Engineering topics

Pierce, Allan D.

Publications and source records attributed to Pierce, Allan D..

18 records

NASA NTRS1996

Turbulence Scales, Rise Times, Caustics, and the Simulation of Sonic Boom Propagation

The general topic of atmospheric turbulence effects on sonic boom propagation is addressed with especial emphasis on taking proper and efficient account of the contributions of the portion oi the turbulence that is associated with extremely high wavenumber components. The recent work reported by Bart Lipkens in his doctoral thesis is reexamined to determine whether the good agreement between his measured rise times with the 1971 theory of the author is fortuitous. It is argued that Lipken's estimate of the distance to the first caustic was a gross overestimate because of the use of a sound speed correlation function shaped like a gaussian curve. In particular, it is argued that the expected distance to the first caustic varies with the kinematic viscosity nu and the energy epsilon dissipated per unit mass per unit time, and the sound speed c as : d(sub first caustic) = nu(exp 7/12) c(exp 2/3)/ epsilon(exp 5/12)(nu x epsilon/c(exp 4))(exp a), where the exponent a is greater than -7/12 and can be argued to be either O or 1/24. In any event, the surprising aspect of the relationship is that it actually goes to zero as the viscosity goes to zero with s held constant. It is argued that the apparent overabundance of caustics can be grossly reduced by a general computational and analytical perspective that partitions the turbulence into two parts, divided by a wavenumber k(sub c). Wavenumbers higher than kc correspond to small-scale turbulence, and the associated turbulence can be taken into account by a renormalization of the ambient sound speed so that the result has a small frequency dependence that results from a spatial averaging over of the smaller-scale turbulent fluctuations. Selection of k(sub c). can be made so large that only a very small number of caustics are encountered if one adopts the premise that the frequency dispersion of pulses is caused by that part of the turbulence spectrum which lies in the inertial range originally predicted by Kolmogoroff. The acoustic propagating wave's dispersion relation has the acoustic wavenumber being of the form k = (omega/c) + F(omega), where c is a spatially averaged sound speed and where, for mechanical turbulence, the extra term F(omega) must depend on only the angular frequency omega, the sound speed c, and the turbulent energy dissipation epsilon per unit fluid mass and per unit time. If the turbulence is weak, then the quantity F(omega) has to be of second order in the portions of the turbulent fluid velocity in the inertial range, so, following Kolmogoroff's reasoning, it must vary with epsilon as epsilon(exp 2/3). Simple dimensional analysis then reveals that F(omega) is K epsilon(exp 2/3) c(exp -7/3) omega(exp l/3), K being a universal dimensionless complex constant.

Pierce, Allan D.↗

NASA NTRS1994

Atmospheric propagation at larger lateral distances from the flight track

Sonic booms received on the ground tend to be restricted to a region of finite lateral extent below the flight track. This occurs because of refraction and because the effective speed of sound, even with winds taken into account, decreases with altitude in the lower atmosphere. Not all rays proceeding initially downwards from the flight track within an allowable range of initial directions will reach the ground. The restricted region which can be reached by rays impacting the ground is known as the primary carpet. However, weak rumbles are heard in the nominal shadow zone beyond the edge of this carpet. A full wave theory is necessary for explaining waveforms in that region, and the present paper gives a matched asymptotic expansion technique for a suitable approximate full wave theory that involves a relatively small number of parameters. The outer solution is derived from the structure of the system of rays that impact near the corridor edge; the inner solution involves a solution of the parabolic equation and results in the special functions encountered in the diffraction of sound over the tops of hills.

Pierce, Allan D.↗

NASA NTRS1994

Progress in modeling atmospheric propagation of sonic booms

The improved simulation of sonic boom propagation through the real atmosphere requires greater understanding of how the transient acoustic pulses popularly termed sonic booms are affected by humidity and turbulence. A realistic atmosphere is invariably somewhat turbulent, and may be characterized by an ambient fluid velocity v and sound speed c that vary from point to point. The absolute humidity will also vary from point to point, although possibly not as irregularly. What is ideally desired is a relatively simple scheme for predicting the probable spreads in key sonic boom signature parameters. Such parameters could be peak amplitudes, rise times, or gross quantities obtainable by signal processing that correlate well with annoyance or damage potential. The practical desire for the prediction scheme is that it require a relatively small amount of knowledge, possibly of a statistical nature, concerning the atmosphere along, the propagation path from the aircraft to the ground. The impact of such a scheme, if developed, implemented, and verified, would be that it would give the persons who make planning decisions a tool for assessing the magnitude of environmental problems that might result from any given overflight or sequence of overflights. The technical approach that has been followed by the author and some of his colleagues is to formulate a hierarchy of simple approximate models based on fundamental physical principles and then to test these models against existing data. For propagation of sonic booms and of other types of acoustic pulses in nonturbulent model atmospheres, there exists a basic overall theoretical model that has evolved as an outgrowth of geometrical acoustics. This theoretical model depicts the sound as propagating within ray tubes in a manner analogous to sound in a waveguide of slowly varying cross-section. Propagation along the ray tube is quasi-one-dimensional, and a wave equation for unidirectional wave propagation is used. A nonlinear term is added to this equation to account for nonlinear steepening, and the formulation has been carried through to allow for spatially varying sound speed, ambient density, and ambient wind velocities. The model intrinsically neglects diffraction, so it cannot take into account what has previously been mentioned in the literature as possibly important mechanisms for turbulence-related distortion. The model as originally developed could predict an idealized N-waveform which often agrees with data in terms of peak amplitude and overall positive phase duration. It is possible, moreover, to develop simple methods based on the physics of relaxation processes for incorporating molecular relaxation into the quasi-one-dimensional model of nonlinear propagation along ray tubes.

Pierce, Allan D.↗

NASA NTRS1993

Progressive wave equations and algorithms for sonic boom propagation

Raspet, Bass, and others have carried out a number of studies of sonic boom propagation using algorithms that can be traced back to a 1973 doctoral thesis written by Pestorius at the University of Texas. The algorithm was originally set forth without a formal derivation from a governing set of partial differential equations. Perusal of Pestorius's report and of subsequent reports from the Texas nonlinear accoustics group suggests that such a derivation has not yet appeared in the archival literature. The author and his colleagues, on the other hand, have been working with an explicit set of approximate partial differential equations analogous to Burgers' equation. The present paper traces through the pertinent statements in the Pestorius reports and shows that the derived algorithm, at least in the limit of sufficiently short steps, is equivalent to the numerical solution of a similar (and nearly equivalent) explicit set of partial differential equations. The tying-together of the two approaches is important because it is possible that some of the basic ideas embodied in the Pestorius algorithm may be incorporated in other contexts where more nearly general models of sonic boom propagation through realistic atmospheres are to be solved numerically.

Pierce, Allan D.↗

NASA NTRS1992

Wave equations and computational models for sonic boom propagation through a turbulent atmosphere

The improved simulation of sonic boom propagation through the real atmosphere requires greater understanding of how the transient acoustic pulses popularly termed sonic booms are affected by atmospheric turbulence. A nonlinear partial differential equation that can be used to simulate the effects of smaller-scale atmospheric turbulence on sonic boom waveforms is described. The equation is first order in the time derivative and involves an extension of geometrical acoustics to include diffraction phenomena. Various terms in the equation are explained in physical terms. Such terms include those representing convection at the wave speed, diffraction, molecular relaxation, classical dissipation, and nonlinear steepening. The atmospheric turbulence enters through an effective sound speed, which varies with all three spatial coordinates, and which is the sum of the local sound speed and the component of the turbulent flow velocity projected along a central ray that connects the aircraft trajectory with the listener.

Pierce, Allan D.↗

NASA NTRS1992

Simulations of sonic boom ray tube area fluctuations for propagation through atmospheric turbulence including caustics via a Monte Carlo method

A theory which gives statistical predictions for how often sonic booms propagating through the earth's turbulent boundary layer will encounter caustics, given the spectral properties of the atmospheric turbulence, is outlined. The theory is simple but approximately accounts for the variation of ray tube areas along ray paths. This theory predicts that the variation of ray tube areas is determined by the product of two similar area factors, psi (x) and phi (x), each satisfying a generic harmonic oscillator equation. If an area factor increases the peak acoustic pressure decreases, and if the factor decreases the peak acoustic pressure increases. Additionally, if an area factor decreases to zero and becomes negative, the ray has propagated through a caustic, which contributes a phase change of 90 degrees to the wave. Thus, it is clear that the number of times that a sonic boom wave passes through a caustic should be related to the distorted boom waveform received on the ground. Examples are given based on a characterization of atmospheric turbulence due to the structure function of Tatarski as modified by Crow.

Sparrow, Victor W.↗

NASA NTRS1992

Relaxation and turbulence effects on sonic boom signatures

The rudimentary theory of sonic booms predicts that the pressure signatures received at the ground begin with an abrupt shock, such that the overpressure is nearly abrupt. This discontinuity actually has some structure, and a finite time is required for the waveform to reach its peak value. This portion of the waveform is here termed the rise phase, and it is with this portion that this presentation is primarily concerned. Any time characterizing the duration of the rise phase is loosely called the 'rise time.' Various definitions are used in the literature for this rise time. In the present discussion the rise time can be taken as the time for the waveform to rise from 10 percent of its peak value to 90 percent of its peak value. The available data on sonic booms that appears in the open literature suggests that typical values of shock over-pressure lie in the range of 30 Pa to 200 Pa, typical values of shock duration lie in the range of 150 ms to 250 ms, and typical values of the rise time lie in the range of 1 ms to 5 ms. The understanding of the rise phase of sonic booms is important because the perceived loudness of a shock depends primarily on the structure of the rise phase. A longer rise time typically implies a less loud shock. A primary question is just what physical mechanisms are most important for the determination of the detailed structure of the rise phase.

Pierce, Allan D.↗

NASA NTRS1990

Weak shock propagation through a turbulent atmosphere

Consideration is given to the propagation through turbulence of transient pressure waveforms whose initial onset at any given point is an abrupt shock. The work is motivated by the desire to eventually develop a mathematical model for predicting statistical features, such as peak overpressures and spike widths, of sonic booms generated by supersonic aircraft. It is argued that the transient waveform received at points where x greater than 0 will begin with a pressure jump and a formulation is developed for predicting the amount of this jump and the time derivatives of the pressure waveform immediately following the jump.

Pierce, Allan D.↗

NASA NTRS1990

The effect of molecular relaxation processes in air on the rise time of sonic booms

A theory is developed to explain the effect of molecular relaxation processes on the rise time of sonic booms. To determine the rise time of sonic booms, both O2 and N2 relaxation processes must be included. The N2 relaxation process delays the shock pressure reaching the maximum pressure, and the O2 relaxation process causes a shock profile to have a gentle slope. The N2 relaxation controls the lower part of overpressure; the O2 relaxation controls the higher part. The constant rise time curves show that the rise times increase as the overpressures and humidity decrease. The present approach gives longer rise times than those acquired by Bass et al. for given shock overpressures.

Kang, Jongmin↗

NASA NTRS1989

Status of sonic boom methodology and understanding

In January 1988, approximately 60 representatives of industry, academia, government, and the military gathered at NASA-Langley for a 2 day workshop on the state-of-the-art of sonic boom physics, methodology, and understanding. The purpose of the workshop was to assess the sonic boom area, to determine areas where additional sonic boom research is needed, and to establish some strategies and priorities in this sonic boom research. Attendees included many internationally recognized sonic boom experts who had been very active in the Supersonic Transport (SST) and Supersonic Cruise Aircraft Research Programs of the 60's and 70's. Summaries of the assessed state-of-the-art and the research needs in theory, minimization, atmospheric effects during propagation, and human response are given.

Darden, Christine M.↗

NASA NTRS1989

The Helmholtz-Kirchhoff integral relation as a framework for developing algorithms for sound propagation through inhomogeneous moving media

Transient sound propagation in an inhomogeneous moving medium is considered. For circumstances in which the medium is slowly varying over distances of a wavelength but possibly varying substantially over the propagation distance, a derivation is given of a new wave equation, which implicitly allows for diffraction and scattering and which also is consistent with earlier geometrical acoustics formulations. This wave equation is used as a starting point to derive a version of the Helmholtz-Kirchhoff integral relation that applies to inhomogeneous moving medium. It is suggested that a good approximation to the Green's function that appears in this relation is that derived from geometrical acoustics, the approximation becoming progressively better the shorter the distance between surfaces in the computation. It should also be at least as good as conventional ray acoustics, but can account for diffraction effects, such as at caustics.

Pierce, Allan D.↗

NASA NTRS1988

Sound propagation over uneven ground and irregular topography

Theoretical, computational, and experimental techniques were developed for predicting the effects of irregular topography on long range sound propagation in the atmosphere. Irregular topography is understood to imply a ground surface that: (1) is not idealizable as being perfectly flat, or (2) that is not idealizable as having a constant specific acoustic impedance. The focus is on circumstances where the propagation is similar to what might be expected for noise from low altitude air vehicles flying over suburban or rural terrain, such that rays from the source arrive at angles close to grazing incidence.

Berthelot, Yves H.↗

NASA NTRS1988

Sound propagation over uneven ground and irregular topography

The goal of this research is to develop theoretical, computational, and experimental techniques for predicting the effects of irregular topography on long range sound propagation in the atmosphere. Irregular topography is understood to imply a ground surface that is not idealizable as being perfectly flat or that is no idealizable as having a constant specific acoustic impedance. The focus is on circumstances where the propagation is similar to what might be expected for noise from low-altitude air vehicles flying over suburban or rural terrain, such that rays from the source arrive at angles close to grazing incidence.

Berthelot, Yves H.↗

NASA NTRS1987

Experiments on the applicability of MAE techniques for predicting sound diffraction by irregular terrains

The sound field diffracted by a single smooth hill of finite impedance is studied both analytically, within the context of the theory of Matched Asymptotic Expansions (MAE), and experimentally, under laboratory scale modeling conditions. Special attention is given to the sound field on the diffracting surface and throughout the transition region between the illuminated and the shadow zones. The MAE theory yields integral equations that are amenable to numerical computations. Experimental results are obtained with a spark source producing a pulse of 42 microsec duration and about 130 Pa at 1 m. The insertion loss of the hill is inferred from measurements of the acoustic signals at two locations in the field, with subsequent Fourier analysis on an IBM PC/AT. In general, experimental results support the predictions of the MAE theory, and provide a basis for the analysis of more complicated geometries.

Berthelot, Yves H.↗

NASA NTRS1987

Sound propagation over uneven ground and irregular topography

The acoustic impedance of the surface coverings used in the laboratory experiments on sound diffraction by topographical ridges was determined. The model, which was developed, takes into account full wave effects and the possibility of surface waves and predicts the sound pressure level at the receiver location relative to what would be expected if the flat surface were not present. The sound pressure level can be regarded as a function of frequency, sound speed in air, heights of source and receiver, and horizontal distance from source to receiver, as well as the real and imaginary parts of the surface impedance.

Berthelot, Yves H.↗

NASA NTRS1987

Sound propagation over uneven ground and irregular topography

The goal of this research is to develop theoretical, computational, and experimental techniques for predicting the effects of irregular topography on long range sound propagation in the atmosphere. Irregular topography here is understood to imply a ground surface that is not idealizable as being perfectly flat or that is not idealizable as having a constant specific acoustic impedance. The interest of this study focuses on circumstances where the propagation is similar to what might be expected for noise from low-attitude air vehicles flying over suburban or rural terrain, such that rays from the source arrive at angles close to grazing incidence. The activities and developments that have resulted during the period, August 1986 through February 1987, are discussed.

Berthelot, Yves H.↗

NASA NTRS1987

Diffraction of sound by a smooth ridge

The propagation of sound over uneven terrain and irregular topography is considered. Laboratory scale experiments were conducted on a 4.88 x 2.44 m plywood bench top with a cylindrical ridge whose radius of curvature is about 2.5 m. Experimental results include the insertion loss on the diffracting surface, and in the penumbra region, at several distances from the apex of the ridge.

Berthelot, Yves H.↗

NASA NTRS1986

Sound propagation over curved barriers

Wide barriers with curved tops are studied with emphasis placed on circumstances whereby the local radius of curvature R of the barrier is continuous along the surface and is large compared to a wavelength. Results analogous to those given by Hayek et al. (1978) are reviewed and extended to cases where the radius of curvature and the surface impedance may vary with position. Circumstances not easily interpreted within the framework of the model proposed by Keller (1956) and Hayek et al. are also considered.

Pierce, Allan D.↗