Maria M. Karzova

Mechanisms for saturation of nonlinear pulsed and periodic signals in focused acoustic beams

Acoustic fields of powerful ultrasound sources with Gaussian spatial apodization and initial excitation in the form of a periodic wave or single pulse are examined based on the numerical solution of the Khokhlov-Zabolotskaya-Kuznetsov equation. The influence of nonlinear effects on the spatial structure of focused beams, as well as on the limiting values of the acoustic field parameters is compared. It is demonstrated that pressure saturation in periodic fields is mainly due to the effect of nonlinear absorption at a shock front, while in pulsed fields is due to the effect of nonlinear refraction. The limiting attainable values for the peak positive pressure in periodic fields turned out to be higher than the analogous values in pulsed acoustic fields. The total energy in a beam of periodic waves decreases with the distance from the source faster than in the case of a pulsed field, but it becomes concentrated within much smaller spatial region in the vicinity of the focus. These special features of nonlinear effect manifestation provide an opportunity to use pulsed beams for more efficient delivery of wave energy to the focus and to use periodic beams for attaining higher values of pressure in the focal region.

Acoustic field characterization of the Duolith: Measurements and modeling of a clinical shock wave therapy device

Extracorporeal shock wave therapy (ESWT) uses acoustic pulses to treat certain musculoskeletal disorders. In this paper the acoustic field of a clinical portable ESWT device (Duolith SD1) was characterized. Field mapping was performed in water for two different standoffs of the electromagnetic head (15 or 30 mm) using a fiber optic probe hydrophone. Peak positive pressures at the focus ranged from 2 to 45 MPa, while peak negative pressures ranged from -2 to -11 MPa. Pulse rise times ranged from 8 to 500 ns; shock formation did not occur for any machine settings. The maximum standard deviation in peak pressure at the focus was 1.2%, indicating that the Duolith SD1 generates stable pulses. The results compare qualitatively, but not quantitatively with manufacturer specifications. Simulations were carried out for the short standoff by matching a Khokhlov-Zabolotskaya-Kuznetzov equation to the measured field at a plane near the source, and then propagating the wave outward. The results of modeling agree well with experimental data. The model was used to analyze the spatial structure of the peak pressures. Predictions from the model suggest that a true shock wave could be obtained in water if the initial pressure output of the device were doubled.

Mach stem formation in reflection and focusing of weak shock acoustic pulses

The aim of this study is to show the evidence of Mach stem formation for very weak shock waves with acoustic Mach numbers on the order of 10(-3) to 10(-2). Two representative cases are considered: reflection of shock pulses from a rigid surface and focusing of nonlinear acoustic beams. Reflection experiments are performed in air using spark-generated shock pulses. Shock fronts are visualized using a schlieren system. Both regular and irregular types of reflection are observed. Numerical simulations are performed to demonstrate the Mach stem formation in the focal region of periodic and pulsed nonlinear beams in water.

Mach-Zehnder interferometry method for acoustic shock wave measurements in air and broadband calibration of microphones

A Mach-Zehnder interferometer is used to measure spherically diverging N-waves in homogeneous air. An electrical spark source is used to generate high-amplitude (1800 Pa at 15 cm from the source) and short duration (50 μs) N-waves. Pressure waveforms are reconstructed from optical phase signals using an Abel-type inversion. It is shown that the interferometric method allows one to reach 0.4 μs of time resolution, which is 6 times better than the time resolution of a 1/8-in. condenser microphone (2.5 μs). Numerical modeling is used to validate the waveform reconstruction method. The waveform reconstruction method provides an error of less than 2% with respect to amplitude in the given experimental conditions. Optical measurement is used as a reference to calibrate a 1/8-in. condenser microphone. The frequency response function of the microphone is obtained by comparing the spectra of the waveforms resulting from optical and acoustical measurements. The optically measured pressure waveforms filtered with the microphone frequency response are in good agreement with the microphone output voltage. Therefore, an optical measurement method based on the Mach-Zehnder interferometer is a reliable tool to accurately characterize evolution of weak shock waves in air and to calibrate broadband acoustical microphones.

Characterization of spark-generated N-waves in air using an optical schlieren method

Accurate measurement of high-amplitude, broadband shock pulses in air is an important part of laboratory-scale experiments in atmospheric acoustics. Although various methods have been developed, specific drawbacks still exist and need to be addressed. Here, a schlieren optical method was used to reconstruct the pressure signatures of nonlinear spherically diverging short acoustic pulses generated using an electric spark source (2.5 kPa, 33 μs at 10 cm from the source) in homogeneous air. A high-speed camera was used to capture light rays deflected by refractive index inhomogeneities, caused by the acoustic wave. Pressure waveforms were reconstructed from the light intensity patterns in the recorded images using an Abel-type inversion method. Absolute pressure levels were determined by analyzing at different propagation distances the duration of the compression phase of pulses, which changed due to nonlinear propagation effects. Numerical modeling base on the generalized Burgers equation was used to evaluate the smearing of the waveform caused by finite exposure time of the high-speed camera and corresponding limitations in resolution of the schlieren technique. The proposed method allows the study of the evolution of spark-generated shock waves in air starting from the very short distances from the spark, 30 mm, up to 600 mm.

Shock formation and nonlinear saturation effects in the ultrasound field of a diagnostic curvilinear probe

Newer imaging and therapeutic ultrasound technologies may benefit from in situ pressure levels higher than conventional diagnostic ultrasound. One example is the recently developed use of ultrasonic radiation force to move kidney stones and residual fragments out of the urinary collecting system. A commercial diagnostic 2.3 MHz C5-2 array probe has been used to deliver the acoustic pushing pulses. The probe is a curvilinear array comprising 128 elements equally spaced along a convex cylindrical surface. The effectiveness of the treatment can be increased by using higher transducer output to provide a stronger pushing force; however nonlinear acoustic saturation can be a limiting factor. In this work nonlinear propagation effects were analyzed for the C5-2 transducer using a combined measurement and modeling approach. Simulations were based on the three-dimensional Westervelt equation with the boundary condition set to match low power measurements of the acoustic pressure field. Nonlinear focal waveforms simulated for different numbers of operating elements of the array at several output power levels were compared to fiber-optic hydrophone measurements and were found to be in good agreement. It was shown that saturation effects do limit the acoustic pressure in the focal region of a diagnostic imaging probe.

Irregular reflection of weak acoustic shock pulses on rigid boundaries : Schlieren experiments and direct numerical simulation based on a Navier-Stokes solver

The local interactions occurring between incident and reflected shock waves in the vicinity of rigid surfaces are investigated. Both regular and irregular — also called von Neumann — regimes of reflection are studied, via experimental and numerical simulations. Shock waves are produced experimentally with a 20 kV electrical spark source which allows the generation of spherically diverging acoustic shocks. The behaviour of the resulting weak acoustic shocks near rigid boundaries is visualized with a Schlieren optical technique which allows the spatial structure of the shocks to be studied. In particular, the evolution of the Mach stem forming above a flat surface is examined, and its height is observed to be directly linked to the angle of incidence and the pressure amplitude of the incident shock. The propagation of an acoustic shock between two parallel rigid boundaries is also studied. It is shown that the strong interactions between the Mach stems emerging from the two boundaries can lead to a drastic ...

Application of a Mach-Zehnder Interferometer to the Observation of Mach Stem Formation When a Shock Wave is Reflected from a Rigid Surface

The reflection of a nonlinear spark-generated N-wave in air from a flat rigid surface is investigated experimentally. The pressure profile of the N-wave is reconstructed from optical measurements performed using a Mach–Zehnder interferometer. An irregular reflection is observed in the experiment; the evolution of a Mach stem is investigated, and the trajectory of a triple point during the N-wave propagation along the surface is investigated.

Calibration of high frequency MEMS microphones and pressure sensors in the range 10 kHz–1 MHz

In the context of both nonlinear acoustics, and downscaled acoustic or aero-acoustic experiments, the characterization of the high frequency response of microphones and pressure sensors remains a critical challenge. In the case of the design of new MEMS microphones and shock pressure sensors with response in the frequency range of 10 kHz–1 MHz, this question was addressed by the definition of a new calibration method based on a spark source that generates spherical weak shock acoustic pulse. Waves are short duration non-symmetric N-waves with duration of about 40 microseconds and front shock rise time of the order of 0.1 microsecond. Taking advantage of recent works on the characterization of such pressure waves using an optical interferometer, and considering non linear propagation of weak shockwaves, we were able to estimate the incident pressure wave in the range of 10 kHz–1MHz. Hence, from the output voltage of the microphones, the frequency response was obtained in this range. The method applies what...

Development of a freely available simulator with graphical interface for modeling nonlinear focused ultrasound fields with shocks

Measurement-based modeling is gaining acceptance as a standard tool for characterizing the nonlinear fields of existing therapeutic ultrasound devices and designing new ones. Here, a freely available simulation tool is presented for modeling axially symmetric, strongly nonlinear HIFU beams with shocks in a layered propagation medium such as water and different types of tissue. Two nonlinear wave equations are included in the simulator: the KZK equation generalized to include an equivalent source boundary condition for strongly focused beams and the Westervelt equation in a nonlinear wide-angle parabolic representation. Both equations are solved in the frequency domain and permit definition of the HIFU transducer as an annular array. The geometrical parameters and power output of the array, electronic focus steering along the beam axis, and acoustic properties of the layered propagation medium can be configured via graphical interface. Visualization and output of various acoustic field parameters such as peak positive and negative pressures, shock amplitude, intensity, heat deposition rate, and total power of a beam are also provided. The simulator can be used for transducers without ideal symmetry through definition of an equivalent radially symmetric source. Widespread availability of such simulation tools will help advance standardized utilization of measurement-based modeling and facilitate the adoption of such approaches for HIFU treatment planning. [Work supported by NIH R01EB7643, R01EB025187, and RSF №14-12-00974.]Measurement-based modeling is gaining acceptance as a standard tool for characterizing the nonlinear fields of existing therapeutic ultrasound devices and designing new ones. Here, a freely available simulation tool is presented for modeling axially symmetric, strongly nonlinear HIFU beams with shocks in a layered propagation medium such as water and different types of tissue. Two nonlinear wave equations are included in the simulator: the KZK equation generalized to include an equivalent source boundary condition for strongly focused beams and the Westervelt equation in a nonlinear wide-angle parabolic representation. Both equations are solved in the frequency domain and permit definition of the HIFU transducer as an annular array. The geometrical parameters and power output of the array, electronic focus steering along the beam axis, and acoustic properties of the layered propagation medium can be configured via graphical interface. Visualization and output of various acoustic field parameters such as p...