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Effective Gol'dberg number for diverging waves
Abstract
An effective Gol'dberg number is proposed for determining the degree of nonlinear distortion achieved in a diverging wave field. For values that are large compared with unity, the degree of nonlinear waveform distortion is virtually the same as that for a plane wave characterized by the traditional Gol'dberg number having the same numerical value. Expressions for the effective Gol'dberg number are proposed for spherical and cylindrical waves, Gaussian beams, and exponential horns.
... (15.2) and (15.3), with the assumption of a rightward traveling wave in Eq. (15.1), leads to a dispersion relation that generates the equilibrium values for propagation speed, c grav , of a shallow-water gravity wave. 2 þjω Àjh o k Àjgk þjω ...
... The distance traveled by the wave once the slope first becomes infinite is known as the shock inception distance, D S . 2 The exact result for the propagation speed at all depths reduces to c grav in Eq. (15.4) in the limit that kh o ! 0. Since this result depends upon k, it is dispersive, so the phase speed, c grav , will not be equal to the group speed except in the "shallow water" kh o ! ...
... the "Grüneisen parameter" and designating it as Γ is not a common choice in other treatments of nonlinear acoustics. For example, in a recent paper by Hamilton [2], Γ represents the Gol'dberg number that is abbreviated as G in this textbook (see Sect. 15.1.4). In Eq. (15.9), the general amplitude variable is simply written as "y," and the equilibrium sound speed is designated c o to distinguish it from the local amplitude-dependent sound speed, c(v) ¼ c o +Γv. ...
A fundamental assumption of linear acoustics is that the presence of a wave does not have an influence on the properties of the medium through which it propagates. By extension, the assumption of linearity also means that a waveform is stable since any individual wave does not interact with itself. Small modifications in the sound speed due to wave-induced fluid convection (particle velocity) and to the wave’s effect on sound speed through the equation of state can lead to effects that could not be predicted within the limitations imposed by the assumption of linearity. Although a wave’s influence on the propagation speed may be small, those effects are cumulative and create distortion that can produce shocks. These are nonlinear effects because the magnitude of the nonlinearity’s influence is related to the square of an individual wave’s amplitude (self-interaction) or the product of the amplitudes of two interacting waves (intermodulation distortion). In addition, the time-average of an acoustically induced disturbance may not be zero. Sound waves can exert forces that are sufficient to levitate solid objects against gravity. The stability of such levitation forces will also be examined along with their relation to resonance frequency shifts created by the position of the levitated object.
... Laboratory-scale jets have been studied to examine cumulative nonlinear effects, [1][2][3][4] coalescence in the sound field, 5 dependence on Reynolds number, 6 pressure skewness and pressure-derivative skewness values, 7,8 as well as analysis with other nonlinearity metrics. 9,10 The nonlinear evolution 11 and annoyance 12 of legacy commercial transports have also been characterized using a normalized quadspectral nonlinearity indicator. Noise waveforms from military jets have been used both to characterize near-field shock formation 13,14 and to model far-field shock formation, [15][16][17] along with comparison of computationally predicted and experimentally measured data. ...
... importance of cumulative nonlinearity in propagation. 1,10 The GBE has also been used to determine local rates of spectral changes due to nonlinearity and other linear effects. [26][27][28] The local analysis stems from the use of the "Morfey-Howell" indicator, 11 which has been previously studied for connection to nonlinear effects. ...
A single-point, frequency-domain nonlinearity indicator is calculated and analyzed for noise from a model-scale jet at Mach 0.85, Mach 1.8, and Mach 2.0. The nonlinearity indicator, νN, has been previously derived from an ensemble-averaged, frequency-domain version of the generalized Burgers equation (GBE) from Reichman, Gee, Neilsen, and Miller [J. Acoust. Soc. Am. 139, 2505–2513 (2016)]. The indicator gives the spatial rate of change due to nonlinear processes in sound pressure level (SPL) spectrum, and two other indicators from the GBE—νS and να—give the same quantity due to geometric spreading and absorption, respectively. Trends with frequency, angle, distance, and jet condition—supported both by spectral analysis and by calculation of the GBE-derived indicators—reveal a concentration of nonlinear effects along radials close to the plume with large overall SPLs. The calculated indicators for nonlinearity and absorption effects far from the source combine to give the same decay predicted by nonlinear theory for monofrequency sources. Trends in the νN indicator are compared with trends observed for other indicators such as pressure-derivative skewness and bicoherence, revealing both the qualitative and quantitative advantages of the νN indicator.
... Analytical expressions for effective Gol'dberg numbers applicable to diverging wave fields have been developed [29] and provide the necessary framework for problems in jet aeroacoustics because the sound intensity from round jets experiences both cylindrical and spherical decay. To determine the likelihood of cumulative nonlinear distortion, one must consider the strengths of two competing effects (i.e., nonlinear distortion and energy absorption); the strengths of these are commonly expressed in terms of their individual length scales. ...
... Although the evolution of a plane wave is determined completely by the single parameter Γ, two dimensionless parameters (length scale ratios Γ and σ 0 ) determine the likelihood of nonlinear distortion in diverging wave fields. Following Hamilton [29], it can be shown how effective Gol'dberg numbers for cylindrically and spherically diverging waves can be written as, respectively, ...
A method for calculating the effective Gol’dberg number for diverging waveforms is presented, which leverages known features of a high-speed jet and its associated sound field. The approach employs a ray tube situated along the Mach wave angle where the sound field is not only most intense, but advances from undergoing cylindrical decay to spherical decay. Unlike other efforts, a “piecewise-spreading regime” model is employed, which yields, separately, effective Gol’dberg numbers for the cylindrically and spherically spreading regions in the far field. The new approach is applied to a plethora of experimental databases, encompassing both laboratory- and full-scale jet noise studies. The findings demonstrate how cumulative nonlinear distortion is expected to form in the acoustic near field of laboratory-scale round jets where pressure amplitudes decay cylindrically; waveform distortion is not expected in the acoustic far field where waveform amplitudes diverge spherically. On the other hand, where full-scale jet studies are concerned, effective Gol’dberg number calculations demonstrate how cumulative waveform distortion is significant in both the cylindrical- and spherical-spreading regimes. The laboratory-scale studies also reveal a pronounced sensitivity to humidity conditions, relative to the full-scale counterpart.
... Similar substitutions can be made for expressions of the SFD for generally diverging waves in lossless media (Hamilton, 2016). ...
The nonlinear evolution of high-amplitude broadband noise is important to the psychoacoustic perception, usually annoyance, of high-speed jet noise. One method to characterize the nonlinear evolution of such noise is to consider a characteristic nonlinear waveform distortion length for the signal. A common length scale for this analysis is the shock formation distance of an initially sinusoidal signal. However, application of this length scale to broadband noise, even with the amplitude and source frequency replaced with characteristic values, may lead to underestimates of the overall nonlinear waveform distortion of the noise as indicated by the skewness of the time derivative of the acoustic pressure (or derivative skewness). This paper provides an alternative length scale derived directly from the evolution of the derivative skewness of Gaussian noise that may be more appropriate when analyzing the nonlinear evolution of broadband noise signals. This Gaussian-based length scale is shown to be a useful metric for its relative consistency and its physical interpretation. Various analytical predictions of the evolution of the derivative skewness for an ensemble of numerical simulations of noise propagation are used to highlight various aspects of this new length scale definition.
... Different acoustic pressures p 0 (0.35 MPa, 1.55 MPa and 4.5 MPa) at the transducer surface were modelled to study the power-dependence of the harmonic generation. Gol'dberg number was obtained to determine the propagation regime in each case, calculated as follows:22 ...
3D-printed holographic lenses coupled to an ultrasound transducer shape the acoustic field into arbitrary images. Predefined thermal patterns can be created with this technology in absorbing media, which can be used to generate localized and specific hyperthermia treatments. In this work we study how non-linearities in the acoustic field can affect the produced thermal pattern at hyperthermia intensity powers. A ‘2’-shaped acoustic hologram was designed with time-reversal methods and both acoustic and thermal numerical simulations were performed considering linear and non-linear propagation of the resulting acoustic field in liver-like medium. Experimental validation of the acoustic hologram was performed in degassed water, while the corresponding thermal pattern was evaluated using magnetic resonance thermometry and infra-red imaging in ex-vivo cow liver tissue and tissue-mimicking phantom. Simulations and experiments were in complete agreement, demonstrating that at hyperthermia intensities most of the energy is carried by the fundamental frequency and thus nonlinear propagation have little impact on resulting thermal pattern.
... In the far-field, the nonlinear wave propagation competes with the linear atmospheric absorption effects. The interplay between the nonlinear and linear dissipation effects is expressed via the inverse acoustic 30 Reynolds number (the Goldberg number), which is not only strongly dependent on the flow conditions such as the jet nozzle pressure and temperature ratio but also the effective distance from the jet [16]. ...
A triple-scale computational model is implemented to simulate noise generated by a supersonic under-expanded screeching jet corresponding to the LTRAC (Laboratory for Turbulence Research in Aerospace and Combustion) experiment at different distances from the source. The investigation is focused on the broadband-associated noise, which is a prominent feature of the acoustic field of the LTRAC jet. In the jet near-field, the compressible Navier-Stokes equations are solved using the high-resolution CABARET Large Eddy Simulation (LES) method accelerated on Graphics Processing Units. The LES solution is substituted in the Ffowcs Williams-Hawkings (FW-H) model to obtain the noise solution in the acoustic mid field at 20 initial jet diameters from the jet nozzle exit. The mid-field acoustic solution is used as the input for the spherical generalised Burgers' equation. The general form of Burgers' equation is solved numerically in the frequency domain for a wide range of observer distances up to 18 million initial jet diameters, where viscous dissipation fully dominates for most frequencies. To answer the question if the nonlinear acoustic wave propagation effects for the LTRAC jet are important, the nonlinear and linear solutions of Burgers' equation are compared.
... While the evolution of a plane wave is governed completely by the single dimensionless parameter Γ , two dimensionless parameters -length scale ratios Γ and σ 0 -determine how spherical waves evolve. It has been recently shown by Hamilton (2013) that an effective Gol'dberg number for diverging spherical waves may be expressed as ...
For some time now it has been theorized that spatially evolving instability waves in the irrotational near-field of jet flows couple both linearly and nonlinearly to generate far-field sound [Sandham and Salgado, Philos. Trans. R. Soc. Am. 366 (2008); Suponitsky, J. Fluid Mech. 658 (2010)]. An exhaustive effort at The University of Texas of Austin was initiated in 2008 to better understand this phenomenon, which included the development of a unique analysis technique for quantifying their coherence [Baars et al., AIAA Paper 2010–1292 (2010); Baars and Tinney, Phys. Fluids 26, 055112 (2014)]. Simulated data have shown this technique to be effective, albeit, insurmountable failures arise when exercised on real laboratory measurements. The question that we seek to address is how might jet flows manifest nonlinearities? Both subsonic and supersonic jet flows are considered with simulated and measured data sets encompassing near-field and far-field pressure signals. The focus then turns to considering nonlinearities in the form of cumulative distortions, and the conditions required for them to be realized in a laboratory scale facility [Baars, et al., J. Fluid Mech. 749 (2014)].
Prior measurements of the sound field produced by a laboratory-scale, Mach 3 jet flow (Baars and Tinney, Journal of Sound and Vibration, Vol. 333, No. 12, 2014, pp. 2539–2553; Fiévet et al., AIAA Journal, Vol. 54, No. 1, 2016, pp. 254–265) suggest that acoustic waveforms steepen early on in their development. This explained the discrepancy between theoretical predictions, based on effective Gol’dberg numbers, that shocks should not form, and observations of steepened Mach waves close to laboratory-scale jets. The present work continues studying this phenomenon by exploring coalescence processes that occur when neighboring waveforms intersect, forming larger-amplitude waveforms with increased cumulative nonlinear distortion. A numerical model based on the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation is developed to show that coalescence-induced steepening is sensitive to the intersection angle between adjacent waveforms, waveform duration, and cylindrical spreading effects. High frame-rate schlieren images of sound waves propagating from the post-potential core region of a laboratory-scale Mach 3 jet are then captured along an angle following the ridge of most intense noise to study the development and evolution of coalescence. A shock detection algorithm isolates shock-like events, which are tracked using a translating coordinate system and decomposed using proper orthogonal decomposition. Reduced-order reconstructions of both schlieren images and the KZK model identify common patterns that characterize the shock coalescence process.
Рассмотрены условия истечения недорасширенной сверхзвуковой струи из эксперимента LTRAC (Laboratory for Turbulent Research in Aerospace and Combustion, университет Monash, Австралия). Проведен анализ характерных параметров линейного и нелинейного переноса для струи LTRAC с использованием решений LES из ближнего и дальнего акустического поля. В обоих случаях показано выполнение условий линейного сценария переноса звука на расстояния, характерные для акустического эксперимента LTRAC. Для верификации теоретических оценок также получены численные решения сферического уравнения Бюргерса, используя начальные данные из расчета LES. Эволюционное уравнение Бюргерса решается численно в частотной области в дальней зоне до тех пор, пока эффект линейной диссипации не станет преобладающим. Решение выполняется с учетом и без учета нелинейного акустического члена, что важно для оценки влияния нелинейности на спектры аэродинамического шума в дальней зоне.
This study aims to numerically investigate the noise radiated by a reduced-scale rocket engine jet at lift-off conditions including a flame trench. An over-expanded Mach 3.1 hot jet entering a flame duct where it impinges on a deflector before being guided in a horizontal duct is considered. The computation is performed with a two-way coupled approach on unstructured grids. This methodology relies on a large-eddy simulation of the jet and the acoustic near field, associated with a full Euler simulation of the acoustic far field. The aerodynamic and acoustic results are compared to a previous computation involving the Ffowcs Williams and Hawkings approach and show a better agreement with the measurements conducted at the MARTEL facility. A more careful analysis of the pressure field suggests that the noise is strongly influenced by the flame trench geometry. Nonlinear propagation effects, natively taken into account by the full Euler solver, are finally highlighted and discussed. Based on appropriate metrics, a good agreement with the experiment is obtained.
Wave steepening and shock coalescence due to nonlinear propagation effects are investigated for a cold Mach 3 jet. The jet flow and near pressure fields are computed using large-eddy simulation. The near acoustic field is propagated to the far field by solving the linearized or the weakly nonlinear Euler equations. Near the angle of peak levels, the skewness factors of the pressure fluctuations for linear and nonlinear propagations display positive values that are almost identical. Thus, the positive asymmetry of the fluctuations originates during the wave generation process and is not due to nonlinear propagation effects. Compressions in the signals are much steeper for a nonlinear than for a linear propagation, highlighting the crucial role of nonlinear distortions in the formation of steepened waves. The power transfers due to nonlinear propagation are examined for specific frequencies by considering the spatial distribution of the Morfey–Howell indicator in the near and far acoustic fields. They are in good agreement with the direct measurements performed by comparing the spectra for nonlinear and linear propagations. This shows the suitability of the Morfey–Howell indicator to characterize nonlinear distortions for supersonic jets.
Since the Morfey-Howell Q/S was introduced as a single-point frequency-domain nonlinearity indicator for propagation of intense broadband noise [AIAA J. 19, 986-992 (1981)], there has been debate about its validity, utility, and interpretation. In this Letter, the generalized Burgers equation is recast in terms of specific acoustic impedance along with linear absorption and dispersion coefficients, normalized quadspectral density (Q/S), and newly proposed normalized cospectral density (C/S). The formulation leads to a rather straightforward interpretation in which Q/S and C/S, respectively, represent the additional absorption and dispersion at a locale, produced by the passage of a finite-amplitude wave.
The spatial evolution of acoustic waveforms produced by a laboratory-scale Mach 3 jet are investigated using both 1∕4 and 1∕8 in. pressure field microphones located along rays emanating from the postpotential core where the peak sound emission is found to occur. The measurements are acquired in a fully anechoic chamber, where ground or other large surface reflections are minimal. Various statistical metrics are examined along the peak emission path, where they are shown to undergo rapid changes within 2 m from the source region. An experimentally validated wave-packet model is then used to confirm the location where the pressure amplitude along the peak emission path transitions from cylindrical to spherical decay. Various source amplitudes, provided by the wave-packet model, are then used to estimate shock formation distance and Gol'dberg numbers for diverging waves. The findings suggest that cumulative nonlinear distortion is likely to occur at laboratory scale near the jet flow, where the waveform amplitude decays cylindrically, but less likely to occur farther from the jet flow, where the waveform amplitude decays spherically. Direct inspection of the raw time series reveals how steepened waveforms are generated by roguelike waves that form from the constructive interference of waves from neighboring sources as opposed to classical cumulative nonlinear distortion.
A model is proposed for predicting the presence of cumulative nonlinear distortions in the acoustic waveforms produced by high-speed jet flows. The model relies on the conventional definition of the acoustic shock formation distance and employs an effective Gol'dberg number A for diverging acoustic waves. The latter properly accounts for spherical spreading, whereas the classical Gol'dberg number F is restricted to plane wave applications. Scaling laws are then derived to account for the effects imposed by jet exit conditions of practical interest and includes Mach number, temperature ratio, Strouhal number and an absolute observer distance relative to a broadband Gaussian source. Surveys of the acoustic pressure produced by a laboratory-scale, shock-free and unheated Mach 3 jet are used to support findings of the model. Acoustic waveforms are acquired on a two-dimensional grid extending out to 145 nozzle diameters from the jet exit plane. Various statistical metrics are employed to examine the degree of local and cumulative nonlinearity in the measured waveforms and their temporal derivatives. This includes a wave steepening factor (WSF), skewness, kurtosis and the normalized quadrature spectral density. The analysed data are shown to collapse reasonably well along rays emanating from the post-potential-core region of the jet. An application of the generalized Burgers equation is used to demonstrate the effect of cumulative nonlinear distortion on an arbitrary acoustic waveform produced by a high-convective-Mach-number supersonic jet. It is advocated that cumulative nonlinear distortion effects during far-field sound propagation are too subtle in this range-restricted environment and over the region covered, which may be true for other laboratory-scale jet noise facilities.
The role of nonlinearity in the propagation of noise radiated from
high-performance jet aircraft has not been a well-understood phenomenon
in the past. To address the problem of finite-amplitude noise
propagation, a hybrid time-frequency domain model has been developed to
numerically solve the generalized Mendousse-Burgers equation, which is a
parabolic model equation that includes effects of quadratic
nonlinearity, atmospheric absorption and dispersion, and geometrical
spreading. The algorithm has been compared against analytical theory and
numerical issues have been discussed. Three sets of experimental data
have been used to evaluate the model: model-scale laboratory jet data,
field data using a large loudspeaker, and static engine run-up
measurements of the F/A-22 Raptor. Comparison of linearly- and
nonlinearly-predicted spectra demonstrates that nonlinearity does, in
fact, impact the noise propagation in all three sets of data.
Additionally, the extensive comparison with the Raptor data shows that
the model is successful in predicting the measured spectrum over
multiple angles and engine conditions, demonstrating that the model
captures much of the physics of the propagation, despite its current
neglect of multipath interference and atmospheric refraction and
turbulence effects. Two additional studies have been carried out in
order to address fundamental questions relevant to the nonlinear
propagation of jet noise: ''What is the impact of nonlinearity on
perceived levels?" and ''At what point does the propagation become
linear?" An investigation of the perceived impact of nonlinearity shows
that there are only minor differences between nonlinear and linear
predictions in calculations of power-based, single-number metrics, such
as A-weighted overall sound pressure level. On the other hand, the
actual perceived differences between nonlinear and linear waveforms are
substantially greater and consequently do not correlate well with
calculated metrics. This investigation suggests the need for further
research in order to better quantify the perceptual differences between
nonlinear and linear propagation. Study of the additional generation of
nonlinearity in the long-range propagation of a noise waveform suggests
that the high-frequency energy decays very slowly and the spatial rate
of change of the difference between nonlinearly- and linearly-predicted
sound pressure levels does not go to zero, which would indicate linear
propagation, but appears to asymptotically approach nonzero constant
behavior. This implies that a finite-amplitude noise waveform may never
truly decay as linear theory would indicate, which could be important
from a human-perception standpoint.
A special analytical method, which combines the parabolic approximation (KZ equation) with nonlinear geometrical acoustics, is developed to model nonlinear and diffraction effects near the axis of a finite amplitude sound beam. The corresponding system of nonlinear equations describing waveform evolution is derived. For the case of an initially sinusoidal wave radiated by a Gaussian source, an analytic solution of the coupled equations is obtained for the paraxial region of the beam. The axial solution is expressed in both the time and frequency domains, and is analyzed in detail for both unfocused and focused beams in their preshock regions. Harmonic propagation curves are compared with finite difference solutions of the KZ equation, and good agreement is obtained for a variety of parameter values.
A method for calculating the effective Gol’dberg number for diverging waveforms is presented, which leverages known features of a high-speed jet and its associated sound field. The approach employs a ray tube situated along the Mach wave angle where the sound field is not only most intense, but advances from undergoing cylindrical decay to spherical decay. Unlike other efforts, a “piecewise-spreading regime” model is employed, which yields, separately, effective Gol’dberg numbers for the cylindrically and spherically spreading regions in the far field. The new approach is applied to a plethora of experimental databases, encompassing both laboratory- and full-scale jet noise studies. The findings demonstrate how cumulative nonlinear distortion is expected to form in the acoustic near field of laboratory-scale round jets where pressure amplitudes decay cylindrically; waveform distortion is not expected in the acoustic far field where waveform amplitudes diverge spherically. On the other hand, where full-scale jet studies are concerned, effective Gol’dberg number calculations demonstrate how cumulative waveform distortion is significant in both the cylindrical- and spherical-spreading regimes. The laboratory-scale studies also reveal a pronounced sensitivity to humidity conditions, relative to the full-scale counterpart.
The acoustic transmitting capacity of a medium is limited by nonlinear propagation effects. As the source amplitude is increased to very high levels, acoustic saturation sets in, a state in which the amplitude at a field point approaches a limiting value, independent of the source amplitude. A theoretical and experimental study of saturation of spherical waves is presented in this paper. Predicted saturation amplitudes for the fundamental component in an originally sinusoidal wave are obtained by matching the well known weak‐shock solution to a linear‐theory solution at the old‐age distance r max. The saturation formulas are verified by experiments done in fresh water with a 3‐in.‐diam piston projector operating at 454 kHz at source levels up to 135 dB re 1 μbar at 1 yd. Amplitude response curves were measured on the acoustic axis at six distances ranging from 0.76 yd to 111 yd. At the longer ranges the curves become isotonic, thereby demonstrating saturation. High‐amplitude beam patterns show the effect of finite‐amplitude attenuation in that the major lobe is broadened and minor lobe suppression is reduced. Finally, the saturation formulas are used to develop curves of maximum SPL as a function of range, with frequency a parameter, for both fresh and sea water.
The propagation of a plane progressive wave of finite amplitude in a thermoviscous fluid is considered. The wave is taken to be purely sinusoidal in shape at its source. The approach is through Burgers equation, which is a very good approximation of the exact equations of fluid motion when effects of nonlinearity and dissipation are relatively small but definitely not negligible. A complicated but exact solution of Burgers' equation is analyzed. The nature of the solution is found to depend strongly on a parameter Γ, which represents the importance of nonlinearity relative to dissipation. Nonlinear effects prove to be significant when Γ>1, a finding in agreement with the criterion proposed by Gol'dberg (Akust. Zh. 2, 325 (1956) [English transl.: Soviet Phys.—Acoust. 2, 346(1956)]) concerning the appearance of shock waves. When Γ>>1, simple asymptotic representations of the exact solution in terms of Fourier series may be obtained. One of these corresponds to Fay's solution [J. Acoust. Soc. Am. 3, 222 (1931)]. An equivalent “time‐domain” representation shows clearly the sawtooth behavior of the wave. The sawtooth region is found to extend approximately to the point x=O.6/α, where α is the dimensional small‐signal attenuation coefficient. Curves of the extra attenuation suffered by the fundamental as a result of nonlinear effects are given for values of Γ in the range 1–100 000. If Γ is large, the extra attenuation in decibels approaches the asymptotic value −12+20 log10Γ as the distance from the source becomes large. A value 1 dB below the asymptote is attained at a distance x=1/α. Application to waves in argon and air is discussed. It is found that nonlinear effects may be employed to increase the efficiency of long‐range sound transmission. A possible application to low‐frequency sound in the ocean is also discussed.
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Interest in characterizing nonlinearity in jet noise has motivated consideration of an effective Gol'dberg number for diverging waves [Baars and Tinney, Bull. Am. Phys. Soc. 57, 17 (2012)]. Fenlon [J. Acoust. Soc. Am. 50, 1299 (1971)] developed expressions for the minimum value of Γ, the Gol'dberg number as defined for plane waves, for which shock formation occurs in diverging spherical and cylindrical waves. The conditions were deduced from a generalized Khokhlov solution and depend on the ratio xsh/r0, where r0 is source radius, and xsh the plane-wave shock formation distance for Γ=∞. Alternatively, by taking the ratio of the nonlinear and thermoviscous terms in Fenlon's Eq. (2), it is proposed here that effective Gol'dberg numbers may be identified for spherical and cylindrical waves: Λ=Γexp(-πxsh/2r0) and Λ=Γ/(1 + πxsh/4r0), respectively. For a given value of Λ, the diverging waves achieve approximately the same degree of nonlinear distortion as a plane wave for which the value of Γ is the same. Conversely, to achieve the same degree of nonlinear distortion as a plane wave with a given value of Γ, the value of Γ for, e.g., a spherical wave must be larger by a factor of exp(πxsh/2r0). Extensions to other spreading laws are presented.
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 e t a l., J. Acoust. Soc. Am. 7 8, 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.
A method for computing nonlinear interactions between the spectral lines of progressive finite‐amplitude waves in homogeneous media is developed via Burgers' equation. By means of Fourier analysis, this equation is reduced to a coupled set of ordinary nonlinear differential equations, which are then solved recursively using Airy's algorithm. The solution thus obtained has the form of a vector which initially contains the spectral amplitudes of the source waveform and is subsequently enriched by nonlinearly generated spectral components as the signal propagates through the medium. The utility of the method consists in the ease with which it can be implemented on a digital computer and its applicability to a wide variety of source waveforms.
A new technique recently used for solution of the plane‐wave Burgers' equation in a lossless medium has led to the development of an approximate analytical solution to Burgers' equation with dissipation for plane, cylindrical, and spherical sinusoidal waves. The solution is valid for the initial propagation zone prior to sawtooth formation for Gol'dberg numbers larger than 5 as indicated by comparison with a previous numerical solution of Burgers' equation for a plane wave. The approximate solution is expressed as a series that converges rapidly enough to permit calculations of harmonic levels by hand. For more viscous cases, where the approximate solution fails, the above technique leads easily to a numerical solution of Burgers' equation. By comparing the harmonic levels predicted by the numerical solution with empirical data taken on distorting spherical waves, it is concluded that numerical results tend to be below the measured values. It is felt that the numerical method can be of use in predicting distortion levels for practical underwater systems that exploit finite‐amplitude effects.
This Letter is an extension of an earlier Letter by Basset al., ‘‘Atmospheric absorption of sound: Update’’ [J. Acoust. Soc. Am. 88, 2019–2021 (1990)]. Errors in a formula for saturation vapor pressure are corrected, and an alternative, much simpler formula is given. The role of atmospheric pressure is emphasized by giving formulas in which the absorption, frequency, and relative humidity are all scaled with respect to atmospheric pressure. Also presented are new, more readable and useful figures showing atmospheric absorption as a function of frequency, relative humidity, and atmospheric pressure. The new figures make it possible to estimate accurately the absorption at any value of atmospheric pressure.
Thesis (Ph. D.)--University of Texas at Austin, 1995. Vita. Includes bibliographical references (leaves 205-216).
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- C L Morfey