Linear Acoustic Theory Research Articles

Shock wave formation in the exhaust pipe of an internal combustion engine.

In automobile exhaust systems, compression waves develop into shock waves under a certain condition and then seem to generate large and jarring exhaust noises. This paper describes the mechanism of the transition of compression waves into shock waves in detail. In the present experiment, a periodic wave generator was designed and constructed for this purpose. Pressure measurements were carried out along the straight pipe connected to the wave generator in order to observe compression waves developing into shock waves. The numerical simulation was also conducted using the TVD scheme. The results were compared with the present experiment. A good agreement was obtained between the numerical simulation and the experiment. Subsequently it was found that the entropy is not conserved near shock wave formation and the application of a linear acoustic theory is thus restricted in this case.

Shocks and singularities in the pressure field of a supersonically rotating propeller

When linear acoustic theory is applied to the thickness noise problem of a supersonic propeller, it can give rise to a surface on which the pressure is discontinuous or singular. A method is described for obtaining the equation of this surface (when it exists), and the pressure field nearby; jumps, logarithms and inverse square roots occur, and their coefficients may be calculated exactly. The special case of a blade with a straight radial edge gives a cusped cone, whose sheets, each with a different type of discontinuity or singularity in pressure, are separated by lines of cusps; the coefficients in formulae for the pressure near the surface tend to infinity as a cusp line is approached, in proportion to the inverse quarter power of distance from the line. These results determine regions of space where nonlinear effects are important, and they suggest a strong analogy with sonic boom.

Validity of linear acoustics for prediction of waveforms caused by sonically moving laser beams

The question is raised as to whether the analysis of the generation of sound by a laser beam moving over a water surface at the sound speed c for an interminable time period requires consideration of nonlinear effects. A principal consideration in this regard is whether the linear acoustics theory predicts a pressure waveform that is bounded in the asymptotic limit when the laser irradiation time is arbitrarily large. It is shown that a bounded asymptotic limit exists when the upper boundary condition corresponds (as is more nearly appropriate) to that of a pressure release surface, but not when it corresponds to that of a rigid surface. The asymptotic solution to the appropriate inhomogeneous wave equation is given exactly for the former case, and it is shown that the highest asymptotic amplitudes, given specified laser power and beam radius a, occur in the limit of a very small light absorption coefficient mu. In this limit, the peak amplitude is independent of mu and occurs at a depth of 0.88/mu. An approximate solution for the pressure waveform at intermediate times establishes that the characteristic time for buildup to the asymptotic limit is of the order of 2.5/(c mu 2a). If this time is substantially shorter than the time that a plane-wave pulse with the asymptotic waveform would take to develop a shock wave, then accumulative nonlinear effects are of minor importance.

Effect of ambient pressure on the pressure Wave from the rapidly expanding bubble

Calculation on the intensity of evaporation and the rapidly expanding bubble formed from the droplet at the superheat limit is calculated. An analysis is presented for the amplitude of the pressure wave generated by the rapidly expanding bubble. For estimating the amplitude, linear acoustics theory for the evaporating droplet and the Rayleigh equation for the expanding bubble are employed. The result shows that the amplitude and the frequency of the pressure wave depend crucially on the difference in pressure between vapor within the bubble and the surrounding liquid. This result is in qualitative agreement with previous experiment.

Sources and observers in motion, III: Acoustic radiation from non-compact rigid bodies moving at high speed

This paper analyzes the linear acoustic solution for the radiation from a non-compact rigid body in motion. The concept and onset of source non-compactness is considered in relation to a non time-varying source, and the effect of finite source distributions investigated. An efficient method is described which calculates the effect of finite chord and span distributions typical of helicopter and propeller rotors. Finally, a linear acoustic theory with discontinuous boundary conditions is used to explain non-linear wave steepening (shock waves).

Vapour Cloud Explosions-The Directional Blast Wave from an Elongated Cloud with Edge Ignition

Abstract A simple model has been developed for the calculation of overpressure generated by the rapid deflagration of an elongated vapour cloud with ignition at one end. By means of linear acoustic theory, analytic solutions have been derived for the wavefield. Thus the principal features of the pressure field—a marked directionality for high-speed deflagrations and a very strong hydrodynarnic (and flow) field local to the cloud—are shown in much greater clarity than in the recently published numerical results of other workers. The overpressures for an elongated cloud are, however, significantly lower than for a hemispherical cloud with central ignition for the same flame speed and cloud volume even in the least favourable direction.

An experimental investigation of two dimensional flame-vortex interactions

Experiments have been performed to study the interaction between a flame and a single vortex impulsively generated by the flame. The objectives are to investigate the amplitude of the cylindrically diverging acoustic wave produced by the interaction and to develop a quantitative approach to determine the degree of violence of the flame-vortex interaction. With the Reynolds number, defined as the ratio of the strength of the vortex to the kinematic viscosity of the unburned gas, characterizing the unburned gas flow condition quantitatively, the acoustic pressure Δ p(t) generated by the interaction has been measured at distances 1.5, 1.75 and 2.0 meters away from the interaction zone using a microcomputer-based data acquisition system and a storage oscilloscope. The dimensionless time rate of change in the heat addition rate, Q( t ), of the flame-vortex interaction, is deduced from the acoustic pressure data via the line source equation in linear acoustic theory. The Fast Fourier Transform has been used to perform the analysis. For the thirty experiments performed, the amplitudes of Q( t ) and Δ p(t) have been found to be dependent on Re, burning speed and two dimensionless geometric parameters. The results show that Q max increases with Re when the geometric parameters are fixed. An empirical logarithmic equation has been obtained to describe the variation of Q max for Re between 10 5 and 4.5×10 5 . The maximum Δ p max recorded was 72 mbar at 1.5 m from the interaction zone. The increase in the “violence” of the interaction has been found to be associated with an increase in the heat addition rate during the period of flame-vortex interaction.

Numerical investigations of nonideal explosions

This study focusses on nonideal free-field blast waves arising from the finite rate of energy addition in a distributed nonspherical source region. The behavior of the blast wave is determined in a compressible medium surrounding a centrally ignited flammable mixture during and after the propagation of a heat addition wave which models the detonative or deflagrative combustion process. Numerical integration of the nonsteady, two-dimensional Euler equations is performed by incorporating a heat addition “two-gamma” working fluid model to represent the flame or detonation wave. This study also investigated the behavior of a blast wave generated by the burst of a high-pressure ellipsoid. The effect of the heat addition waves on the near and far field blast waves was studied by determining the relevant blast parameters, peak overpressure, and positive phase impulse, as well as the detailed evolution of the gas dynamic variables with time and distance. The blast parameters which correspond to low velocity flames compare favorably with those obtained using linear acoustic monopole theory. The results indicate that pressure relief associated with the presence of a compressible inert medium surrounding the combustion products after the flame propagates to the interface of nonspherical clouds sevrely restricts the intensity of the explosion and thereby reduces the damages that can be produced by the blast wave.

Weak spherical shock-wave transitions of N-waves in air with vibrational excitation

In order to clarify the effects of vibrational excitation on shock-wave transitions of weak, spherical N-waves, which were generated by using sparks and exploding wires as sources, the compressible Navier-Stokes equations were solved numerically, with a vibrational-relaxation equation for oxygen. An explosion from a small pressurized sphere filled with air was used to simulate the N-waves generated from the actual sources. By employing the random-choice method (r. c. m.) with an operator-splitting technique, the effects of artificial viscosity appearing in finite-difference schemes were eliminated and accurate profiles of the shock transitions were obtained. However, a slight randomness in the variation of the shock thickness remains. It is shown that a computer simulation is possible by using a proper choice of initial parameters to obtain the variations of N-wave overpressure and half-duration with distance from the source. The calculated rise times are also shown to simulate both spark and exploding-wire data. It was found that, in addition to the vibrational-relaxation time of oxygen, both the duration (N-wave effect) and the attenuation rate (non-stationary effect) of a spherical N-wave are important factors controlling its rise time. These effects are discussed in more detail in relation to Lighthill’s analytical solutions and the r. c. m. solutions for non-stationary plane waves and spherical N-waves. It is also shown that the duration and the attenuation rate of a spherical N-wave are affected by viscosity, and vibrational non-equilibrium, so that they can deviate from the results of classical, linear acoustic theory for very weak spherical waves.

Impact of a liquid mass on a perfectly plastic solid

The use of steady, normal and oblique shock configurations is explored in calculating the pressure and deformation produced by the impact of a liquid mass on a plane solid surface. Since pressures generated are very large, the change in bulk modulus of the liquid (water) is accounted for by using an equation of state following Field, Lesser & Davies (1979). The impact of a plane-ended liquid mass is analysed using a normal shock for the cases of a rigid surface and a perfectly plastic surface. For the former, it is found that pressures somewhat in excess of the ‘water-hammer’ pressure of linear acoustic theory are predicted, and for the latter there is a critical impact velocity below which no deformation occurs. Above this velocity the surface deforms at a constant rate, producing a pit with maximum depth at the centre.If the liquid mass is wedge-shaped then an oblique shock is formed, which is attached to the contact point provided that the impact Mach number is large enough, as originally shown by Heymann (1969). Pressure and deformation velocity can again be calculated for the cases of rigid and perfectly plastic surfaces respectively. For a rigid surface it is confirmed that pressures considerably in excess of the plane-ended case are produced at shock detachment. For the plastic surface, it is found that there is no critical impact velocity and deformation can occur at any velocity as shock detachment is approached. For a cylindrical liquid mass with a conical tip, the pit produced again has maximum depth at the centre, but with a considerably increased value. The possible use of these models for pitting caused by microjets associated with cavitation bubbles and by impact of liquid drops is discussed.

On linear acoustic solutions of high speed helicopter impulsive noise problems

The nature of linear acoustic solutions for a helicopter rotor blade with a blunt leading edge operating at high transonic tip Mach number is studied. As a part of this investigation a very efficient computation procedure for helicopter rotor blade thickness noise according to linear theory is developed. Numerical and analytical results reveal that as the blade tip Mach number approaches unity, the solution develops singularities and a radiating discontinuity. It is shown that these characteristic features are caused by the contributions of the higher harmonics which decrease in magnitude only as n 1 2 in the limit n → ∞. These higher harmonics are generated by the blunt leading edge. The far field wave form at sonic tip Mach number for a blade with a NACA 0012 airfoil section has a singularity of the inverse root type at its front and a logarithmic singularity near its end. Thus caution must be exercised in applying linear acoustic theory to high speed helicopter impulsive noise problems.

Linear Acoustic Model for the Exhaust Duct of a Pulsed Chemical Laser

The photo-initiated reaction in a typical pulsed chemical laser produces transient disturbances in the medium with peak pressure magnitudes on the order of 10 atmospheres and temperatures of up to several thousands of degrees Kelvin. The quality of the emitted beam is influenced by the chemical and optical homogeneity of the medium at the time of initiation; it is thus necessary to attentuate the reaction-generated disturbances in the short time between laser pulses. Typical requirements are reduction of disturbance magnitudes to the range of 10-3 to 10-5 atmospheres in times of 10-2 sec. While the initial disturbance is decidedly nonlinear, the acceptable perturbation levels are within the limits of traditional small disturbance theory, and complete analyses of the fluid dynamics of the clearing process must span the entire range of magnitudes. This paper deals with one end of that range, and describes a computational model, based on linear acoustic theory, that has been used to gain insight into the effects of conventional acoustic attenuation devices on laser cavity clearing times. The model developed is for one-dimensional wave propagation in channels with nonuniform geometry, nonuniform wall acoustic-impedance properties and nonuniform medium properties. The method of inner and outer asympotic expansions has been used to develop a transfer matrix computational model that produces a reaction-cavity reflection coefficient spectrum. Results are shown which illustrate the effects of the axial location of a slug of heated gas on the reflection coefficient spectrum of an acoustically-treated exhaust duct. The results suggest that the presence of temperature nonuniformities in the exhaust duct impede the rapid attenuation of disturbances from the reaction cavity. Pulsed laser devices are under development for applications requiring high power and high energy output. The major problems associated with pulsed chemical lasers involve reagent flow technology; the large amplitude, violent transient flow that results from a pulse must be cleared from the reaction

A note on sound radiation into a uniformly flowing medium

The influence of a uniform cross flow on the power output from an idealized mechanical source, namely a vibrating strip set in a coplanar rigid wall, is studied within the framework of linear acoustic theory, and the time-average power output is characterized by appropriate expansions (of exact integral representations), for both small and large wave length/strip width ratios, in the subsonic flow regime. Since boundary layers are ignored, the model source envisaged furnishes only a limited accounting of fluid-acoustical coupling effects.

On the singular behavior of linear acoustic theory in near-sonic duct flows

The propagation of sound in a converging-diverging duct containing a quasi-one-dimensional steady flow with a high subsonic throat Mach number is studied. The behavior of linearized acoustic theory at the throat of the duct is shown to be singular, and the explicit form of the singularity is determined for two special types of area variation. Numerically computed results showing the development of the singularity are presented. The singularity implies that linearized acoustic theory is invalid for sound propagating through a throat carrying a near-sonic flow.

The mechanical excitation of an acoustic resonance in a constant volume cavity

A standing half wave was excited in the column of ambient air within a long cylindrical tube, closed at both ends, by a longitudinal oscillation of the tube at the fundamental resonance frequency. For low amplitudes of oscillation the resonant behavior was adequately described by using linear acoustic theory. At high amplitudes shock waves appeared and the preliminary results generally agree with results for stationary tubes excited with a piston. This excitation apparatus is easy to build and operate. The constant volume cavity feature permits an accurate means of studying the non-linear shock wave behavior in various fluids at different pressures in the tube.

Nonlinear Unsteady Transonic Flow Theory-Local Linearization Solution for Two-Dimensional Flow

4ALYSIS is described for the prediction of pressure distributions on aerodynamic configurations undergoing both rigid body and elastic oscillations. The theory is based on dividing the flow into steady and unsteady components and solving the resulting equations by the local linearization method. The analysis is developed generally, but specific applications are confined here to airfoils in two-dimensional flow with freestream Mach number M^ — 1. The results are shown to converge correctly to nonlinear quasi-steady theory as the reduced frequency of oscillation based on chord, k -»0, and to linear acoustic theory as k becomes large (k ^ 2). For k ^ 0.1, the results display significant nonlinear thickness effects coupled to the steady-state solution, and demonstrate the complete inadequacy of linear theory in this frequency regime of prime importance in many flutter applications.

Nonlinear Acoustic Phenomena in the Presence of Absorptive Liners in Circular Ducts

High acoustic intensities are often encountered in acoustically lined ducts, such as in jet aircraft intake ducts and with gas turbine power-generating installations. Many acoustically absorptive materials have characteristics at high intensity which differ from those in the low-amplitude linear region. There are two basic nonlinear phenomena which are encountered with lined ducts at high intensities. First, the absorption properties of the material become nonlinear and, second, the propagation of the acoustic wave in the duct can no longer be described by linear acoustic theory. This is, of course, true even for a hard-walled duct where there is no liner present. A hard-walled circular duct has been employed which is driven by a horn-coupled high-intensity sound source and is anechoically terminated. Sound pressure measurements, waveform photographs, and spectra were obtained at a number of locations in the duct for the unlined case and for several absorptive liners. A previously reported high-order harmonic ringing effect has been further studied at various intensities and for different linear lengths. Possible theoretical explanations for the effect will be presented along with the experimental data and the additional complications caused when higher-order duct modes are present will also be discussed. [This work is supported under NSF Grant No. GK-32584.]

Non-linearity, relaxation and diffusion in acoustics and ultrasonics

As they propagate through a gas, fluctuating pressure signals of moderate amplitude and of ultrasonic frequency are affected by amplitude dispersion, by relaxation damping and, particularly in ‘shock layers’, by diffusive damping. We derive a ‘high frequency’ theory including all these effects, for disturbances of arbitrary wave form excited by a wide variety of boundary conditions. By introducing a phase variable α, and taking account of non-linearity, we show how the signal propagates along the rays of linear acoustics theory, with constantly changing wave profile.Relaxation dampens the signal, as for linear acoustics, and also diminishes amplitude dispersion. A criterion for shock formation is given, and the importance of non-linearity for signal attenuation exhibited. As shocks form, α surfaces coalesce and diffusive mechanisms are accentuated. Whitham's area rule is shown to be relevant for unsteady three-dimensional flows in relaxing gases, and is used to compute the attentuation of an ultrasonic beam. Supersonic relaxing flow over a wavy wall is also analyzed, and focusing effects are discussed.

Focusing of Underwater Step-Waves near a Caustic

The pressure field near a caustic (convergence zone) is analyzed for transient pulses with vanishing rise time. On the caustic, linear acoustic theory predicts a singularity of the order of (t−ta)−1/6 at the shockfront, ts being the arrival time of the front at the caustic. The corresponding total impulse remains finite. Factors limiting the peak pressure are discussed. Theoretical results are compared with data obtained by the Naval Ordnance Laboratory with explosive sources in a flooded quarry. [This work was supported by the U. S. Office of Naval Research.]

HF radio measurements of high-altitude acoustic waves from a ground-level explosion

Linear acoustic theory predicts that a large ground-level explosion—several hundred tons of TNT—should produce a pressure wave amounting to several per cent of ambient in the atmosphere 200 km above the explosion. Analysis of the interaction of such a pressure disturbance with the ionosphere implies that the passage of the wave should be readily measurable by means of vertical-incidence radio phase-path soundings. Radio measurements made at the time of a 500-ton explosion have confirmed the hypothesis. The shape of the observed phase disturbance agrees well with that predicted, and the observed onset times check to within 10 seconds with acoustic ground-to-ionosphere travel times. The amplitude and duration of the disturbances were approximately 100 radians and 2 minutes, respectively. The amplitude is slightly smaller than predicted; however, the observed duration is approximately twice that calculated. Refining and fitting the model to obtain better agreement—especially in disturbance onset time, shape, and duration—may represent a technique for improving present estimates of the upper-atmosphere density molecular weight and temperature.