the journal of the acoustical society of america volume 73 issue 6 1983 [doi 10.1121%2f1.389585]...

Propagation in air of N waves produced by sparks Wayne M. Wright Department of Physics, Kalamazoo College, Kalamazoo, Michigan 4900 7and Applied Research Laboratories, The University of Texas at Austin, Austin, Texas 78712

(Received 30 September 1982; accepted for publication 8 March 1983} Weak sparks, of length 0.5-1.0 cm and energy per discharge 0.01-0.1 J, served to produce intense acoustic transients resembling N waves. Amplitude decay and waveform elongation were studied, for propagation distance up to 2 m, through the use of a wideband capacitor microphone with essentially uniform response from dc to 1 MHz. Within the range of propagation distances for which the first (compression) phase of the N wave was completely formed, the duration of this compression phase T and its amplitude Ps were found to agree with the theoretical relations T= To[1 + ro ln(r/ro)] 1/2 andps = (roPso/r)[1 + ro ln(r/ro)] -/2, where r o is a parameter that depends upon the values Ofps and T at a reference distance from the source to. The time required for the amplitude of the head shock to increase from 5% to 95% of peak value was observed to vary from 0.45 ps (imposed by the microphone response) to greater than 2.0 ps as the wave traveled outward and as its amplitude decreased. Finally, the microphone was calibrated through use of the variation with distance of measured values of T; this new method has led to calculation of a free-field sensitivity that agrees within + 1 dB with the results of other calibrations. PACS numbers: 43.25.Cb

INTRODUCTION

The transient acoustic signal called an "Nwave" gets its name from the resemblance of its time waveform to the letter N. The pressure signature begins with a very sharp rise (the head or bow shock), continues with an approximately linear drop to a value about as far below zero acoustic pressure as the original rise, and finishes with another sharp step upward (the tail shock) that restores the pressure to zero. An idealized representation is shown in Fig. 1. Sources of N waves include explosions, electric sparks, bursting balloons, exploding wires, and objects in supersonic motion. Probably the most famous example is the sonic boom.

According to the linear theory of sound in gases, a sinu- soidal wave should propagate without distortion and an acoustic transient should experience no change in form other than that associated with the frequency-dependent attenuation of its spectral components. In an early application of what has since become known as weak-shock theory, Lan- dau 2 showed that both cylindrical and spherical N waves should elongate as they travel and should decay more rapidly than would be expected on the basis of geometrical spreading alone. DuMond et al., independently and by a different method, obtained equivalent results. They confirmed some of their theoretical predictions through measurements on cylindrical N waves from supersonic bullets. 3 The elongation and accelerated decay are produced by nonlinear effects. There is a general tendency for transient disturbances made up of any sort of condensation followed by a rarefac- tion to distort into an N wave. 4

Another feature of the N wave that is of interest is the shock rise time. In a shock there is a quasibalance between nonlinear effects, which tend to steepen the shock and make it thinner, and dissipation, which tends to smooth out or disperse the discontinuity. 5 In a spherically propagating N wave, dissipation rapidly becomes predominant. Zero rise

time is assumed in weak-shock theory, and as a result the formulas derived from weak-shock theory become inappli- cable when the shock rise time is an appreciable fraction of the half-duration T.

For the present experimental study, acoustic radiation in air from an electric Spark 6 was observed along a normal to the spark. Such propagation is effectively spherical so long as the observation point is more than a few spark lengths from the discharge. Near the spark the pressure signature resembles that of a blast wave, 7 with its characteristic sharp, nar- row positive phase, headed by a shock, and its shallow, stretched-out negative phase, which contains no shock. Be- cause of nonlinear effects, the head shock decays rapidly and the compression part of the negative phase steepens to form the tail shock (if dissipative effects don't become the major influence before that stage is reached). The N-wave stage is relatively long lasting because, despite the elongation and extra decay, the general shape of the wave remains the same. Eventually, however, when the shock rise time becomes sig-

P$

t-r/%

FIG. 1. Idealized N wave.

1948 J. Acoust. Soc. Am. 73 (6), June 1983 0001-4966/83/061948-08500.80 @ 1983 Acoustical Society of America 1948

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nificant, the wave enters its final, small-signal stage and more closely resembles a single cycle of a sine wave than an N.

Included here are results that have been described oral- ly 8 and in a technical report. 9 To our knowledge no comparison of measurements to theoretical expectations, for the spherical propagation of N waves, has been reported in the regular literature. In recent years Nwaves from weak sparks have formed the basis for a number of research projects and have become a commercially available tool for architec- tural acoustics and noise control model studies. This paper surveys the characteristics of these important transient sig- nals. Waveform features that deviate from idealizations are noted, as are the effects of using nonideal measuring apparatus such as a microphone of finite size or limited bandwidth.

I. THEORY

A. Propagation o! the idealized V wave Detailed arguments for the attainment and mainte-

nance of the N-shaped waveform have been presented by several authors. '2'2-4 As is indicated in Fig. 1, the acoustic pressurep is assumed to vary linearly with time (at distance r from the wave source) between discontinuities (shocks)of amplitudeps, with 2T defined to be the duration of the waveform. It is assumed thatps is much smaller than the ambient pressure Po, but large enough so that first-order terms in p/ Po are important in the analysis. The wave will tend to elongate as a result of convection and a nonlinear contribution to the sound speed, with the zero-pressure point in the center of the profile traveling with the small-signal sound speed Co. Energy is assumed to be lost from the wave only at the shocks, within which the waveform-steepening tendencies associated with finite amplitudes are balanced by the blur- ring tendencies of dissipative mechanisms. If the analysis is limited to distances for which the radius of curvature of a spherical wave front is very much greater than the thickness of a shock, results from the theory of plane-wave propagation of weak shocks will be expected to be valid. 2

The resulting theoretical dependence of N-wave length L = 2coTand shock overpressure = ps/Po upon the propagation distance r is described by the following equations from Sec. D.V of Ref. 1'

6 = Cr-l[ln(r/al)] -1/2, (1) L = (y + 1)y-lC[ln(r/al)] 1/2. (2)

Here y is the ratio of specific heats for the gas, ,while C and al are constants of integration.

Equations more suitable for comparison with experimental data are obtained if C and a l are eliminated in favor of the shock amplitude Po and half-duration To which exist at a reference distance ro from the center of curvature of the spherical wave fronts. Equation (1) then becomes

p = roPo r-l[ 1 q- ao ln(r/ro)] -1/2, (3) where the constant ao = [ln(ro/a)]-. The half-duration of the N wave can be expressed similarly as

T= To[1 +tro ln(r/ro)] /2. (4) 1949 J. Acoust. Soc. Am., Vol. 73, No. 6, June 1983

Finally, the dimensionless parameter ao is evaluated in terms of measurable quantities by dividing Eq. (1) by Eq. (2), giving each variable its value at r- ro, and using L = 2cot and ao = [ln(ro/a )]- '

Co = (y + 1)roPso/(2yPocoTo). (5) For Nwaves in air at 25C and normal atmospheric pressure, and at a reference distance ro = 0.10 m, this has the value ao = (2.47 X 10-9)ps o/To (with shock amplitude in pascals and positive-phase duration in seconds).

Examination of Eq. (4) suggests a method for determining the value of ao experimentally. One can expect to mea- sure T with reasonable precision using a calibrated oscilloscope, and a plot of T 2 versus log(r) should yield a straight line with slope proportional to To2ao . From this line and the arbitrarily chosen reference distance ro, one can obtain To and ao. The N-wave amplitude then is determined, with the use of Eq. (5), independent of microphone calibration. Stated in another way, one has here a convenient means of calibration for a microphone with a uniform response over a suit- ably wide frequency range.

The broad frequency content of an N wave, and the effects on this waveform of removing high- or low-frequency components, have been discussed with reference to sonic boom studies. 5,6 The amplitude spectrum (i.e., the absolute value of the Fourier transform of the pressure versus time waveform) is proportional to the spherical Bessel function of first kind and first order,

j(coT) = (sin coT)/(coT) 2 -- (cos coT)/(coT). (6) For the representative case T = 10 kts, this function has a maximum at 33 kHz; it decreases monotonically at lower frequencies and with oscillations at higher frequencies. The envelope of this particular amplitude spectrum is greater than 10% of its maximum value within the frequency range 2.1-360 kHz; as the waveform elongates, this range and the location of the maximum move to lower frequencies.

B. Shock profile Thus far we have assumed idealized N waves, as repre-

sented by the pressure profile of Fig. 1, which are terminated by two negligibly thin discontinuities. It is of interest now to consider the actual profile of the initial transition region at large enough distances from the source that the wave fronts are essentially plane.

In a gas for which relaxation effects are unimportant, the pressure within a weak, plane shock separating equilibri- um regions is shown theoretically to vary with distance x to either side of the midpoint of the shock as tanh(x/A ). 3,4 The thickness of a shock is predicted to depend upon the change in particle velocity across it as well as upon the viscosity and thermal conductivity of the gas, through A. Lighthill 3 de- fines shock rise time r as the time required for the pressure to increase from 5% to 95% of its final value; for air and the conditions of our experiment, his equations and parameter values lead to the result 7

= 172/ps its. (7)

Wayne M. Wright: Propagation in air of N waves 1949


FIG. 2. More realistic model of an N wave.

As the amplitude of an N wave decreases with pr6paga- tion, the rise time eventually will become significant in comparison with the positive-phase duration. We expect the propagation laws as presented above to cease to be valid in this region, where in fact the waveform will only vaguely resemble the profile of Fig. 1. For the shock pressure some- what below 200 Pa, nonlinear effects are expected to become of less importance than dissipation and dispersion, with the major influence under the conditions of our experiment being the relaxation of vibrational energy of O: molecules. s We thus anticipate an associated distortion of the wave profile in the form of a relatively slow rounding at the end of the shock. A more realistic waveform than that of the idealized N wave is sketched in Fig. 2.

(the necessity of which is discussed in the Appendix}, the microphone had a free-field sensitivity of -- 71.5 q- 1 dB re: 1 V/Pa, and there was no evidence of nonlinearity of the response for pressure amplitudes in excess of 3000 Pa.

A battery-operated preamplifier with a voltage gain of -4.5 dB was mechanically attached to the microphone and was followed by a Tektronix type 535A oscilloscope with type W preamplifier. Display of a stable, expanded voltage waveform was obtained by triggering the delayed-sweep scope mode wi,th electromagnetic radiation from the spark. This display was photographed with Polaroid type 47 or 410 film, and all measurements related to the acoustic waveforms were taken from the resulting photographs.

The microphone, with its integral plane baffle, was mounted above an optical bench in the equatorial plane of the spark. Alignment of the microphone, which was espe- cially critical for measurements of shock rise time, was as- sured through the use of a He-Ne laser beam that passed through an aperture directly behind the center of the spark gap, was reflected from the microphone diaphragm, and re- turned to the aperture. There was no evidence of waveform distortion as a result of acoustic diffraction from either the microphone housing or the electrodes and their 20-mil sup- porting wires.

Minor fluctuations in the discharge path of the spark, which occurred even though the ends of this path were fixed at the electrode tips, led to jitter of magnitude as great as 1/s in the time-of-arrival at the microphone of the acoustic wave; corresponding variations in the amplitude of the observed shock ranged up to 10%. The major features of

II. DESCRIPTION OF EXPERIMENT

Our experimental apparatus was similar to that which we have described previously. 9'2 The energy source for the spark was a variable dc power supply with a high internal impedance, which charged a mica capacitor in parallel with the spark gap. The power supply was adjusted to provide an open-circuit voltage slightly greater than the 10 kV per cen- timeter gap length required to produce a discharge between the 3-mil tungsten wire electrodes. Discharges with 0.01-0.1 J initially stored in the capacitor produced acoustic disturbances of sufficient amplitude for N waves to be formed. Greater values of the discharge energy CV 2 were accompa- nied by uncomfortable audible levels and resulted in rapid deterioration of the electrodes.

A wideband condenser microphone provided the response that was essential for this study of short-duration

1. acoustic transients. These microphones 9 use a thin (e.g., mil) Mylar-film diaphragm with an electrically conducting coating of aluminum evaporated on one surface and the other side in contact with a nominally smooth back electrode. A thin layer of air trapped in microscopic surface irregularities provides the mechanical compliance to control the diaphragm motion. Fo,r the present measurements the microphone had a back electrode of 1.23 cm diam and a capacitance at 1 kHz, with 135-V bias applied, of 470 pF. The rise time and resonance frequency were about 0.4/s and 1.2 MHz, respectively. Mounted flush in a large, plane baffle

(a) '- ' ' ,' 100

' 200

(b)

L 20p. s- J F 3

FIG. 3. Sample acoustic waveforms produced by weak sparks, showing the changes of shape which occur as the signal is propagated over a distance r. Pressure (vertical) scale arbitrary and different for each waveform. (a) Ca- pacitance C = 2 nF, spark length 1.0 cm, discharge energy 0.1 J; (b) C = 1 nF, spark length 0.5 cm, discharge energy 0.012 J.

1950 J. Acoust. Soc. Am., Vol. 73, No. 6, June 1983 Wayne M. Wright: Propagation in air of N waves 1950


successive N waves were essentially identical, and it was be- lieved that the results would not be improved significantly by the addition of a device for triggering the discharges. Such a modification would have permitted a shorter spark and thus a better approximation to a point source for the N waves, but this would have sacrificed the directionality and signal en- hancement associated with a line source (that have been discussed in a previous paper2). III. EXPERIMENTAL RESULTS

Representative examples of N waves generated by weak sparks are displayed in Fig. 3. Here the alterations of the waveforms with increased distance of propagation are readi- ly apparent: initial rise time increases, duration 2 Tincreases, the decrease of amplitude with time in the central portion achieves a uniform slope, and the negative {expansion} peak becomes more similar to the positive {compression} peak. A tail shock was observed to form for the more energetic sparks. The propagation of each phase of the profile should be unaffected by departures of the waveform from the N- wave shape on the other side of the central zero-pressure point, and our measurements have been limited to the initial compression phase.

Figure 4 indicates how data were extracted from the waveform photographs. The observed profile was approxi- mated by the idealized form, with the initial discontinuity assumed to coincide with the center of the head shock. When he rise time was essentially that of the microphone, the discontinuity was assumed to coincide with the knee at the bot- tom of the sudden voltage increase.

For 1.0-cm-long, 0.1-J sparks, about 20 cm of propagation were required for the compression peak to overtake the head shock. The half-duration T was measured for each of a large number of oscilloscope traces, and T 2 was plotted against the logarithm of the distance from the spark {Fig. 5}. A straight line was fitted to the data for the range 0.20

T

T

!( T

FIG. 4. Illustration of method used for determining half-duration and amplitude of N waves from waveform photographs.

I I I I I I I II

o o

200

IOC

o IO

T2=114.1+137.6 log (r/10)

, I I o

I I I I I II , _ 40 ) 80 I00 150 200 r (cm).

FIG. 5. Experimental data for the square of the N-wave half-duration, for C = 2 nF, spark length 1.0 cm, energy 0.1 J. The parameter r o = (AT2/ A log r}/(2.3 Tg } = 0.524, and from this the reference shock pressure at the reference distance of ro = 10 cm is calculated to be Po = 2.26 X 103 Pa.

m

Ps (10 -: volt)

5C)0 ' I ! I , , , ',r/lo)Vl'l'o.24 In (r/10)

1.14 I00 r

50

20-

IO to

, I 20 50 I00 200

r(m)

FIG. 6. N-wave amplitude as a function of distance from the center of the spark, for C = 2 nF, spark length 1.0 cm. The solid curve shows the theoretical behavior for r o = 0.524, while the dashed line corresponds to an amplitude decay as r- .4.

each quantity a constant value (i.e., horizontal line} would be associated with lossless, small-amplitude theory. Similar behavior was observed for each of the spark sizes and energies that we studied. Note that although there is an appreciable increase in the half-duration of the wave, our equation indicates that a doubling of To should require a propagation distance of 32 m; weak-shock theory clearly ceases to be valid well before that point, however.

Over a wide range of spark sizes and energies, the product of N-wave amplitude, half-duration, and propagation distance was found to be essentially constant, in agreement with theory. Values of the product {rTps } for the data of Fig. 7, when normalized through division by the average value of this product, were distributed about unity with a standard deviation of 0.03.

Changes in the observed profile of the head shock, as the N wave traveled outward in the equatorial plane of the spark and its amplitude decreased, are indicated in Fig. 8. Overshoot and tinging, which were introduced by the microphone, disappeared as the rise time of the actual shock be- came greater than that of the microphone.

The time required for the oscilloscope trace to increase from 5% to 95% of its value at the end of its initial rise has been plotted in Fig. 9. Here the horizontal variable is proportional to the inverse of the magnitude of the initial pressure rise, rather than of the extrapolated amplitude of the N wave (as was the case in Fig. 6}. Data are included here for two spark sizes (with a single point for a third size}; the associated half-duration for each is noted at the point where the head- shock amplitude was 65 Pa. The straight line on this graph

corresponds to the theoretical amplitude dependence of the rise time for a plane shock between steady-state levels, which was given by Eq. {7}. Uncertainty in the calibration of the microphone was equivalent to a shift in the slope of the theoretical line of about 10%. The lower limit to the rise time values is that imposed by the microphone, while the falling of ' below predicted values for the weaker shocks is consistent with the limited half-duration of the N waves. Orenstein and Blackstock 22 recently have reported on an extensive. set of rise-time measurements {using the definition of Pierce7}. They claim good agreement with the results of a computer

. propagation algorithm that enables them to include both finite-amplitude and atmospheric absorption effects. Hoist- Jensen and his collaborators 23 have obtained similar results for weak N waves at large distances from very intense spark and exploding-wire sources.

16

o o

4

(b)

14 o

12

10.7 I-I-0.52 In (r/10) IC '-=

T , I I I I ! lilt , io 20 5o ioo

r(m)

7 o ' 1+0.52 In (r/10)

20 50 I00 r (cm) "'-

200

200

FIG. 7. Comparison of experimental data with theory for C = 2 nF, spark length 1.0 cm: (a) waveform half-duration; (b) product of N-wave amplitude and distance from the spark. The theoretical curve for Twas calculated for the values of To and cr o determined from Fig. 5, while that for rps was calculated for the same value of cr o fitted by eye to the data.

1 g52 J. Acoust. Soc. Am., Vol. 73, No. 6, June 1 g83 Wayne M. Wright: Propagation in air of N waves 1 g52


r= IOcm

60cm

120cm

L J

FIG. 8. Observed profile of the head shock at three distances from the spark, for C = 2 nF and spark length 1.0 cm. With the microphone slightly misa- ligned, so that the initial shock arrived at different portions of the microphone diaphragm at slightly different times, the ringing was eliminated for the closest distance and one could see that the shock did not extend to the maximum pressure of the wave.

IV. DISCUSSION

An unbounded, homogeneous medium in the neighbor- hood of a sudden localized release of energy can be divided into three separate regions. Close to the source of acoustic disturbance there is little effect of dissipation, and nonlinear effects tend to encourage the formation of an N-shaped wave profile. In the second region the wave amplitude decreases and finite-amplitude effects eventually become less important than the effects of viscosity, heat conduction, and molecular relaxation. In the third region weak-shock theory is no longer applicable.

Our experimental results have been obtained in the second of these regions. For distances from the spark at least 20 times the gap length, where Schlieren photographs support

the assumption of spherical wave fronts about the center of the spark, agreement with the theoretical laws for spherical propagation of N waves is excellent. In summary'

(a) An N wave elongates as it travels outward, with the most rapid stretching when the amplitude is greatest.

(b) The amplitude decay is slightly more rapid than the r- variation of small-amplitude acoustic waves i.n a lossless medium.

(c) The product of propagation distance, shock amplitude, and positive-phase duration is invariant.

Dissipative characteristics of the medium control the structure of the head shock, although they do not appear to influence the rates of attenuation and elongation of N waves. Our wideband microphone has permitted the direct observation of extremely rapid pressure changesand has led to mea-' surements in good agreement with theory for rise times of our more intense shocks. It is not unreasonable, given the rapid decrease of amplitude associated with spherical propagation and the negative pressure slope immediately behind the shock, that with further travel the rise time would not keep up with the theoretical value for a plane, step shock of the same amplitude. Our photographs clearly show the initial pressure jump in the shock being followed by a more gradual increase, which can be attributed to the effects of molecular relaxation. For the conditions of our experiment (25 C, 40% relative humidity), the calculated relaxation time of 4/s due to O2 molecules TM is consistent with the waveform rounding observed at the greater propagation distances.

ACKNOWLEDGMENTS

The continuing interest, encouragement, and advice of Dr. David T. Blackstock is noted with gratitude. This work was supported in part by the U.S. Office of Naval Research. A portion was carried out while the author was a visiting professor at the U.S. Naval Postgraduate School.

E.O

'-.I- ;.L. -

0 5 I0 15 20

FIG. 9. Observed rise time of the N waves, plotted as a function of the reci. procal of the shock amplitude (arbitrary scale). El: C = 2 nF, 1.0 cm; : C = 1 nF, 0.5 cm; A: C = 2.0 nF, 1.5 cm. The straight line shows the theoretical dependence of rise time upon shock amplitude, from Lighthill. 13

1953 J. Acoust. Soc. Am., Vol. 73, No. 6, June 1983

APPENDIX: DETAILS ABOUT EXPERIMENTAL TECHNIQUES

We have obtained N waves with a modest range of dura- tions by varying the spark length and discharge energy. With spontaneous breakdown between pointed electrodes when the capacitor voltage reached the necessary level, however, independent variation of N-wave amplitude and duration has not been possible. Measurements were carried out for several spark lengths in the range 0.25-1.50 cm, with capacitance of 0.5-5.0 nF {and stored energy of 0.003-0.23 J). At the arbitrarily chosen distance of 1.0 m, the extrapolated shock amplitude was found to be related to the N-wave duration (in/s) by the empirical formula, Ps = 2.2T' Pa. (In addition, at this same distance Ps -- 110 d '82C'3 Pa, where spark length d is in centimeters and capacitance C is in nano- farads.} Klinkowstein 24 used a 0.4-cm gap, 4.5-J spark energy, and a separate triggering discharge; his measurements at 1.0-m distance were Ps -- 300 Pa and T- 28/s, which are consistent with our equation above within microphone calibration accuracy.

Wayne M. Wright: Propagation in air of N waves 1953


OBSERVED

illill = ""'nlv

20/s

COMPUTED

20/s

i i i i i t i

FIG. A 1. Response to an Nwave normally incident on the microphone face of (a) locally constructed microphone with active area of 0.20-cm diam and 9.9-cm baffle diam; (b) Bruel & Kjaer -in. capacitor microphone, type 4136, with active area of 0.32-cm diam and effective baffle diam of 0.60 cm.

Observations such as have been reported here are possible only with a microphone having a frequency response that is flat up to around 1 MHz and without a pronounced resonance peak. 25 With commercially available microphones, one typically will observe a waveform that is strongly in- fluenced by the microphone impulse response. In addition, diffraction from the microphone housing can substantially modify the free-field waveform. For the present work the microphone was mounted flush with a rigid, plane circular baffle of radius great enough that the diffracted arrival occurred well after the sensing of the direct N wave. Lockwood and Wright 26 have studied this situation and found the observed waveforms that propagated radially inward, after diffraction at the edge of a circular plate, to agree well with waveforms that were calculated with the use of linear theory. At the center of the baffle, where the normally incident N

0

wave experiences pressure doubling, the diffracted wave is inverted and has one-half the amplitude of the direct signal. With a locally constructed microphone of finite size at the center of the circular baffle, as was the case for the oscilloscope display of Fig. A1 (a), a temporal spreading of the initial shock is evident. This apparent increase of rise time is attributable to integration of the pressure signal over the active area of the transducer. Figure A1 (b) indicates how a high-quality, commercially available microphone, a Bruel & Kjaer type 4136, responds to the superposition of the direct N wave and the spread-out diffracted signal. For the calculated waveform shown here, the microphone was considered to have a baffle with diameter equal to that of the microphone housing. The differences between observed and computed waveforms are attributable to the limited bandwidth of the B&K microphone, which was not taken into account in computing the predicted response.

I j. W. M. DuMond, E. R. Cohen,'W. K. H. Panofsky, and E. Deeds, "A determination of the wave forms and laws of propagation and dissipation of ballistic shock waves," J. Acoust. Soc. Am. 18, 97-118 { 1946}. The name "N wave" was first used in this article.

2L. D. Landau, "On shock waves," J. Phys. {Moscow) 6, 229-230 {1942); "On shock waves at large distances from the place of their origin," J. Phys. (Moscow) 9, 496-500 (1945).

3A thorough summary of work in this area over a 20-year period is provided by W. B. Snow, "Survey of acoustic characteristics of bullet shock waves," IEEE Trans. Audio Electron. AU-15, 161-176 (1967).

4See, for. example, G. B. Whitham, "The flow pattern of a supersonic pro- jectile," Commun. Pure Appl. Math. 5, 301-348 (1952). SA number of analytical studies of rise time have been-done as a result of sonic boom measurements, which show that sonic boom rise times are far too long to be explained on the basis of viscosity and heat conduction. See, for example, A.D. Pierce and D. J. Maglieri, "Effects of atmospheric irregularities on sonic boom propagation," J. Acoust. Soc. Am. 51, 702-721 (1972). The atmospheric effects responsible for anomolous sonic boom rise times play no part in the study reported here.

6It is interesting to note that more than a century ago A. Tfpler, who with Foucalt is credited with developing the Schlieren technique, applied his apparatus to observe the acoustic disturbance generated by a spark [Ann. Phys. 131, 180-215 (1867)]. Ernst Mach gives a fascinating account of Tfpler's investigation, as well as his own on "head waves" from bullets, in his book Popular Scientific Lectures, translated by T. J. McCormack {Open Court, Chicago, 1898), 3rd ed. pp. 309-337.

7See, for example, G. F. Kinney, Explosive Shocks in Air (Macmillan, New York, 1962), p. 77.

8D. T. Blackstock, J. L. McKittrick, and W. M. Wright, "Laboratory modeling of the sonic boom," Am. J. Phys. 35, 679 (A)(1967); W. M. Wright, "Propagation laws and shock profile for N waves from sparks," J. Acoust. Sac. Am. 49, 119 (1971).

9W. M. Wright, "Studies of Nwaves from weak sparks in air," Final Report under Contract Nonr-3932(00), Kalamazoo College, Kalamazoo, MI (June 1971 ) (AD 725 865).

lB. A. Davy and D. T. Blackstock, "Measurements of the refraction and diffraction of a short N wave by a gas-filled soap bubble," J. Acoust. Soc. Am. 49, 732-737 (1971); D. T. Blackstock, W. M. Wright, B. A. Davy, J. C. Lockwood, E. P. Cornet, M. O. Anderson, D. R. Kleeman, W. N. Cobb, and R. D. Essert, "Finite-amplitude behavior of N-shaped pulses from sparks," J. Acoust. Sac. Am. Suppl. 1 68, S29 (1980); D. T. Black- stock et al., "Experiments with N waves from sparks" (in preparation).

R. G. Cann and R. H. Lyon, "New acoustical modeling instrumentation," J. Acoust. Soc. Am. 61, 1094-1097 (1977).

2D. T. Blackstock, "Nonlinear Acoustics (Theoretical)," in American In- stitute of Physics Handbook, edited by D. E. Gray (McGraw-Hill, New York, 1972), Chap. 3n, pp. 3-194 to 3-197.

13M. J. Lighthill, "Viscosity effects in sound waves of finite amplitude," in Surveys in Mechanics, edited by G. K. Batchelor and R. M. Davies (Cam- bridge U. P., Cambridge, United Kingdom, 1956).

4A. D. Pierce, Acoustics: An Introduction to Its Physical Principles and Ap- plications (McGraw-Hill, New York, 1981), Chap. 11.

sp. B. Oncley and D. G. Dunn, "Frequency spectrum of N waves with finite rise time," J. Acoust. Soc. Am. 43, 889-890 (1968).

6M. J. Crocker, "Measurement of sonic boom with limited frequency response instrumentation--A theoretical study," Wyle Labs, Huntsville, AL, Rep. WR 66-20 (April 1966).

7pierce's definition of shock rise time (Ref. 14, p. 591), in terms of the slope of the waveform at the half-peak point, leads to a value about 10% smaller than that given by Eq. (7).

SReference 14, pp. 555 and 593. 9W. M. Wright and J. L. McKittrick, "Diffraction of spark-produced

acoustic impulses," Am. J. Phys. 35, 124-128 (t967). eW. M. Wright and N. W. Medendorp, "Acoustic radiation from a finite

line source with N-wave excitation," J. Acoust. Soc. Am. 43, 966-971 (1968).

See, for example, K. Gjaevenes and A. Homstol, "Measurements of the attenuation of muzzle shock waves in air," J. Acoust. Sac. Am. 49, 1688- 1690 (1971); L. B. Poch6, Jr., "Underwater shock-wave pressures from small detonators," J. Acoust. Soc. Am. 51, 1733-1737 (1972).

22L. B. Orenstein and D. T. Blackstock, "Experimental investigation of the rise time of N waves," J. Acoust. Soc. Am. Suppl. 1 71, S29 (1982); L. B. Orenstein, "The rise time of N waves produced by sparks," Appl. Res. Labs. Tech. Rep. 82-51, The University of Texas at Austin, Austin, TX (October 1982).



230. Hoist-Jensen, "An experimental investigation of rise times of very weak shock waves," U. of Toronto Inst. of Aerosp. Stud. Tech. Note No. 229, Toronto (March 1981 ); H. Honma, I. I. Glass, O. Hoist-Jensen, and Y. Tsumita, "Weak spherical shock-wave transitions of N-waves in air with vibrational excitation," 13th Int. Symp. Shock Tubes and Waves, Niagara Falls, NY (July 1981 ).

24R. E. Klinkowstein, "A study of acoustic radiation from an electrical

spark discharge in air," Acoust. Vib. Lab. Rep., MIT, Cambridge, MA (July 1974).

2SFor a description of a successful alternative approach to visualizing spark N waves, see G. Smeets, "Laser interference microphone for ultrasonics and nonlinear acoustics," J. Acoust. Soc. Am. 61, 872-875 (1977).

26j. C. Lockwood and W. M. Wright, "Effect of edge diffraction on microphone response to transients," J. Acoust. Soc. Am. 51, 106 (1972).



the journal of the acoustical society of america volume 73 issue 6 1983 [doi 10.1121%2f1.389585]...

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