconducting qubits presents challenges, typically requiring either a stronger coupling of the drive line or an excessively strong microwave signal to the qubit. To address this, we introduce on-chip filters of the qubit drive exhibiting a stopband at the qubit frequency, thus enabling long coherence times and strong coupling at the subharmonic frequency, facilitating fast single-qubit gates, and reduced thermal load. The filters exhibit an extrinsic relaxation time of a few seconds while enabling sub-10-ns gates with subharmonic control. Here we show up to 200-fold improvement in the measured relaxation time at the stopband. Furthermore, we implement subharmonic driving of Rabi oscillations with a π pulse duration of 12 ns. Our demonstration of on-chip filters and efficient subharmonic driving in a two-dimensional quantum processor paves the way for a scalable qubit architecture with reduced thermal load and noise from the control line.

On possibility of anomalous microwave power absorption and gyrotron frequency sub-harmonics emission in X2-mode ECRH experiments at the TCV tokamak

In this paper, we analyze theoretically the low-threshold parametric decay instability (PDI) that can be excited at the tokamak à configuration variable (TCV tokamak, Lausanne, Switzerland) during X2 electron cyclotron resonance heating experiments producing a two-dimensionally localized upper hybrid (UH) wave and a daughter extraordinary wave running from the decay layer outward to the plasma edge. The primary instability is then saturated due to the cascade of secondary decays into two-dimensionally localized UH and ion Bernstein waves. The level of plasma microwave emission in the frequency range substantially below half the pump wave frequency and of anomalous power losses of the pump are predicted.

Molding the acoustic cavity–analyzing the influence of toroidal vortex development on acoustic multi-bubble macrostructures under different ultrasonic horn tip diameters

The acoustic cavity structure typically experiences a sequence of transfigurations during its sinusoidal growth–collapse cycle. However, upon examining the cavity structure in aqueous bodies, it appears that the growth structure attained falls between two geometrical structures, namely, mushroom-like structure (MBS) and cone-like bubble structure (CBS), based on the actuated ultrasonic horn tip diameter. With the recurring observations of the emergence of proximal toroidal vortices, the present investigation conducts a numerical analysis exploring the vortex development under 3, 6, 13, 16, and 19 mm horn tips to establish a potential correlation between the vortex and the cavity structure. The study presents a computational fluid dynamic investigation to capture the nature of the vortex evolution, in terms of size and position, and its respective cavitation development. The first indicator of potential correlation was the equivalency of the vortex expansion–contraction frequency and the cavity's sub-harmonic frequency. It has been found that the cavity structure is molded into MBS by the presence of a symmetric locomotive vortex structure that extends up to 1.5 times the horn tip diameter. Meanwhile, CBS is observed to take shape in the presence of an eccentric locomotive vortex that attains a size within 0.2–0.6 times the horn tip diameter. The significance of the vortex size and position is also observed in the cavity's collapse, as the vortex appears to govern the ability of the cavity impinging jet to initialize the collapse phase.

Resonant response of a flexible semi-submersible floating structure: experimental analysis and second-order modelling

The dynamics and nonlinear wave forcing of a flexible floating structure are investigated experimentally and numerically. The floater was designed to match sub-harmonic rigid-body natural frequencies of typical floating wind turbine substructures, with the addition of a flexible bending mode. Experiments were carried out for three sea states with phase-shifted input signals to allow harmonic separation of the measured response. We find for the weakest sea states that sub-harmonic rigid-body motion is driven by even-harmonic difference frequency forcing, and by linear forcing for the strongest sea state. The flexible mode was tested in a soft, linearly forced layout, and a stiff layout, forced by second-, third- and fourth-harmonic frequency content, for increasing severity of the sea state. Further insight is gained by analysis of the amplitude scaling of the resonant response. A new simplified approach is proposed and compared with the recent method of Orszaghova et al. (J. Fluid Mech., vol. 929, 2021, A32). We find that resonant surge and pitch motions are dominated by even-harmonic potential-flow forcing and that odd-harmonic response is mainly potential-flow driven in surge and mainly drag driven in pitch. The measured responses are reproduced numerically with second-order forcing and quadratic drag loads, using a recent and computationally efficient calculation method, extended here for the heave, pitch and flexible motions. We are able to reproduce the response statistics and power spectra for the measurements, including the subharmonic pitch and heave modes and the flexible mode. Deeper analysis reveals that inaccuracies in the even-harmonic forcing content can be compensated by the odd-harmonic loads.

Observations of Parametric Subharmonic Instability of Diurnal Internal Tides in the Northwest Pacific

Abstract Based on yearlong observations from three moorings at 12°, 14°, and 16°N in the northwest Pacific, this study presents observational evidence for the occurrence and behavior of parametric subharmonic instability (PSI) of diurnal internal tides (ITs) both in the upper and abyssal ocean around the critical latitudes (O1 IT: 13.44°N; K1 IT: 14.52°N), which is relatively less explored in comparison with PSI of M2 ITs. At 14°N, near-inertial waves (NIWs) feature a “checkerboard” pattern with comparable upward- and downward-propagating components, while the diurnal ITs mainly feature a low-mode structure. The near-inertial kinetic energy at 14°N, correlated fairly well with the diurnal KE, is the largest among three moorings. The bicoherence analysis, and a causality analysis method newly introduced here, both show statistically significant phase locking between PSI triads at 14°N, while no significant signals emerge at 12° and 16°N. The estimated PSI energy transfer rate shows a net energy transfer from diurnal ITs to NIWs with an annual-mean value of 1.5 × 10−10 W kg−1. The highly sheared NIWs generated by PSI result in a 2–6 times larger probability of shear instability events at 14°N than 12° and 16°N. Through swinging the local effective inertial frequency close to either O1 or K1 subharmonic frequencies, the passages of anticyclonic and cyclonic eddies both result in elevated NIWs and shear instability events by enhancing PSI efficiency. Particularly, different from the general understanding that cyclonic eddies usually expel NIWs, enhanced NIWs and instability are observed within cyclonic eddies whose relative vorticity can modify PSI efficiency. Significance Statement Parametric subharmonic instability (PSI) effectively transfers energy from low-mode internal tides (ITs) to high-mode near-inertial waves (NIWs), triggering elevated mixing around critical latitudes. This study provides observational evidence for the occurrence of PSI of diurnal ITs in the northwest Pacific and its role in enhancing shear instability. Generally, anticyclonic eddies act to trap NIWs while cyclonic eddies tend to expel NIWs. Here we document elevated NIWs and shear instability within both anticyclonic and cyclonic eddies, which shift the local effective inertial frequency close to either O1 or K1 subharmonic frequencies, thereby enhancing PSI efficiency. Processes associated with PSI and the modulation of PSI efficiency by mesoscale eddies have significant implications for improving mixing parameterizations in ocean circulation and climate models.

Primary and secondary resonance phenomenon for two-layer liquid sloshing in a rectangular container under horizontal excitation

Laboratory experiments were conducted to study primary and secondary resonant sloshing in a laterally excited rectangular container containing two-layer stratified liquids. The findings revealed that primary resonance of the free surface or the separation surface occurs when the forcing frequency closely matches the corresponding natural frequency. Some intriguing phenomena, such as the sudden wave amplitude increase and the downward shift in resonant peaks, can be observed due to the soft-spring effect. Secondary resonance arises when superharmonic or sub-harmonic frequencies associated with liquid sloshing closely align with the natural frequencies of the system. For the free surface, the secondary resonances of the first five modes occur at forcing frequencies closely related to a third of the first mode, half of the second mode, a third of the third mode, a quarter of the fourth mode, and a fifth of the fifth mode natural frequency of the free surface, respectively. Furthermore, secondary resonance can also manifest under other forcing conditions, such as when the harmonic at a fractional multiple is near the second mode natural frequency of the free surface. Regarding the separation surface, it is worth noting that secondary resonance occurs due to the dominant contribution of fractional harmonics of the forcing frequency. To the best of the authors' knowledge, this is the first systematic investigation of primary and secondary resonance behaviors in a two-layer liquid system.

Subharmonic resonance of phospholipid coated ultrasound contrast agent microbubbles

Phospholipid encapsulated ultrasound contrast agents have proven to be a powerful addition in diagnostic imaging and show emerging applications in targeted therapy due to their resonant and nonlinear scattering. Microbubble response is affected by their intrinsic (e.g. bubble size, encapsulation physics) and extrinsic (e.g. boundaries) factors. One of the major intrinsic factors at play affecting microbubble vibration dynamics is the initial phospholipid packing of the lipid encapsulation. Here, we examine how the initial phospholipid packing affects the subharmonic response of either individual or a system of two closely-placed microbubbles. We employ a finite element model to investigate the change in subharmonic resonance under ‘small’ and ‘large’ radial excursions. For microbubbles ranging between 1.5 and 2.5 µm in diameter and in its elastic state (σ0 = 0.01 N/m), we demonstrate up to a 10 % shift towards lower frequencies in the peak subharmonic response as the radial excursion increases. However, for a bubble initially in its buckled state (σ0 = 0 N/m), we observe a maximum shift of 8 % towards higher frequencies as the radial excursion increases over the same range of bubble sizes – the opposite trend. We studied the same scenario for a system of two individual microbubbles for which we saw similar results. For microbubbles that are initially in their elastic state, in both cases of a) two identically sized bubbles and b) a bubble in proximity to a smaller bubble, we observed a 6 % and 9 % shift towards lower frequencies respectively; while in the case of a neighboring larger bubble no change in subharmonic resonance frequency was observed. Microbubbles that are initially in a buckled state exert no change, 5 % and 19 % shift towards higher frequencies, in two-bubble systems consisting of a) same-size, b) smaller, and c) larger neighboring bubble respectively. Furthermore, we examined the effect of two adjacent bubbles with non-equal initial phospholipid states. The results presented here have important implications in ultrasound contrast agent applications.

Harmonic current suppression method of virtual DC motor based on fuzzy sliding mode

Abstract In the operation of power grid, the low frequency subharmonics in power lines are easy to cause harmonic current problems. A harmonic current suppression method is constructed based on fuzzy sliding mode and virtual DC motor (VDC). First, the adaptive linear network is introduced to compare and analyze the output value and the actual value in the power line, and the possible harmonic current is obtained, which is detected and discretized. After that, this study adopts Fuzzy sliding mode control (FSMC) strategy for VDC, uses FSMC to participate in on-off control of VDC, and then realizes the suppression of harmonic current in VDC ports. In addition, the model also uses the circuit superposition theorem and the amount of harmonic data to be compensated by FSMC. The whole method uses VDC control technology to ensure the voltage stability of DC bus and load side when the load changes. In this physical experiment, harmonic current generated by load disturbance is eliminated by reducing DC harmonic voltage, so as to achieve the suppression of DC port harmonic current. Experimental results show that the third and fifth harmonics in power lines can be eliminated by this method to a large extent. Therefore, the proposed method can accurately detect and eliminate the harmonic current caused by sudden load change, and then realize the suppression of VDC harmonic current.

Optical frequency comb generation using cascaded injection of semiconductor lasers

We study optical frequency comb (OFC) generation using cascaded injection of semiconductor lasers in this work. The OFC generation system is operated in two cascaded optical injection stages. When a master laser optically injects into the first stage with proper injection power and frequency, period-one (P1) dynamics are invoked in an optically injected semiconductor laser of the first stage. Another semiconductor laser in the second stage is then optically injected by the P1 dynamics. With proper injection power adjusted in the second stage, the P1 dynamics are regenerated, and the semiconductor laser relaxation oscillations (ROs) become undamped so that subharmonic oscillations appear. Because a subharmonic oscillation frequency is half of an oscillation frequency of the P1 dynamics, extra optical frequency components appear in the middle of the adjacent optical frequency components of the P1 dynamics, thus signaling OFC generation. The OFC signals exhibit at least 15 comb lines, resulting in a bandwidth greater than 140 GHz. Microwave comb signals are obtained after photodetection, although the microwave linewidth is on the order of a few megahertz because of the semiconductor laser noise. Thus, we propose a cascaded injection-locking scheme to stabilize the P1 dynamics and OFC signals. We have demonstrated pure microwave generations with a linewidth of less than 3 Hz and low phase noise.

Flow-induced vibrations of ten tandem cylinders at low Reynolds number

Flow-induced vibrations of ten cylinders in tandem arrangement are numerically investigated by using the immersed boundary method with a low Reynolds number (Re = 100). Seven spacing ratios L/D (where L = center–center spacing between tandem cylinders and D = diameter of the cylinder) are selected from 1.1 to 4.0, and the reduced velocity Ur ranges from 2.0 to 13.5 with an increment of 0.5. Small (L/D < 2.0) and large (L/D > 2.0) spacing ranges are identified, both including two types of responses: wake-induced vibration (WIV; Ur = 2.0–9.0∼10.5 for a small L/D and Ur = 2.0–6.0∼6.5 for a large L/D) and wake-induced galloping (WIG; Ur > 9.0∼10.5 for a small L/D and Ur > 6.0∼6.5 for a large L/D). The largest vibration amplitude of each cylinder is obtained in the WIG region for the small L/D condition. The presence of downstream cylinders suppresses the vortex shedding of upstream cylinders and thus postpones the vibration of upstream cylinders at a small Ur, whereas the downstream cylinder enhances the vibration at a larger Ur due to the wake interference. For a small L/D, three flow regimes with the extended-body, reattachment, and co-shedding patterns are successively presented as Ur increases. For a large L/D, four types of flow regimes, namely, EB-2S (the extended-body with “2S” pattern), RT-2S (the reattachment with “2S” pattern), TR-2S (two-row vortex street with “2S” pattern), and CS-VS (co-shedding with variation shedding), are classified. Two new vortex shedding patterns, “2G (two counter-rotating vortices shed from each side per vibration cycle)” and “2C (two co-rotating vortices shed from each side per vibration cycle),” have been identified. In the WIV region, there is only one dominant vibration frequency for upstream cylinders (C1–C7), while a sub-harmonic frequency emerges and dominates C8–C10 when L/D is large. The fluctuating lift force spectra show a broad-band frequency distribution due to the irregular positions of the vortex generation and merging, and the dominant frequency in the WIG region decreases consecutively from C1 to C10.

Flow structure and impinging interactions of two confined turbulent converging jets in crossflow

In this work, time-resolved particle image velocimetry (TR-PIV) measurements are carried out to study the flow structure and impinging interactions of two confined turbulent converging jets in crossflow, with the upstream and downstream jets inclined at 20° and 170° with respect to the oncoming crossflow direction. Experiments are conducted for fixed dimensionless nozzle separation distance of s=L/D=3, fixed dimensionless nozzle-to-target surface distance of H/D=6.73, fixed duct Reynolds number of Re=3500 and two jets' Reynolds numbers of Rej=3000 and 6000 that correspond to jets'-to-crossflow velocity ratios of rj=3.45 and 6.91, respectively. Flow visualization images showing ensemble-averaged and instantaneous flow distributions and turbulent characteristics including Reynolds stresses and turbulence intensities for equal and non-equal discharge velocities of the jets are presented on several orthogonal planes of the test section. The swirling strength criterion has been employed to identify coherent structures, and cross spectral analysis of the fluctuating velocity signals detecting subharmonic and harmonic sideband frequencies identify the merging and breakup interactions of the evolving coherent structures. The snapshot proper orthogonal decomposition technique (POD) has been applied to extract the dominant vortical structures and to quantify their relative and cumulative energy contributions to the total kinetic energy fluctuation. The spatial structure of the main four POD modes and the temporal projections of these POD coefficients have been explored to understand the spatio-temporal characteristics of the dominant turbulent fluctuations and their spectral characteristics. Our results show the effect of the velocity ratios on the jets' trajectories, penetration, spreading characteristics, crossflow entrainment through the inner and outer shear layers, transverse velocity decay, size of the converging region and location of the combining point. In addition, the dimensionless shedding frequencies of the oscillating combined jet, the developing wall jets and wake vortices have been obtained. Furthermore, it has been shown that for all values of the jets' velocity ratios employed in this study, a recirculation region does not develop in the converging region.

Physical Characterization to Improve Scalability and Potential of Anesthetic-Loaded Nanodroplets.

Drug-loaded perfluorocarbon nanodroplets (NDs) can be activated non-invasively by focused ultrasound (FUS) and allow for precise drug-delivery. Anesthetic-loaded NDs and transcranial FUS have previously achieved targeted neuromodulation. To assess the clinical potential of anesthetic-loaded NDs, in depth physical characterization and investigation of storage strategies and triggered-activation is necessary. Pentobarbital-loaded decafluorobutane nanodroplets (PBNDs) with a Definity-derived lipid shell (237 nm; 4.08 × 109 particles/mL) were fabricated and assessed. Change in droplet stability, concentration, and drug-release efficacy were tested for PBNDs frozen at -80 °C over 4 weeks. PBND diameter and the polydispersity index of thawed droplets remained consistent up to 14 days frozen. Cryo-TEM images revealed NDs begin to lose circularity at 7 days, and by 14 days, perfluorocarbon dissolution and lipid fragmentation occurred. The level of acoustic response and drug release decreases through prolonged storage. PBNDs showed no hemolytic activity at clinically relevant concentrations and conditions. At increasing sonication pressures, liquid PBNDs vaporized into gas microbubbles, and acoustic activity at the second harmonic frequency (2 f0) peaked at lower pressures than the subharmonic frequency (1/2 f0). Definity-based PBNDs have been thoroughly characterized, cryo-TEM has been shown to be suitable to image the internal structure of volatile NDs, and PBNDs can be reliably stored at -80 °C for future use up to 7 days without significant degradation, loss of acoustic response, or reduction in ultrasound-triggered drug release.

Numerical simulations of internal gravity wave resonant triads

We investigate the triadic resonance interactions using numerical simulations of freely propagating internal waves at near-inertial frequency. We force multiple combinations of normal modes at primary frequency ω0 from the left boundary of the computational domain and study the spatiotemporal evolution of the superharmonic waves. Both the resonant and off-resonant simulations show distinct peaks of energy in the higher harmonics at 2ω0,3ω0, and 4ω0 close to the forcing region. The spatial evolution of modal amplitudes shows that higher normal modes of primary wave decay faster and the higher normal modes of superharmonic waves grow faster. Away from the forcing region, these distinct peaks in the resonant simulations weaken and redistribute the energy into continuous spectra, whereas in the off-resonant simulations, these distinct peaks at superharmonic frequencies sustain throughout the domain. The off-resonant simulations also show significant energy in sub-harmonic frequencies, and we see a buildup of energy in the near-inertial frequency regime far from the forcing region. The frequency and the wavenumber spectra for resonant simulations reveal a −2 power law consistent with the Garrett–Munk spectrum (E(ω,m)∝ω−2m−2).

Flow control of a plunging cylinder based on resolvent analysis

We design an open-loop active flow control for separated flows around a plunging circular cylinder based on resolvent analysis. The cylinder is plunging at a Strouhal number of 0.36 and a Reynolds number of 500. A linear time-periodic system for control is obtained by linearizing the non-inertial incompressible vorticity equation in the cylinder-fixed frame about a time-averaged base flow. Using the Lyapunouv–Floquet transformation, the linear time-periodic system is transformed into a similar linear time-invariant system, whose resolvent is analysed to obtain an optimal actuating Strouhal number of 0.1464 for the transformed linear system. Simulations show that the active control with tangential actuations is capable of reducing the lift fluctuation by up to 25.7 % when the flow is actuated near the predicted harmonic and subharmonic frequencies.

Dynamics near the edge-of-chaos in a fiber laser model

We consider the dynamics in a fiber laser model for pump modulation frequencies much smaller than that of the relaxation oscillations of the model. We find that for low enough subharmonic pump frequencies and those near the Edge-of-Chaos, long-term correlations arise in the discrete time series of the laser intensity. Moreover, within this low frequency range we demonstrate that both frequency locking and, imperfect phase synchronization occur. These are determined through the notion of mean-frequency. We conjecture that these dynamical effects may be experimentally observed in suitable time series.

Biomechanical characteristics in the carotid artery: Noninvasive assessment using subharmonic emissions from microbubbles.

Stroke is closely related to carotid atherosclerotic plaques, which tend to occur in specific parts of the arteries, especially at the bifurcations, and are considered to be caused by biomechanical factors. Quantitative analysis of hemodynamic stress characteristics of the carotid sinus in vivo provides a mechanical basis for the development of atherosclerotic plaque in the carotid sinus. Previous studies found that ultrasound (US) contrast agent microbubbles would vibrate nonlinearly under the excitation of sound pressure, generating subharmonics (transmission fundamental frequency, i.e., f0 and subharmonic frequency at f0 /2), which have the highest sensitivity to pressure changes and exhibit an inverse linear relationship with environmental pressure. This study employed subharmonic aided pressure estimation (SHAPE) technology to reflect carotid artery hydrodynamic characteristics in the carotid lumen. From May 2021 to December 2021, this prospective study reviewed a total of 26 normal carotid arteries of 13 participants, all of whom received bilateral carotid artery routine US and SHAPE US examinations. During this study, the lumen of the bilateral distal segment of the common carotid artery (Distal-CCA), carotid artery bifurcation (CAB), and carotid bulb (CB) were scanned section by section from bottom to top in longitudinal and transverse sections. Subsequently, the subharmonic amplitudes in the lumen of normal carotid arteries were collected and analyzed. This study found that the amplitude of subharmonic amplitude in the carotid was distributed unevenly, with the amplitudes of subharmonic at the CAB being higher. Specifically, the subharmonic gradient of the carotid artery bifurcation apex plane was maximum (9.72±4.31dB), while the average subharmonic amplitude of the outer lateral layer of the carotid artery was higher (-56.40±6.31dB) (p<0.001). The SHAPE technique is capable of indirectly reflecting the pressure changes of vascular system tissues, which may provide a new monitoring method for evaluating mechanical characteristics obviating invasion.

Statistical analysis of observed Faraday wave patterns

We present experimental observations of Faraday wave patterns when a liquid in a square recipient is submitted to vertical vibrations with forcing frequencies and amplitudes in the intervals 10 ≤ F ≤ 80 Hz and 0.1 ≤ A ≤ 3 mm, respectively. A sweep of the forcing parameter space is performed to investigate the surface patterns that form for two different sizes of the recipient. The waveforms are obtained by means of a Pearson correlation analysis of a sequence of liquid surface images for each experiment. The Pearson correlation coefficients are then fast Fourier transformed to obtain the power spectra. For 10 ≤ F ≤ 15 Hz, tessellation patterns consisting of ordered arrays of rhomboid-shaped tiles are observed; whereas, for 20 ≤ F ≤ 25, the tessellation patterns become irregular over the entire surface where square, rhomboidal, andhexagonal tilings coexist. At 30 ≤ F ≤ 45 Hz, the patterns correspond to a mixed state where ordered and irregular quadrilateral tessellation patterns coexist. At frequencies between 50 and 80 Hz, the tessellated patterns that form consist of irregularly distributed square-shaped tiles. In all cases, the excited waves are periodic with a dominant subharmonic frequency f = F/2. However, for given forcing parameters, the waveforms and their amplitudes differ markedly when the size of the recipient is changed.