Finite Amplitude Wave Propagation Research Articles

Finite amplitude wave propagation through bubbly fluids

The existence of only a few bubbles could drastically reduce the acoustic wave speed in a liquid. Wood’s equation models the linear sound speed, while the speed of an ideal shock waves is derived as a function of the pressure ratio across the shock. The common finite amplitude waves lie, however, in between these limits. We show that in a bubbly medium, the high frequency components of finite amplitude waves are attenuated and dissipate quickly, but a low frequency part remains. This wave is then transmitted by the collapse of the bubbles and its speed decreases with increasing void fraction. We demonstrate that the linear and the shock wave regimes can be smoothly connected through a Mach number based on the collapse velocity of the bubbles.

Conserved vectors and solutions of the two-dimensional potential KP equation

Abstract This article investigates the potential Kadomtsev–Petviashvili (pKP) equation, which describes the evolution of small-amplitude nonlinear long waves with slow transverse coordinate dependence. For the first time, we employ Lie symmetry methods to calculate the Lie point symmetries of the equation, which are then utilized to derive exact solutions through symmetry reductions and with the help of Kudryashov’s method. The solutions obtained include exponential, hyperbolic, elliptic, and rational functions. Furthermore, we provide one-parameter group of transformations for the pKP equation. To gain a better understanding of the nature of each solution, we present 3D, 2D, and density plots. These obtained solutions, along with their associated physical characteristics, offer valuable insights into the propagation of small yet finite amplitude waves in shallow water.In addition, the pKP equation conserved vectors are derived by utilizing the multiplier method and the theorems by Noether and Ibragimov.

Three-dimensional simulations of sound propagation in a trumpet with accurate mouthpiece shank geometry

The length and bore geometry of musical instruments directly influences the quality of sound that can be produced. In brass instruments, nonlinear effects from finite-amplitude wave propagation can lead to wave distortion giving sounds a brassy timbre [3,5,16,22,29]. In this paper, we propose a three-dimensional model to describe nonlinear wave propagation in a trumpet and investigate the importance of the mouthpiece shank geometry. The time pressure waveforms corresponding to B3b and B4b notes at the mouthpiece were recorded at the mouthpiece shank are recorded and used as inputs for our model. To incorporate compressibility and nonlinear effects, the compressible Euler equations are used to describe the evolution of the nonlinear waves. The system was solved numerically using the discontinuous Galerkin method. The numerical results were compared with the waveforms of the musical notes, which were also measured outside the bell of the trumpet, to validate our approach. Simulations were run for both notes in computational trumpets where different bore geometries were modelled. Our results demonstrate that the shape of the narrow region near the mouthpiece greatly influences the wave propagation and must be considered in a trumpet model.

Burgers equation and convexification method: Propagation of wave packets and narrowband noise.

The Burgers equation is a standard model for the propagation of progressive finite amplitude waves in a lossy medium. This paper is mainly devoted to the one-dimensional Burgers equation and its solution deduced from a convexification method. Its starting point is the Hopf-Cole solution in the inviscid limit, then the Legendre-Fenchel transform and the convex envelope construction are employed to obtain the single-value waveform. A fractional step method is used to numerically solve the Burgers equation in a segregated manner (nonlinearity and dissipation). Numerical results are obtained for nonlinear propagation of a wave packet with a Gaussian envelope and a narrowband noise (deterministic and non-deterministic signals). The convexification method provides a tool to solve the problems of nonlinear propagation, especially in the case of noise propagation with the formation and interaction (coalescence) of a large number of discontinuities (shocks).

Illuminating finite-amplitude wave propagation: Hargrove’s use of recurrence relations

A classic problem in the theory of finite-amplitude wave propagation is that of the spectral evolution of a plane wave. An initial sinusoidal wave becomes triangular, with a vertical face at the shock distance. The wave in the pre-shock regime was represented by a Fourier series in the seminal work by Fubini, i.e., Fubini Ghiron [Alta Frequenza, 4, 530 (1935)]. Independently, Hargrove performed a similar derivation, also invoking recurrence relations for cylindrical Bessel functions, but presenting this with exceptional heuristic clarity and brevity [J. Acoust. Soc. Am. 32, 511 (1960)]. This was coincident with a third paper, by Keck and Beyer [Phys. Fluids 3, 346 (1960)], which knowingly followed Fubini. All of this work has facilitated understanding, providing cogent reminders of the power of recurrence relations when dealing with special functions.

Wave propagation of resonance multi-stripes, complexitons, and lump and its variety interaction solutions to the (2+1)-dimensional pKP equation

This study deals with the (2+1)-dimensional potential Kadomtsev-Petviashvili (pKP) equation, which is used to describe the dynamics of a wave of small but finite amplitude in two dimensions in diverse areas of physics and applied mathematics. Through symbolic computations with Maple, the resonance multi-stripe solutions in real fields, and multi-stripe complexiton solutions in complex fields are derived via the simplified linear superposition principle in conjunction with Hirota's bilinear method (HBM). Furthermore, lump, lump-stripe, and lump-triangular periodic wave solutions are derived by employing the HBM. A positive quadratic function with exponential, hyperbolic cosine and trigonometric cosine functions are considered to reach such aims. For lump solutions, it is found that the shape and amplitude of a lump wave remain unchanged during its propagation. On the other hand, lump-stripe interaction solitons present the fission and fusion interaction phenomena between the lump and stripe solitons. To illustrate the dynamical characteristics of the attained solutions, the three-dimensional (3D) and contour plots of some of the representative attained solutions are displayed with the particular choice of the free parameters. The obtained solutions and their physical features might be helpful to understand the propagation of small but finite amplitude waves in shallow water.

1D Numerical Study of Nonlinear Propagation of Finite Amplitude Waves in Traveling Wave Tubes with Varying Cross Section

The nonlinear acoustic problem of a finite amplitude plane wave propagating along the axial direction in a traveling wave tube is studied. Based on the one-dimensional Westervelt equation, a one-dimensional nonlinear wave equation is derived in which the cross section of the traveling wave tube is considered. The two-order finite difference scheme is used to solve the nonlinear wave equation. The nonlinear propagation characteristics of a finite amplitude wave in the traveling wave tube is analyzed. In the expanding transition section, the acoustic pressure amplitude of the acoustic wave decreases with the increase of the cross-sectional area of the pipeline. The nonlinear characteristics of the acoustic wave show waveform distortion and harmonic growth. The waveform distortion becomes more serious in the rear of traveling wave tube than in the front of the tube. Considering the acoustic reflection condition at the mouth, the influence of differently shaped diffusion sections on the acoustic pressure distribution in the test section is investigated. The larger the change rate of the diffusion section in an area, the less amplitude of the sound pressure, and the nonlinear effect of the sound wave propagation is weakened. These nonlinear wave propagation characteristics in a travelling wave tube provide important guidance for both designing a uniform sound pressure distribution in the test section and determining the optimal measuring points for different sizes of structures in spacecraft.

Simulation of Nonlinear Resonance, Amplitude-Frequency, and Harmonic Generation Effects in SAW and BAW Devices.

Nonlinearly coupled sets of piezoelectric field equations in the frequency domain were derived for the nonlinear propagation of finite-amplitude waves in piezoelectric bulk acoustic wave (BAW) and surface acoustic wave (SAW) devices. To verify their accuracy, we have embedded these sets of equations in the finite-element method (FEM) of COMSOL Multiphysics software and compared the FEM results with both the analytical and experimental results found in the published literature. The nonlinear frequency responses for both plano- and contoured-plate resonators of AT-cut quartz were investigated under various voltage drives, circuit resistances, and quality factors. The proposed equations with FEM have also been employed to study 33.3-MHz very-high-frequency (VHF) quartz resonators, showing that under different conditions of how well the third overtone mode (f3) matches the third harmonic (3f) and how the fractional frequency shift of the third overtone mode (f3) occurs as a function of the fundamental mode current. Furthermore, we have studied the nonlinear harmonic generation of an 840-MHz 128° Y-cut X-propagating (128° YX) LiNbO3 SAW resonator. The second-harmonic (H2) and third-harmonic (H3) modes were observed to occur, respectively, at two-time (2f) and three-time (3f) frequencies of fundamental frequency (f) when such resonators were driven with high power. The effects of substrate thickness, bottom surface conditions of the substrate, and different circuit connections on the H2 and H3 generations were simulated and compared with available measurements. Current proposed sets of equations are general and could be used for the study of nonlinear resonance, amplitude-frequency effect, and harmonic generation in any piezoelectric devices, provided that the necessary nonlinear material constants are known.

Infrasound Propagation of Finite Amplitude Wave in the Big Size Horn

The analytical expressions, allowing analyzing the laws of increase and decrease of the second harmonic at excitation of the finite amplitude wave in the air-horn, are obtained. The distance, where the second harmonic propagating along the air-horn reaches its maximum, is determined. Comparative analysis for air-horns of different shapes is performed. All calculations were made for the horn, whose length is 10 meters and the diameter of the outlet section 4 meters. The resonant frequency of the acoustic signal in such a mouthpiece is 20 Hz.

Fully non-linear wave models in fiber-reinforced anisotropic incompressible hyperelastic solids

Many composite materials, including biological tissues, are modeled as non-linear elastic materials reinforced with elastic fibers. In the current paper, the full set of dynamic equations for finite deformations of incompressible hyperelastic solids containing a single fiber family are considered. Finite-amplitude wave propagation ansätze compatible with the incompressibility condition are employed for a generic fiber family orientation. Corresponding non-linear and linear wave equations are derived. It is shown that for a certain class of constitutive relations, the fiber contribution vanishes when the displacement is independent of the fiber direction.Point symmetries of the derived wave models are classified with respect to the material parameters and the angle between the fibers and the wave propagation direction. For planar shear waves in materials with a strong fiber contribution, a special wave propagation direction is found for which the non-linear wave equations admit an additional symmetry group. Examples of exact time-dependent solutions are provided in several physical situations, including the evolution of pre-strained configurations and traveling waves.

On an acoustics–thermal–fluid coupling model for the prediction of temperature elevation in liver tumor

The present study is aimed at predicting liver tumor temperature during a high-intensity focused ultrasound (HIFU) thermal ablation using the proposed acoustics–thermal–fluid coupling model. The linear Westervelt equation is adopted for modeling the incident finite-amplitude wave propagation. The nonlinear hemodynamic equations are also taken into account in the simulation domain that contains a hepatic tissue domain, where homogenization dominates perfusion, and a vascular domain, where blood convective cooling may be essential in determining the success of HIFU. Energy equation for thermal conduction involves two heat sinks to account for tissue perfusion and forced convection-induced cooling. The effect of acoustic streaming is also included in the development of the current HIFU simulation study. Convective cooling in large blood vessel and acoustic streaming were shown to change the temperature near blood vessel. It was shown that acoustic streaming effect can affect the blood flow distribution in hepatic arterial branches and leads to the mass flux redistribution.

Nonlinear progressive waves in a slide trombone resonator

The propagation of finite-amplitude waves inside a slide trombone is studied through direct pressure measurements corresponding to dynamic extremes. A two-microphone method is used to separate left-moving and right-moving waves inside the trombone, permitting the detection of nonlinear effects associated with progressive waves. It is found that a redistribution of energy across the spectrum toward the higher-frequencies occurs for large distances and high initial pressure levels. These results are consistent with the theory of weakly nonlinear acoustics and also with those reported in this same context by other authors, but which have been obtained mostly through examination of standing-waves.

Superposition of transverse and longitudinal finite-amplitude waves in a deformed Blatz—Ko Material

In this paper, special Blatz—Ko nonlinear elastic materials are considered, which are characterized by a constitutive constant and a constitutive function. We here deal with the propagation of finite-amplitude waves in such materials subjected to an arbitrary static homogeneous deformation. In a previous paper, it was shown that linearly polarized transverse damped inhomogeneous plane waves may propagate. The orthogonal propagation and polarization directions are arbitrary. The special Blatz—Ko materials are compressible so that homogeneous longitudinal waves may also propagate. Here it is shown that the superposition of a transverse damped inhomogeneous wave and of a longitudinal wave is also a solution, in the case when the propagation direction of the longitudinal wave is orthogonal to the polarization direction of the transverse wave. Also, results are obtained for the energy density and the energy flux of the superposition of these waves.

Investigation of a three-phase medium with a negative parameter of nonlinearity

Nonlinear acoustic properties of a composite medium, consisting of hollow microspheres suspended in Castor Oil, at elevated hydrostatic pressure are experimentally investigated. The acoustic nonlinear parameter B∕A of the medium is found to be highly dependent upon hydrostatic pressure. B∕A varies from a small negative value near ambient pressure to a very large negative value (around −6000) in the vicinity of 7×104 Pascals, above ambient pressure. With a further increase in hydrostatic pressure the magnitude of B∕A decreases passing through zero and finally becomes positive. Finite amplitude wave propagation in this medium at low hydrostatic pressures is characterized by waveform steepening in the backward direction leading to rarefactive shockwaves and, at high hydrostatic pressures, by steepening in the forward direction leading to compressive shockwaves. Estimates of B∕A are obtained from both the measurement of thermodynamic properties and from waveform distortion during propagation.

Numerical simulation of finite amplitude wave propagation in air using a realistic atmospheric absorption model

A nonlinear system of fluid dynamic equations is modeled that accounts for the effects of classical absorption, nitrogen and oxygen molecular relaxation, and relative humidity. Total variables rather than acoustic variables are used, which allow for the inclusion of frequency-independent terms. This system of equations is then solved in two dimensions using a fourth-order Runge-Kutta scheme in time and a dispersion-relation-preserving scheme in space. It is shown that the model accurately simulates wave steepening for propagation up to one shock formation distance. For a source amplitude of 157dB re 20μPa, the Fourier component amplitudes of the analytical and computed waveforms differ by 0.21% at most for the first harmonic in a lossless medium and 0.16% at most for the first harmonic in a medium that includes thermoviscous losses. It is shown that the absorption due to classical effects and molecular relaxation demonstrated by the model is within 1% of the analytical model and the computed dispersion due to molecular relaxation of nitrogen and oxygen is within 0.7% over a large frequency range. Furthermore, it is shown that the model can predict the amplification factor at a rigid boundary, which in a nonlinear system will be greater than two, to within 0.015%.

Finite amplitude elastic waves due to non-uniform traction at the surface of a cylindrical cavity

This paper is concerned with two spatial dimension, finite amplitude wave propagation emanating from the surface of an initially circular cylindrical cavity in an unbounded isotropic compressible isotropic hyperelastic solid. The solid is initially in the natural reference configuration and the wave propagation is due to an azimuthally non-uniform, sudden application of compressive nominal traction at the surface of the cavity. Governing equations for the problem are obtained in Lagrangian form in terms of cylindrical polar coordinates and two different classes of strain energy functions are considered. Numerical solutions, for a particular application of traction, are obtained from a fully explicit finite difference scheme. It was found that the responses to the particular application of traction differed negligibly for the various strain energy functions considered.

Propagation of finite amplitude sound through turbulence

Pestorius and Blackstock [‘‘Propagation of finite amplitude noise,’’ Finite-amplitude wave effects in fluids, Proceedings of the 1973 International Symposium on Nonlinear Acoustics, edited by L. J. Bjo/rno/, 1973] developed a method to simulate the propagation of finite amplitude noise in a tube. The method separates the effects of nonlinear distortion and absorption and diffraction. The nonlinear steepening of an arbitrary waveform is performed in the time domain. The absorption and diffraction effects are implemented in the frequency domain. This method has been used widely for finite amplitude wave propagation. Lipkens and Blanc-Benon [‘‘Propagation of finite-amplitude sound through turbulence: a geometrical acoustics approach,’’ C. R. Acad. Sci. Paris, Ser. II<th>b 320, 477–484 (1995)] used the Pestorius model to simulate the propagation of N waves through turbulence. Linear geometrical acoustics is used to trace rays through individual realizations of a turbulent field. A nonlinear transport equation describes the propagation of sound through turbulence. The equation is solved using the Pestorius algorithm. The results indicate that the equivalent nonlinear distortion after propagation through turbulence is always less than that for the homogeneous case. In addition, the effect of a random velocity field is more pronounced that that of a random temperature field.

Some aspects of thermodynamic effects in finite amplitude wave propagation in rubber-like solids

In this study, we apply the modified entropic elasticity theory to investigate the thermodynamic effects in finite amplitude wave propagation in a stretched hyperelastic string. In formulating the problem, we assume that the string is composed of a non-conducting rubber-like material, and that the deformation is isochoric. Adiabatic stress–stretch relations are obtained for strictly entropic and piezotropic materials, which are the two limiting cases of modified entropic elasticity. The Lagrangian system of governing equations for finite amplitude wave propagation in a stretched hyperelastic string is derived in conservation form. Similarity solutions are presented for two strain energy functions. The errors in adopting an isothermal stress–stretch relation instead of an adiabatic stress–stretch relation, and the validity of the isentropic approximation, are investigated. The treatise developed in this study, and the results obtained, appear to be new.

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.

The dynamic response of an incompressible non-linearly elastic membrane tube subjected to a dynamic extension

The dynamic response of an isotropic hyperelastic membrane tube, subjected to a dynamic extension at its one end, is studied. In the first part of the paper, an asymptotic expansion technique is used to derive a non-linear membrane theory for finite axially symmetric dynamic deformations of incompressible non-linearly elastic circular cylindrical tubes by starting from the three-dimensional elasticity theory. The equations governing dynamic axially symmetric deformations of the membrane tube are obtained for an arbitrary form of the strain-energy function. In the second part of the paper, finite amplitude wave propagation in an incompressible hyperelastic membrane tube is considered when one end is fixed and the other is subjected to a suddenly applied dynamic extension. A Godunov-type finite volume method is used to solve numerically the corresponding problem. Numerical results are given for the Mooney–Rivlin incompressible material. The question how the present numerical results are related to those obtained in the literature is discussed.