Khokhlov Zabolotskaya Kuznetsov Equation Research Articles

Nonlinear wave propagation from a discrete annular array: Theory

Theoretical investigations of finite amplitude wave propagation from bounded beam sources has been considered by Lee and Hamilton [J. Acoust. Soc. Am. 97, 906–917 (1995)] as well as others. These researchers have restricted their analyses to a source geometry of either an unfocused circular piston or a weakly, spherically focused circular piston. When the peak pressure does not exceed approximately 150 MPa, the Khokhlov–Zabolotskaya–Kuznetsov (KZK) nonlinear parabolic wave equation yields numerical solutions in agreement with experiments. However, little research has been reported on finite amplitude wave propagation from an array of discrete sources. Time domain simulations, based on the KZK equation, will be discussed for an eight element discrete annular array immersed in fresh and sea water. The source signal is typically a short sine wave tone burst with a center frequency of 500 kHz. The transitions from linear acoustics to nonlinear acoustics will be discussed, and comparisons between a spherically focused circular source and a properly phased array will be presented. While some data from recent experiments may be shown, the scope of this talk will be restricted to the theoretical treatment and the numerical simulations for the discrete annular array. [Work supported by ONR.] by Edson

Investigation of a high intensity sound beam from a plane transducer. Experimental and theoretical results

Results of a systematic investigation of a high intensity sound beam from a plane circular transducer are presented for a large range of excitation levels. The ratio of the shock formation distance to the Rayleigh distance ranges from 1.7 to 0.06. A detailed description of the experimental setup is given. Problems encountered when calibrating highly intense and highly directive sound sources are discussed, and a novel method of calibration is proposed. Experimental results for the case of cw excitation are presented and compared with a simulation using the Bergen Code, thus allowing to assert the usefulness of the model (Khokhlov–Zabolotskaya–Kuznetsov equation) and of the numerical algorithm (spectral decomposition and finite difference method of the transformed beam equation) for high intensity sound beams. A new effect, nonlinear defocusing of the beam, is uncovered.

Generation of harmonics in a focused Gaussian sound field

A general solution of the lossy nonlinear KZK (Khokhlov–Zabolotskaya–Kuznetsov) equation is derived for any order harmonic of a Gaussian beam. The result shows that in a weakly focused Gaussian field all harmonics maintain a Gaussian profile in the axial direction at any distance. Moreover, the Gaussian coefficient of the nth harmonic is exactly n times as great as that of the fundamental; i.e., the beamwidth of the harmonics decreases as n−1/2.

Self-demodulation of amplitude- and frequency-modulated pulses in a thermoviscous fluid

The self-demodulation of pulsed sound beams in a thermoviscous fluid is investigated experimentally and theoretically. Experiments were performed in glycerin at megahertz frequencies with amplitude- and frequency-modulated pulses. The theory is based on the Khokhlov–Zabolotskaya–Kuznetsov (KZK) nonlinear parabolic wave equation. Numerical results were obtained from an algorithm that solves the KZK equation in the time domain [Y.-S. Lee and M. F. Hamilton, Ultrasonics International 91 Conference Proceedings (Butterworth–Heinemann, Oxford, 1991), pp. 177–180]. A quasilinear analytic solution, which describes the main features of the waveform at all axial locations, is developed in the limit of strong absorption. Theory and experiment are in good agreement throughout the near- and far fields.

Harmonic generation in finite amplitude sound beams from a rectangular aperture source

Theoretical analysis and some experiments are performed on nonlinearly generated harmonic components in bounded sound beams emitted from a rectangular aperture source. The Khokhlov–Zabolotskaya–Kuznetsov equation, which takes account of nonlinearity, dissipation, and diffraction effects in the beams, is numerically solved by means of the alternating direction implicit difference method. Using a planar source of size 24×44 cm, axial sound pressures and beam patterns of the first three harmonics are measured in air for initially sinusoidal ultrasounds of 25- and 30-kHz frequency, and are compared with the theory. They are in relatively good agreement. Deformation of the source face from circular to rectangular shape results in the unclear appearance of pressure peaks and dips with propagation. Within the framework of these studies, the harmonic pressure levels in the far field are almost the same as from a circular aperture source with equal face area and equal initial pressure, independent of the source levels.

Diffraction of light by a focused ultrasonic wave

The measurement of the acoustic pressure of a planar ultrasonic wave by light diffraction is well established. The ability to do similar measurements in the case of spherical waves has been doubted. However, we show that the range of validity can be extended to the focal region of a spherical concave piezoelectric transducer. Light is passed through the focal plane of a spherical concave transducer and is diffracted as a result of the variation in the index of refraction. The peak pressure can be calculated from the diffraction intensity by making the following simplification. We assume that in the focal plane the ultrasound can be approximated by a profiled planar wave, which in turn can be modeled by a wave of constant amplitude and effective width. The experimental results for moderate-pressure amplitudes in water compare favorably with the calculations using the Khokhlov–Zabolotskaya–Kuznetsov equation, which incorporates both nonlinearity and diffraction effects of the acoustic field.

A time domain computer algorithm for solving the KZK equation

A time domain computer algorithm for solving the KZK (Khokhlov-Zabolotskaya-Kuznetsov) nonlinear parabolic wave equation is being developed for pulsed, axisymmetric, finite-amplitude sound beams. The sound beams may be weakly focused. A coordinate transformation is introduced in the KZK equation [Hamilton et al., J. Acoust. Soc. Am. 78, 202–216 (1985)], and the resulting parabolic equation is integrated once analytically with respect to time. The diffraction and thermoviscous dissipation terms are integrated numerically with implicit backward finite-difference methods, and the Earnshaw solution [Pestorius and Blackstock, J. Acoust. Soc. Am. 53, 383 (1973)] is used to account for the nonlinearity. Weak shock theory may be incorporated in the algorithm. Numerical results for several limiting cases are compared with analytical solutions. Comparisons are also made with numerical results obtained with a frequency domain computer algorithm for solving the KZK equation [see, e.g., Berntsen, in Frontiers of Nonlinear Acoustics—12th ISNA, edited by Hamilton and Blackstock (Elsevier, London, 1990), pp. 191– 196]. [Work supported by the Office of Naval Research and the Texas Advanced Research Program.]

Nonlinear effects in focused sound beams

Nonlinear effects in focused sound beams are investigated with numerical solutions of the Khokhlov–Zabolotskaya–Kuznetsov (KZK) parabolic wave equation. The KZK equation accounts consistently for the combined effects of nonlinearity, diffraction, and absorption. To improve the efficiency of the calculations, a coordinate transformation is introduced that follows the convergent geometry of the focused beam. The transformation is similar to one introduced earlier to accommodate the divergent geometry of unfocused beams [Hamilton et al., J. Acoust. Soc. Am. 78, 202–216 (1985)]. A Fourier series expansion of the acoustic pressure is used to reduce the transformed KZK equation to a set of coupled parabolic equations that are solved using a finite difference method. Attention is devoted to radiation from circular sources with uniform amplitude distributions and linear focusing gains of order 50. The combined effects of nonlinearity, diffraction, and absorption in the focal region are clearly revealed in both the time and frequency domains. For the range of parameters considered, beam patterns in the focal region are found to be less dependent on source amplitude and absorption than are beam patterns in the farfield of unfocused sources.

Nonlinear effects in focused sound fields

Focused finite amplitude sound fields are investigated with numerical solutions of the Khokhlov‐Zabolotskaya‐Kuznetsov (KZK) equation. The numerical solution is based on the algorithm developed by Aanonsen et al. [J. Acoust. Soc. Am. 75, 749–768 (1984)], who used a Fourier series expansion of the sound pressure to reduce the KZK equation to a set of coupled parabolic equations. The basic algorithm has been modified by introducing a coordinate system that follows the convergent geometry of focused sound fields. In this way, more efficient numerical evaluation of the detailed field structure within the focal region is achieved. Arbitrary axisymmetric sources can be modeled. Here, circular sources having linear focusing gains of order 50 will be considered. The calculated time waveforms, propagation curves, and beam patterns illustrate clearly the combined effects of nonlinearity, diffraction, and absorption on finite amplitude sound that passes through a focal region. Among the new results are power curves ...