Nonlinear acoustic wave equations with fractional loss operators

Fractional derivatives are well suited to describe wave propagation in complex media. When introduced in classical wave equations, they allow a modeling of attenuation and dispersion that better describes sound propagation in biological tissues. Traditional constitutive equations from solid mechanics and heat conduction are modified using fractional derivatives. They are used to derive a nonlinear wave equation which describes attenuation and dispersion laws that match observations. This wave equation is a generalization of the Westervelt equation, and also leads to a fractional version of the Khokhlov-Zabolotskaya-Kuznetsov and Burgers' equations.     

Nonlinear frequency shift in surface acoustic wave resonator operating as a gas sensor

The shift in the resonance frequency of a two-port quartz surface acoustic wave (SAW) resonator operating as a gas sensor without a selective layer is studied versus the power of an SAW excited in the resonator. At working frequencies of the resonator (≈389 MHz) placed in the flow of moisture-containing nitrogen gas, an anomalously large positive shift of the resonance frequency is observed as the SAW power exceeds 1 mW. This shift is one order of magnitude larger than that due to the nonlinear amplitude-frequency effect, which is known for quartz SAW resonators. Possible physical mechanisms underlying this phenomenon are analyzed. Experimental data indicate that such a shift is associated with the influence of a powerful SAW on sorption processes taking place on the active surface of the resonator rather than being a direct consequence of heating of the SAW substrate by the powerful SAW.     

Propagation of acoustic pulses in material with hysteretic nonlinearity

Evolution equations for propagation of both unipolar and bipolar acoustic pulses are derived by using hysteretic stress-strain relationships. Hysteretic stress-strain loops that incorporate quadratic nonlinearity are derived by applying the model of Preisach-Mayergoyz space for the characterization of structural elements in a micro-inhomogeneous material. Exact solutions of the nonlinear evolution equations predict broadening in time and reduction in amplitude of a unipolar finite-amplitude acoustic pulse. In contrast with some earlier theoretical predictions, the transformation of the pulse shape predicted here satisfies the law of "momentum" conservation (the "equality of areas" law in nonlinear acoustics of elastic materials). A bipolar pulse of nonzero momentum first transforms during its propagation into a unipolar pulse of the same duration. This process occurs in accordance with the "momentum" conservation law and without formation of shock fronts in the particle velocity profile.     

Nonlinear dynamics in a Fabry-Perot resonator

We develop a general formalism to describe the dynamical behavior of an ensemble of two-level systems in a Fabry-Perot cavity. Our main result is a set of space and time-dependent, integro-differential equations for the slowly varying radiation and atomic variables, which we derive through a precise analysis of the slowly-varying amplitude approximation in the presence of counter-propagating fields. With the help of new and properly chosen variables we recast these equations in a form that makes their boundary conditions formally identical to those of an ideal Fabry-Perot resonator, and introduce in a natural way a modal structure even for systems with arbitrary mirror reflectivity. We derive simplified forms of these equations in the uniform field limit and within the more general single-frequency approximation. Finally, we extend our formulation to include driven systems such as optically bistable devices and the laser with an injected signal.     

Nonlinear standing waves in a resonator with feedback control

An experimental study is presented to demonstrate that nonlinear effect on standing waves in a resonator can be reduced by a feedback loop responding to the second harmonic. The resonator was a cylindrical tube sealed at one end and driven by a horn driver unit at another end. The feedback control loop consisted of a pressure sensor, a frequency filter, a phase shifter, and an actuator. The results show that the waveform distortions can be eliminated and large amplitude sinusoidal pressure oscillations are obtained. A simple model is proposed for a qualitative discussion on the control mechanism, which shows that the feedback loop alters the imaginary part of the complex mode frequency so as to suppress (or enhance) the second harmonic.     

Nonlinear Fabry-Perot resonator with a silicon photonic crystal waveguide

We derive an equation that describes the nonlinear operation of a Fabry-Perot resonator with a large group index waveguide. Specifically, a silicon photonic crystal microcavity with two-photon-excited free carrier nonlinearity and Kerr nonlinearity is assumed. The equation clearly explains the bistability of the device and the reduction of the required pump energy for a specific nonlinear phase shift at an appropriate phase detuning from the resonance. We present a simple procedure to predict the required optical pump energy for the modulation and the resulting modulation depth by use of the equation and the device parameters.     

Nonlinear theory of self-oscillations in a phase-conjugate resonator

We present an exact, nonlinear analysis of self-oscillations in a phase-conjugate resonator. Optical bi- and multistability, as well as period doubling to chaos are predicted.     

Laser-generated surface acoustic waves in a ring-shaped waveguide resonator

Surface acoustic wave (SAW) waveguide resonator is formed by a ring-shaped strip of copper 10 μm wide and ∼130 μm in diameter embedded into a 0.8 μm thick layer of silica on a silicon wafer. SAWs are excited at one side of the copper ring by a short laser pulse focused into a spatially periodic pattern and detected via diffraction of the probe laser beam overlapped with the excitation spot. SAW wavepackets with central frequency 460 MHz travel around the ring and are detected each time they make a full circle and pass trough the probe spot. Potential applications of ring resonators for SAWs are discussed.     

Nonlinear stationary acoustic wave in a solid with dislocations

It is shown that a nonlinear stationary acoustic wave can be formed in a solid with dislocations. Such a wave is periodic and moves faster than acoustic signals in a linear medium. The wave has a saw-tooth shape and its wavelength increases with the amplitude.     

锥型热声谐振管内非线性声场的数值模拟研究

从声学角度出发,考虑粘性耗散、非线性效应及管型结构变化的影响,利用伽辽金法,对锥型热声谐振管内的一维声场进行了数值模拟研究,对谐振管结构参数对声场的影响进行了分析,给出了锥型管内压比随谐振管结构参数变化的规律,通过与圆柱型直管的比较,揭示了锥型管在抑制谐波及提高压比等方面的优越性。     

锥形颈部赫姆霍兹共振器声学性能预测

锥形颈部赫姆霍兹共振器具有更好的低频消声能力,而其声学性能尚无准确解析预测方法。为了研究其声学性能,在声学长度修正的基础上,利用一维解析方法建立了用于计算传递损失的一维修正模型。运用分割法计算锥形管内部声传播的声学长度修正,并给出了声学修正长度计算公式。采用得到的锥形管声学修正长度和一维修正模型,计算出的锥形颈部赫姆霍兹共振器频率与有限元及实验测试结果偏差在2 Hz以内,明显优于不修正的计算结果。表明锥形管声学长度修正法提高了一维解析方法的精度,从而可以快捷准确的预测锥形颈部赫姆霍兹共振器的消声性能。     

Nonlinear all-photonic-crystal Fabry-Perot resonator with angle-independent bistability

We propose a nonlinear all-photonic-crystal (PhC) Fabry-Perot cavity tuned to the subdiffractive regime of the interior PhC, and we study angular-resolved nonlinear propagation of monochromatic plane wave excitations. With rigorous numerical simulations, we show that, for sufficiently large negative pump detunings and a focusing nonlinearity, the transmitted field has a bistable dependence on the pump field. Moreover, we reveal that, in contrast to a homogeneous resonator for different inclinations, the hysteresis curve is virtually unchanged for a fairly wide angular range. This may pave the way for obtaining novel kinds of nonlinear localized solutions in driven nonlinear resonators.     

Nonlinear effects of resonance transparency for two-frequency acoustic pulses in a paramagnetic crystal

Two modes of nonlinear propagation of two-frequency acoustic pulses in a low-temperature crystal containing paramagnetic resonance impurities with an effective spin S = 1 in an external magnetic field and a field of the static strain are considered. It is shown that the spin-phonon transitions occurring within spin triplets according to the V scheme are responsible for two-frequency self-induced acoustic transparency. When the spin-phonon transitions follow the Λ scheme, there can arise an acoustic effect similar to electromagnetically induced transparency in a pulsed mode, which is accompanied by trapping of the population of the spin sublevels.     

Interaction of acoustic waves in microinhomogeneous media with hysteretic nonlinearity and relaxation

The propagation of a weak high-frequency pulse in the field of an intense standing low-frequency pump wave excited in an acoustic resonator with quadratic hysteretic nonlinearity and relaxation is theoretically investigated. The nonlinear damping coefficient and the carrier phase delay of the weak high-frequency pulse propagating under the action of an intense low-frequency wave are determined.     

Nonlinear effects and shock formation in the focusing of a spherical acoustic wave

The focusing of acoustic waves is used to study nucleation phenomena in liquids. At large amplitude, nonlinear effects are important so that the magnitude of pressure or density oscillations is difficult to predict. We present a calculation of these oscillations in a spherical geometry. We show that the main source of nonlinearities is the shape of the equation of state of the liquid, enhanced by the spherical geometry. We also show that the formation of shocks cannot be ignored beyond a certain oscillation amplitude. The shock length is estimated by an analytic calculation based on the characteristics method. In our numerical simulations, we have treated the shocks with a WENO scheme. We obtain a very good agreement with experimental measurements which were recently performed in liquid helium. In addition, the comparison between numerical and experimental results allows us to calibrate the vibration of the ceramic used to produce the wave, as a function of the applied voltage.Received: 11 July 2003, Published online: 24 October 2003PACS: 67.40.-w Boson degeneracy and superfluidity of 4He - 43.25. + y Nonlinear acoustics - 62.60. + v Acoustical properties of liquids     

Nonlinear dynamics of complex hysteretic systems: Oscillator in a magnetic field

Complex hysteresis is a well-known phenomenon in many branches of science. The most prominent examples come from materials with a complex microscopic structure such as magnetic materials, shape-memory alloys, or, porous materials. Their hysteretic behavior is characterized by the existence of multiple internal system states for a given external parameter and by a non-local memory. The input-output behavior of such systems is well studied and in a standard phenomenological approach described by the so-called Preisach operator. What is not well understood, are situations, where such a hysteretic system is dynamically coupled to its environment. Since the hysteretic sub-system provides a complicated form of nonlinearity, one expects non-trivial, possibly chaotic behavior of the combined dynamical system. We study such a combined dynamical system with hysteretic nonlinearity. In this original contribution a simple differential-operator equation with hysteretic damping, which describes a magnetic pendulum is considered. We find, for instance, a fractal dependence of the asymptotic behavior as function of the starting values. The sensitivity of the system to perturbations is investigated by several methods, such as the 0–1 test for chaos and sub-Lyapunov exponents. The power spectral density is also calculated and compared with analytical results for simple input-output scenarios.