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Apr 1985

Volume 77, Issue S1, pp. S1-S108

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back to top Session V. Physical Acoustics IV: Nonlinear Acoustics
Contributed Papers
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An experimental study of the effects of turbulence on the rise time of shock waves propagation in the atmosphere (A)

Bruce A. Layton, Henry E. Bass, and Lee N. Bolen

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S46-S46 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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As a shock wave propagates through the atmosphere various effects change the waveform. Nonlinear steepening tends to shorten the rise time of the leading and trailing edges of the shock. Frequency dependent dissipation and scattering from atmospheric turbules tend to lengthen the rise time. Simultaneous measurements of turbulence (i.e., wind velocity and temperature fluctuations) and shock waveforms resulting from supersonic projectiles were obtained and analyzed. The shock wave measurements required dielectric capacitive microphones specifically designed for this application. The simultaneous measurement of turbulence parameters and shock waveforms are the first reported and allow for the comparison between measured rise times and those computed assuming turbulence is the dominant mechanism affecting the rise time. The measured rise times are also compared to those computed considering only relaxation effects (ignoring turbulence). The rise times computed from those two different mechanisms are similar in magnitude; both agree reasonably well with measurements. [Work supported by ARO.]
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Long‐range nonlinear geometric acoustic pulse propagation in the atmosphere (A)

Stephen I. Warshaw and Paul F. Dubois

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S46-S46 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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Linear geometric acoustic ray tracing is useful for determining wavefront propagation in inhomogeneous and moving atmospheres. For propagation over many pulse lengths, or for large amplitude pulses, nonlinear advective, dissipative, and diffusive effects need to be accounted for. We have developed a simple, approximate modeling technique for theoretically predicting small amplitude pulse propagation over long distances in the terrestrial atmosphere, which takes these nonlinear effects into account. It consists of extending Blokhintsev's development [D. Blokhintzev, J. Acoust. Soc. Am. 18, 322 (1946)] into the nonlinear regime by applying classical Riemann invariant methods locally to one‐dimensional propagation of acoustic pulses along geometric acoustic ray tubes, and including viscosity and heat conductivity effects in an elementary way [S. I. Warshaw, LLNL Rep. UCRL‐53055 (1980)]. The advantages and disadvantages of this modeling technique will be briefly discussed. Predictions of the propagation of the pulse and wave front from a surface explosion to ionospheric altitudes based on this approach will be compared with ionospheric radar measurements made during an actual explosion [D. J. Simons et al., EOS Trans. Am. Geophys. Union 62, 979 (1981)]. The agreement between predicted and observed amplitudes, durations, and shapes of the pulse in the ionosphere is good. [Work performed under auspices of U.S. Department of Energy by LLNL under Contract W‐7405‐Eng‐48.]
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Second harmonic of a finite amplitude Gaussian beam in a fluid (A)

Gonghuan Du and M. A. Breazeale

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S46-S46 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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The diffraction of a Gaussian ultrasonic beam in absorbing fluids has been shown to produce a Gaussian field distribution in the fundamental component [J. Acoust. Soc. Am. Suppl. 1 76, S23 (1984)]. Now we have derived an analytical expression for the second harmonic and find that it also is a Gaussian function in the quasilinear approximation. Our expressions for the second harmonic also are free of the nearfield oscillations characteristic of the radiation from a piston transducer [cf. S. I. Aanonsen, T. Barkve, J. N. Tjøtta, and S. Tjøtta, J. Acoust. Soc. Am. 75, 749 (1984)]. Calculated results are compared with measurements taken with the Gaussian transducer described previously. Reasonable agreement between theory and experiment are found for both the fundamental and the second harmonic of an initially sinusoidal Gaussian ultrasonic beam in water at 2 MHz. [Research supported in part by the Office of Naval Research.]
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Use of the Burgers equation to model the farfield of finite amplitude sound beams (A)

Mark F. Hamilton, Jacqueline Naze Tjøtta, and Sigve Tjøtta

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S46-S46 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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The Kuznetsov equation has been solved numerically and shown by comparison with experiments to adequately model the nonlinear propagation of directional sound beams in dissipative fluids [S. I. Aanonsen, M. F. Hamilton, J. Naze Tjøtta, and S. Tjøtta, 10th ISNA, Kobe (1984)]. However, such numerical computations are generally quite time consuming, particularly in the farfield where asymptotic formulas are frequently applied. The simplest equation that accounts consistently for nonlinearity and absorption in spherical wave propagation is the Burgers equation. It is shown by way of multiple scale expansion that the Kuznetzov equation is in the farfield asymptotically equivalent to the Burgers equation. Numerical solutions of the Kuznetsov equation show that when an appropriate matching condition is imposed at the Rayleigh distance, the Burgers equation adequately describes the axial field when nonlinearity is not too high. Off axis, however, significant nonlinear effects generated in the nearfield can propagate out to tens of Rayleigh distances. Not until these nearfield effects become insignificant does the Burgers equation correctly predict beam patterns. [Support of MFH was provided by the F. V. Hunt Post‐doctoral Research Fellowship.]
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Measurement of harmonics generated by finite amplitude sound radiated by a circular piston in water (A)

Thomas L. Riley

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S46-S46 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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A circular piston with diameter selectable as 25, 50, or 76 mm (with values of ka of about 25, 50, and 75) was used to radiate 470‐kHz sound in a fresh water lake. Source levels ranged from 195–230 dB re: 1 μPa at 1 m. Beam patterns and axial sound pressure levels at about 7 and 11 m were recorded for the fundamental through fourth harmonic frequencies. The data were used to draw amplitude response curves and curves of beamwidth versus source level. Predictions based on two models of finite amplitude sound radiated by a piston, the equivalent spherical source model [J. A. Shooter, T. G. Muir, and D. T. Blackstock, J. Acoust. Soc. Am. 55, 54–62 (1974)], and the quasilinear approximation model [J. N. Tjøtta and S. Tjøtta, J. Acoust. Soc. Am. 67, 484–490 (1980)], agree well with the data. [Work supported by Office of Naval Research.]
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Matched asymptotic expansions for steady second‐order effects in acoustics (A)

Richard S. McGowan

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S47-S47 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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Matched asymptotic expansions have been used to solve linear scattering problems for farfield information in the long wavelength limit [D. G. Crighton and F. G. Leppington, Proc. R. Soc. London Ser. A 335, 319–339 (1973)]. They can also supply the necessary nearfield information to solve for the streaming field and the steady radiation pressure. The viscous boundary layer is an important component for streaming. Matched asymptotic expansions can be used here to develop a higher order acoustic boundary‐layer theory. This helps to separate forcing terms responsible for streaming and radiation pressure in the boundary‐layer region. Examples of application include streaming over a blunt end of a slab, viscous effects on steady pressure, and the effect of wave curvature on radiation pressure. Also, an investigation of the importance of imposed mean field gradients is presented. [Work supported by ONR.]
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Propagation and attenuation of intense tones in air‐filled porous materials (A)

D. A. Nelson and D. T. Blackstock

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S47-S47 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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In a previous presentation [J. Acoust. Soc. Am. Suppl. 1 74, S59 (1983)] it was demonstrated that the nonlinear mechanism for propagation in bulk porous material is an amplitude‐dependent resistivity of the form r  =  r1 + r2u sgn(u), where u is the instantaneous particle velocity and r1 and r2 are constants that depend on porosity and material geometry. An approximate Helmholtz equation was derived that governs the change in signal harmonies with distance. Recently obtained numerical solutions of this equation compare favorably with experimental data. By assuming that harmonic interaction is weak, we have obtained analytical solutions for the amplitude and phase of the fundamental. It is found that the attenuation increases at high intensity, while at the same time the phase velocity decreases. Experiments have been done with Kevlar® 29 in the porosity range 0.94 to 0.98, in the frequency range 500–1500 Hz, at SPLs up to 165 dB. The experimental data confirm the predictions. [Work supported by NASA and ONR.]
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Derivation of Planck's radiation law and thermodynamic state functions from stochastic nonlinear acoustic fields (A)

John H. Cantrell, Jr.

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S47-S47 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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The Boltzmann‐Ehrenfest adiabatic invariance formalism used to derive the acoustic radiation stress equation for solids [J. H. Cantrell, Jr., Phys. Rev. B 30, 3214 (1984)] is combined with recent developments in stochastic classical dynamics to obtain the internal and Helmholtz free energies in terms of random zero‐point nonlinear acoustic modes. The results lead to an expression of the thermal expansion coefficients of crystalline solids in terms of nonlinearity parameters related directly to the acoustic radiation‐induced static strains. When the model acoustic non‐linearity parameters are set to zero, the internal energy expresion reduces to the Planck radiation law obtained from quantum mechanics. If, in addition, the quasiharmonic assumption is invoked for the model frequencies, the thermal expansion equation reduces to that obtained from Debye‐Grüneisen‐Einstein statistical model of a system of quantum oscillators.
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Finite amplitude distortion and dispersion in a hard‐walled waveguide (A)

J. H. Ginsberg and H. C. Miao

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S47-S47 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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The fundamental symmetric two‐dimensional mode in a hard‐walled rectangular waveguide is decomposed into a pair of obliquely propagating planar waves, in order to treat the effect of nonlinearity. A perturbation analysis of the nonlinear wave equation for the velocity potential identifies amplitude dispersion as one source of singularity. The only interaction between the oblique waves attributable to this effect is a change in the distance parameter affecting the magnitude of the higher harmonics. Another singularity arises when the frequency ω or width L is large. The oblique waves are then closely aligned with the axis, resulting in resonant interaction with the true planar mode. Harmonic generation in this case has the appearance of a spatial beating pattern. A set of coordinate transformations make the representation uniformly valid. Analyses of limiting forms are confirmed by quantitative examples. Small values of ωL are well described by an earlier general solution in terms of groups of nondispersive modes [J. H. Ginsberg, J. Acoust. Soc. Am. 65, 1127–1133 (1979)], while large ωL yields a quasiplanar signal. The transition at moderate ωL is characterized by frequency, as well as amplitude, dispersion. The distortion of waveforms then is very close to that obtained in the nearfield of sound beams. [Work supported by ONR.]
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Higher‐order finite amplitude modes in a rectangular waveguide (A)

Mark F. Hamilton and James A. TenCate

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S47-S47 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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Finite amplitude propagation of higher‐order modes in a rectangular waveguide is analyzed by decomposing the modes into plane waves. Two types of nonlinear interactions may then be considered. The self interaction of an individual plane wave generates harmonics that propagate in the same direction. Such interactions are unaffected by dispersion because each harmonic propagates at the same speed, although in a different mode. The second type includes noncollinear interactions between plane waves. In this case geometric dispersion prevents efficient transfer of energy between the interacting components. A single transverse mode excited at a frequency not far from cutoff is composed of two plane waves propagating in very different directions. The noncollinear interactions are then so highly dispersive that as a first approximation they may be ignored. The remaining, nondispersive interactions were modeled using a modified Burgers equation that accounts for tube wall absorption of each mode. Theoretical results for this case agree very well with experiment. If the fundamental frequency is well above cutoff, dispersive interactions can no longer be ignored. Experimental waveforms then resemble those observed within finite amplitude sound beams. [Work supported by ONR.]
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Production of skewness by coupling of nonlinearity and frequency dispersion (A)

D. G. Crighton

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S47-S47 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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In full scale jet noise experiments it has been observed that the skewness factor S = 〈p3〉/〈p2〉3/2 of the pressure fluctuations increases from near zero values near the source to positive values as high as 0.8 in the farfield, with a distinctive change in the typical waveform to one containing high, spiky positive pressure rises followed by more gradual and smaller negative pressure excursions. Further, the value of S at a fixed range and identical running conditions is highly sensitive to meteorological conditions. This paper presents a simple model—based on coupling between nonlinear acoustic distortion and linear frequency dispersion—capable of explaining these observations. Such coupling will be shown to always produce positive skewness, increasing with range, and extremely sensitive to the difference between frozen and equilibrium sound speeds, and hence to atmospheric humidity levels.
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Hydrodynamic attenuation of weak shock waves (A)

W. D. Curtis and C. E. Rosenkilde

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S47-S48 (1985); (2 pages)

Online Publication Date: 12 Aug 2005

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A pair of ordinary differential equations is derived, giving the shock amplitude and pressure gradient at the front as functions of range, for a weak shock propagating in a homogeneous medium. These equations enable us to complete a discussion originally given by G. I. Taylor in his work on the decay of blast waves. It is shown that the equations can be solved asymptotically at large ranges to give the well‐known results for amplitude decay and pulse spreading for plane, cylindrical, and spherical week shocks. [This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. W‐7405‐Eng‐48.]
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Nonlinear stability analysis of a Stokes boundary layer (A)

Charles Thompson and Shawn Burke

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S48-S48 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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The stability of the Stokes boundary layer generated by an acoustic disturbance in a two‐dimensional waveguide having a slowly varying height is examined. It is shown that the stability of the Stokes layer to three‐dimensional disturbances is governed by the value of three small parameters ϵ, 1/R, and 1/S, where ϵ is the wall slope, 1/R is the ratio of the oscillatory boundary layer thickness to the typical duct height, and 1/S is the ratio of the oscillatory particle displacement to the duct height. A stability analysis is presented for the amplitude range ϵ2R/S2  =  O(math). It is shown that instability to infinitesimal disturbances occurs when h″ϵ2math/S2 is greater than 49.3. As the disturbance amplitude becomes finite in value the solution bifurcates from that obtained using linear stability theory. We will show that the first bifurcation satisfies the equation d2A/dx2 + (γ1x + T1γ2)A  =  γ3A3, where A is the disturbance amplitude and, γ1, γ2, and γ3 are parameters dependent on the local behavior of acoustic wave as well as the duct geometry. [Work supported by the National Science Foundation.]
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Nonlinearity parameters from measurement of acoustic radiation‐induced static strains in silicon (A)

W. T. Yost, John H. Cantrell, Jr., and Peter Li

J. Acoust. Soc. Am. Volume 77, Issue S1, pp. S48-S48 (1985); (1 page)

Online Publication Date: 12 Aug 2005

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The radiation stress associated with a finite amplitude acoustic wave propagating in a solid generates a static strain having an amplitude directly proportional to the energy density of the propagating wave. The coupling coefficient in the static strain‐energy density equation is the acoustic nonlinearity parameter. We have determined the nonlinearity parameters in single crystal silicon along the pure mode propagation directions from acoustic radiation stress measurements. We obtain the values 2.12 ± 0.36 along (100), 4.3 ± 0.72 along (110), and 3.87 ± 0.58 along(111). These values are in good agreement with independent measurements obtained from harmonic generation as well as from stress derivative experiments.
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