Mark F. Hamilton

TIME-DOMAIN MODELING OF PULSED FINITE-AMPLITUDE SOUND BEAMS

A time‐domain algorithm that solves the Khokhlov–Zabolotskaya–Kuznetsov (KZK) nonlinear parabolic wave equation is described. The algorithm models the propagation of pulsed finite amplitude sound beams radiated from axisymmetric sources in homogeneous, thermoviscous fluids. Numerical results are presented for waveform distortion and shock formation in directive beams radiated by pulsed circular pistons. Waveforms are calculated through the shock region and out to far‐field locations where they are dominated by the nonlinearly generated low‐frequency components. Effects of pulse duration, frequency modulation, and noise are examined. Methods for including relaxation and focusing are described.

Nonlinear Wave Processes in Acoustics

Preface 1. Nonlinearity, dissipation and dispersion in acoustics 2. Simple waves and shocks in acoustics 3. Nonlinear geometrical acoustics 4. Nonlinear sound beams 5. Sound-sound interaction (nondispersive medium) 6. Nonlinear acoustic waves in dispersive media 7. Self-action and stimulated scattering of sound Conclusion Subject index.

Finite-amplitude waves in isotropic elastic plates

The propagation of finite-amplitude waves in a homogeneous, isotropic, stress-free elastic plate is investigated theoretically. Geometric and weak material non-linearities are included, and perturbation is used to obtain solutions of the non-linear equations of motion for harmonic generation in the waveguide. Solutions for the second-harmonic, sum, and difference-frequency components are obtained via modal decomposition. Ordinary differential equations for the modal amplitudes in the expansion of the second-order solution are obtained using a reciprocity relation. There are no restrictions on the modes or frequencies of the primary waves. Two conditions for internal resonance are quantified: phase matching, and transfer of power from the primary to the secondary wave.

Time‐domain modeling of finite‐amplitude sound in relaxing fluids

A time‐domain computer algorithm that solves an augmented Burgers equation is described. The algorithm is a modification of the time‐domain code developed by Lee and Hamilton [J. Acoust. Soc. Am. 97, 906–917 (1995)] for pulsed finite‐amplitude sound beams in homogeneous, thermoviscous fluids. In the present paper, effects of nonlinearity, absorption and dispersion (both thermoviscous and relaxational), geometrical spreading, and inhomogeneity of the medium are taken into account. The novel feature of the code is that effects of absorption and dispersion due to multiple relaxation phenomena are included with calculations performed exclusively in the time domain. Numerical results are compared with an analytic solution for a plane step shock in a monorelaxing fluid, and with frequency‐domain calculations for a plane harmonic wave in a thermoviscous, monorelaxing fluid. The algorithm is also used to solve an augmented KZK equation that accounts for nonlinearity, thermoviscous absorption, relaxation, and diffraction in directive sound beams. Calculations are presented which demonstrate the effect of relaxation on the propagation of a pulsed, diffracting, finite‐amplitude sound beam.

Nonlinear distortion of short pulses radiated by plane and focused circular pistons.

Detailed measurements of finite-amplitude pulses radiated by plane and focused circular pistons in water are presented. Comparisons of time waveforms and frequency spectra, both on and off axis, are made with numerical calculations based on the nonlinear parabolic wave equation. Emphasis is on nonlinear distortion of amplitude- and frequency-modulated tone bursts. Use of short pulses enabled resolution of the direct and diffracted waves prior to their coalescence and subsequent shock formation along the axis of the source. Because of its relevance to investigations of cavitation inception, attention is devoted to variation of the peak positive (p+) and negative (p-) pressures along the axis of a focused source. It is shown that with increasing source amplitude, the maximum of each shifts away from the focal plane, toward the source. This effect is more pronounced for p- than for p+.

Acoustic streaming generated by standing waves in two-dimensional channels of arbitrary width

An analytic solution is derived for acoustic streaming generated by a standing wave in a viscous fluid that occupies a two-dimensional channel of arbitrary width. The main restriction is that the boundary layer thickness is a small fraction of the acoustic wavelength. Both the outer, Rayleigh streaming vortices and the inner, boundary layer vortices are accurately described. For wide channels and outside the boundary layer, the solution is in agreement with results obtained by others for Rayleigh streaming. As channel width is reduced, the inner vortices increase in size relative to the Rayleigh vortices. For channel widths less than about 10 times the boundary layer thickness, the Rayleigh vortices disappear and only the inner vortices exist. The obtained solution is compared with those derived by Rayleigh, Westervelt, Nyborg, and Zarembo.

50 kHz capacitive micromachined ultrasonic transducers for generation of highly directional sound with parametric arrays

In this study, we examine the use of capacitive micromachined ultrasonic transducers (CMUTs) with vacuum- sealed cavities for transmitting directional sound with parametric arrays. We used finite element modeling to design CMUTs with 40-mum- and 60-mum-thick membranes to have resonance frequencies of 46 kHz and 54 kHz, respectively. The wafer bonding approach used to fabricate the CMUTs provides good control over device properties and the capability to fabricate CMUTs with large diameter membranes and deep cavities. Each CMUT is 8 cm in diameter and consists of 284 circular membranes. Each membrane is 4 mm in diameter. Characterization of the fabricated CMUTs shows they have center frequencies of 46 kHz and 55 kHz and 3 dB bandwidths of 1.9 kHz and 5.3 kHz for the 40-mum- and 60-mum-thick membrane devices, respectively. With dc bias voltages of 380 V and 350 V and an ac excitation of 200 V peak-to-peak, the CMUTs generate average sound pressure levels, normalized to the devices surface, of 135 dB and 129 dB (re 20 muPa), respectively. When used to generate 5 kHz sound with a parametric array, we measured sound at 3 m with a 6 dB beamwidth of 8.7deg and a sound pressure level of 58 dB. To understand how detector nonlinearity (e.g., the nonlinearity of the microphone used to make the sound level measurements) affects the measured sound pressure level, we made measurements with and without an acoustic low-pass filter placed in front of the microphone; the measured sound levels agree with numerical simulations of the pressure field. The results presented in this paper demonstrate that large-area CMUTs, which produce high-intensity ultrasound, can be fabricated for transmitting directional sound with parametric arrays.

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.

Separation of compressibility and shear deformation in the elastic energy density (L)

A formulation of the elastic energy density for an isotropic medium is presented that permits separation of effects due to compressibility and shear deformation. The motivation is to obtain an expansion of the energy density for soft elastic media in which the elastic constants accounting for shear effects are of comparable order. The expansion is carried out to fourth order to ensure that nonlinear effects in shear waves are taken into account. The result is E≃E0(ρ)+μI2+13AI3+DI22, where ρ is density, I2 and I3 are the second- and third-order Lagrangian strain invariants used by Landau and Lifshitz, μ is the shear modulus, A is one of the third-order elastic constants introduced by Landau and Lifshitz, and D is a new fourth-order elastic constant. For processes involving mainly compressibility E≃E0(ρ), and for processes involving mainly shear deformation E≃μI2+13AI3+DI22.

Parametric array in air: Distortion reduction by preprocessing

In a parametric array, highly directional low‐frequency sound is generated by the self‐demodulation of an intense, amplitude‐modulated high‐frequency sound beam as a result of nonlinear propagation effects. The term ‘‘audio spotlight’’ was introduced by Yoneyama et al. [J. Acoust. Soc. Am. 73, 1532 (1983)] for a parametric array in air used to generate directional audio frequency sound with an ultrasonic primary beam. Berktay’s far‐field solution [Berktay, J. Sound Vib. 2, 435 (1965)] predicts a demodulated secondary waveform along the axis of the beam that is proportional to the second time derivative of the square of the modulation envelope. The secondary wave is therefore generated with high levels of harmonic distortion, even at moderate modulation indexes [Blackstock, J. Acoust. Soc. Am. 102, 3106(A) (1997)]. Integrating the modulation signal twice and taking the square root removes this distortion; however, the resulting reduction in distortion due to taking the square root is severely limited by th...