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4 Acoustic Radiations from Lightning ARTHUR A. FEW, JR. Rice University ACOUSTIC SOURCES IN THUNDERSTORMS Electric storms produce a variety of acoustic emis- sions. The acoustic emissions can be broadly divided into two categories those that are related to electric processes (i.e., they correlate with lightning) and those that either do not depend on cloud electricity or for which no correlations with electric changes have been observed. Only the first group will be discussed here (see Few, 1982; Georges, 1982, for reviews of nonelectrical acoustics) . Two types of acoustic emissions correlated with elec- tric processes are thunder, which is produced by the rap- idly heated lightning-discharge channel, and infrasonic emissions produced by electrostatic fields throughout the charged regions of the cloud. Thunder is probably the most common of all loud natural sounds, while other acoustic emissions are not ordinarily observed without special devices. THUNDER THE RADIATION FROM HOT CHANNELS Spectrographic studies of lightning return strokes (Orville, 1968) show that this electric-discharge process heats the channel gases to a temperature in the 24,000 K range. At high temperatures the expansion speed of the 46 shock wave is roughly 3 x 103 m/see and decreases rap- idly as the shock wave expands; in comparison the mea- sured speeds for various lightning-breakdown processes range from 104 to 108 m/see (U~man, 1969; Weber et al., 1982~. Therefore, the electric breakdown process in a discharge event is completed in a given length of the channel before the hydrodynamic responses are fully or- ganized. Other electric processes occur over longer pe- riods (e. g., continuing currents), but the energy input to the hot channel is strongly weighted toward the early breakdown processes when channel resistance is higher (Hill, 1971~. Shock-Wave Formation and Expansion The starting point for developing a theory for a shock- wave expansion to form thunder is the hot ~ ~ 24,000 K), high-pressure ~ > 106 Pa) channel left by the electric dis- charge. Hill's (1971) computer simulation indicated that approximately 95 percent of the total channel en- ergy is deposited within the first 20 ,usec with the peak electric power dissipation occurring at 2 ,usec; during the 20-,usec period of electric energy input, the shock wave can only move approximately 5 cm. This simula- tion may actually be slower than real lightning because Hill used a slower current rise time than indicated by

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ACOUSTIC RADlATlONS FROM LIGHTNING more recent measurements (Weidman and Krider, 1978). The time-resolved spectra of return strokes (Orville, 1968) show the effective temperature dropping from ~ 30,000 to ~ 10,000 K in a period of 40 ,usec and the pressure of the luminous channel dropping to atmo- spheric in this same time frame. During this period the shock wave can expand roughly 0.1 m. Even though channel luminosities and currents can continue for pe- riods exceeding 100 ,usec, the processes that are impor- tant to the generation of thunder occur very quickly ~ < 10 ,usec) and in a very confined volume (radius ~ 5 cm). The strong shock wave propagates outward be- yond the luminous channel, which returns to atmo- sphere pressure within 40 ,usec. The channel remnant cools slowly by conduction and radiation and becomes nonconducting at temperatures between 2000 and 4000 K perhaps 100 msec later (Uman and Voshall, 1968~. Turning our attention now to the shock wave itself we can divide its history into three periods strong shock, weak shock, and acoustic. The division between strong and weak can be related physically to the energy input to the channel, the weak-shock transition to acoustic is somewhat arbitrary. Calculations and measurements have shown that the radiated energy is on the order of 1 percent of the total channel energy (e.g., Uman, 1969; Krider and Guo, 1983), hence most of the available en- ergy is in the form of internal heat energy behind the shock wave. As the strong shock wave expands it must do thermo- dynamic work (PdV) on the surrounding fluid. The ex- pected distance though which the strong shock wave can expand will be the distance at which all the internal thermal energy has been expended in doing the work of expansion. Few (1969) proposed that this distance, which he called the "relaxation radius," would be the appropriate scaling parameter for comparing different sources and different geometries. The expressions for the spherical, Rs' and cylindrical, Rc' relaxation radii are Rs= (3Et14~Po)~/3 and Rc= (EiI~Po)i'2, (4.1) (4.2) where E' is the total energy for the spherical shock wave, Ei is the energy per unit length for the cylindrical shock wave, and P0 is the environmental atmospheric pres- sure. Table 4.1 gives RC over a range of values that have been suggested in the literature for En. Nondimensional distances denoted by X may be defined for spherical problems by dividing by Rs and for cylindrical problems by dividing by RC. Figure 4.1 shows the propagation of the strong shock 47 TABLE 4.1 Relaxation Radii (R..) (in meters) for Different Energies per Unit Length (El) of Cylindrical Shock Waves P(,= lOOkPa ( ~ surface) 0.18 0.25 0.40 0.56 0.80 1.26 1.78 Pi, = 60 kPa Pi, = 30 kPa ( ~ 4-km height) ( ~ 9-km height) 0.23 0.32 0.52 0.72 1.03 1.63 2.30 104 2x 104 5x 104 105 2x 105 5x 105 lO'5 0.33 0.46 0.73 1.02 1.46 2.30 3.25 into the transition region (X ~ 1) and beyond into the weak-shock region. As the shock front passes the relaxa- tion radius (X = 1) the central pressure falls below am- bient pressure as postulated in the definition of the re- laxation radius. The momentum gained by the gas during the expansion carries it beyond X = 1 and forces the central pressure to go momentarily below atmo- spheric. At this point the now weak-shock pulse decou- ples from the hot-channel remnant and propagates out- ward. Figure 4.2 shows on a linear coordinate system the final output from Brode's (1955) numerical solution, the weak-shock pulse at a radius of X = 10.5. Figure 4.3 shows Plooster's (1968) cylindrical shock wave near X = 1 with Brode's (1955) spherical shock wave. The effects of channel tortuosity will be discussed 50 20 It Con 5 _ 14 ly ,! //, / 1 ~ l ~ _ _ I l _ ~ . ~ 1 / 1 / ~ l 1 1 SPHERICAL SHOCK WAVE CYLINDRICAL SHOCK WAVE ,1'} ~1 l id . 1 1 X FIGURE 4.1 The expansion of spherical and cylindrical shock waves from the strong-shock region into the weak-shock region. The radii of both spherical and cylindrical geometries have been nondimensiona- lized using the relaxation radii defined in Eqs. (4.1) and (4.2). The spherical shock wave is that of Brode (1956), and the cylindrical shock wave is from a similarity solution by Sakurai (1954).

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48 p/po . , .03 .02 .01 .00 .99 .98 ~ 7 8 9 X 10 11 FIGURE 4.2 The weak shock wave formed from the spherical strong shock wave. This is the final pressure profile computed by Brode (1956). For an energy input of 105 Jim (R. = 0.56 m for P.`, = 105 Pa) this weak shock wave would be approximately 6 m from the lightning channel. in greater detail later; for now we note that owing to tortuosity we cannot expect the shock wave to continue to perform as a cylindrical wave once it has propagated beyond a distance equal to the effective straight section of the channel that generated it. If the transition from cylindrical to spherical occurs near X = 1 as suggested by Few (1969), then the spherical weak-shock solutions of Brode provide a good means of estimating the wave shapes of lightning-caused acoustic pulses. Figure 4.4 presents a graphical summary of the vari- ous transitions that are thought to take place. The initial strong shock will behave cylindrically following the dashed line based on Plooster's (1968) computations; this must be the case for the line source regardless of the tortuosity because the high-speed internal waves (3 x 103 m/see) will hydrodynamically adjust the shape of the channel during this phase. The transition from strong shock to weak shock occurs near X = 1, and the transi- tions from cylindrical divergence to spherical diver- gence will occur somewhere beyond X = 0.3 and proba- bly beyond X = 1 depending on the particular geometry of the channel at this point. The family of lines labeled x in Figure 4.4 represent transitions occurring at different points. x is the effective length, L, of the cylindrical source divided by Rig (x = LIRC); it is approximately equal to the value of X at which the transition to spheri- cal divergence takes place. Comparisons with Numerical Simulations and Experiments In the numerical solutions of Plooster (1971a, b) and Hill (1971) the energy inputs to the cylindrical problem ARTHUR A. FEW, JR. were computed as a function of time for specified cur- rent wave shapes and channel resistance obtained from the computations in the numerical model. These model results predicted that the energy input to the lightning channel was an order of magnitude or more below the values obtained from electrostatic estimates or from other indirect measurements of lightning energy (Few, 1982~. The major differences might be due to the as- sumed current wave forms used in the models. The re- cent data obtained with fast-response-time equipment yields current rise times for natural cloud-to-ground lightning in the 35-50 kA/,usec range (Weidman and Krider, 1978~. These values are considered as represen- tative of normal strokes; extraordinary strokes have been measured with current rise times in the 100-200 kA/,usec range. By way of comparison, Hill's (1971) cur- rent rise time was 2.5 kA/,usec. Laboratory simulations of lightning have been suc- cessfully performed in a series of experiments conducted at Westinghouse Research Laboratories; these results provide us with our best quantitative information on thunder generation. In these tests a 6.4 x 106 V impulse generator was used to produce 4-m spark discharges in air (Uman et al., 1970~. Circuit instrumentation al- lowed the measurement of the spark-gap voltage and current from which the power deposition can be com- puted. Calibrated microphones were used to measure the shock wave from the spark as a function of distance. The results of the research (Uman et al., 1970) have been compared with the theory of Few (1969) and with other 2.2 t _ _ o CL 1.8 0 - ~ 1.4 a) .= In In ~ 1.0 -Spherical Pressure Wave /: ~~Cylindrical Pressure Wave /, .4 ~ {' ll .8 I.C 1~2 FIGURE 4.3 Comparison of spherical and cylindrical shock-wave shapes near X = 1. These profiles are for the point-source, ideal-gas solutions of Brode (1955) and Plooster (1968). In the transition region of strong shock to `` eak shock, these w eve shapes are nearly identical. From Fee (1969) faith permission of the American Geophysical Union.

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ACOUSTIC RADIATIONS FROM LIGHTNING possible interpretations (Plooster, 1971a). The data were found to be consistent with the theory developed by Few. Figure 4.5 compares a measured spark-pressure pulse with the profile that is predicted from the theory; both represent conditions in the plane perpendicular to the spark channel. Figures 4.6 and 4.7 summarize the exten- sive series of spark measurements. Figure 4.6 is in the same format as Figure 4.4. The center line passing _.Ol through the scattered points and labeled L = 0.5 m cor responds (using the measured energy input of 5 x 103 him, which gives RC = 0.126 m) to x = 4 in Figure 4.4. The two boundary lines L = 6.25 cm and L = 4.0 m would correspond to x values 0.5 and 32. The lower bound is very close to the lower limit value of one third indicated in Figure 4.4. The upper bound of Figure 4.6 (x ~ 32) is too large to be depicted in Figure 4.4, where x = 4 is the last line shown. The data points of Figure 4.6 corresponding to the larger x or L values could represent situations where the shock-wave expansion was following the cylindrical be- havior over a long distance, hence large x. However, if the expansions were truly cylindrical to that extent, then to 100 _ ~ to = ~. - o 10 o in ~ of\ -Spherical Divergence - \ --Cylindrical Divergence Lower bound for ^~ .01 ~ ~ ~ ~ ~ ~ ~ ~ 1 .1 .2 .5 1 X FIGURE 4.4 Line-source shock-wave expansion. The overpressure of the shock front is given for spherical (Brode, 1956) and cylindrical (Plooster, 1968) shock waves. Line sources must initially follow cylin- drical behavior, but on expanding to distances of the same size as line irregularities they change to spherical expansion following curves simi- lar to the depicted curves. From Few (1969) with permission of the American Geophysical Union. 49 .02- |t Pressure Profile Co .ol- ~ \: Spark (5 X 103 j/m) at 3 meters at: ~\\Predicted profile > ~ ~_. TIME .! .2 .3 .4 .5 .6 .7 .S .9 m see FIGURE 4.5 Comparison of theory with a pressure wave from a long spark. The measured pressure wave from a long spark (Uman et al., 1970) is compared with the predicted pressure from a section of a mesotortuous channel having the same energy per unit length. x is assumed to be 4/3. From Few (1969) with permission of the American Geophysical Union. the length of the pulse would be longer, as required by the cylindrical-wave predictions. The data of Figure 4.7 indicate that this cannot be the case. The lengths of the positive-pressure pulses shown in Figure 4.7 are clearly not in the cylindrical regime; if anything, they tend to be even shorter than predicted by the spherical expan- sion. (See also Figure 4.5.) It is obvious from both the spark photographs and wave forms in Uman et al. (1970) that the spark is tortu- ous and produces multiple pulses. They found that the wave shapes, more distant from the spark where pulse- transit times were most similar, showed evidence of an in-phase superposition of pulses; at closer range the pulses exhibited greater relative phase shifts and more multiplicity aspects. The in-phase superposition of spherical waves would reproduce the distributions shown in Figures 4.6 and 4.7. The pressure amplitude would be increased relative to a single pulse, but the wavelength would not be substantially affected. The measured spark wave forms (Uman et al., 1970) were systematically shorter than predicted by the the- ory. As shown in Figure 4.5, the tail of the wave was compressed, and the data of Figure 4.6 indicate that the positive pulse was similarly shortened. This shortening could be due simply to an inadequacy in the numerical shock-wave model; we think, instead, that the differ- ence results from the energy input being instantaneous in the one case (Brode, 1956) and of longer duration for the spark case. If energy, even in small quantities, con- tinues to be input into the low-density channel core after the shock front has moved outward then the core will be kept at temperatures much higher than predicted by the theories, having an instantaneous energy input followed by expansion. Owing to the elevated sound speed associ- ated with the higher core temperature the part of the

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so 1 n .010 nn _ \ 1 1 1 1 1 1 1 1 , 1 1 1 1 1 1 1 1 _ ~ \\ _\ \N \ \ ~ Plooster, 3 - \ <~Cylinder, W=5.0xlO J/m \ i\ _ \- .~. N~ _ I; . a\ ~` Brode, Sphere, - / N;:# IN . \ WL =2.0x 10 J - B rode, \, ~ . < ` L = 4. 0 m Sphere, V ~ +\ ~ / WL=3.1X102J \~. i+ +V~` L =6.25 cm >by If \ ~ B rode, \t it\+ \ ~ `` Sphere, +: ~ '\ \ `~ WL=2.5x103J I \ \ ~ L=0.5m +~ \: \ +N I ~ 1 1 1 1 1 11 1 \ 10 30 1 1 1 1 1 1 1 111 1 1 . 0.1 1.0 Distance, meters FIGURE 4.6 Shock-front overpressure as a function of distance from the spark. The dots represent data obtained with a piezoelectric micro- phone; the crosses data obtained with a capacitor microphone. The total electric energy per unit length computed from measurement of the spark voltage and current is 5 x 103 J/m. Also shown are theoreti- eal values for cylindrical and spherical shock waves. From Uman et al. (1970). 0.7 ~ ~ , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1 0.6 _ =. 0.4 - - c 0.3 to to - `= 0. 2 1 C PlOoster. ~~/ Cylinder / W =5.0xlO3J/m / / / / /~ Brode. 4 a,' Sphere, WL=2.0xlO J L =4.0m 4 _ 1 _ O ....... 0.1 1.0 Distance, meters ~T / / Brode, + /// Sphere, WL=2.5xlO J + L=0.5 ~ 1 ~ + + + + B rode, 2 Sphere, WL = 3.1 x 10 J L=6.25cm 1 1 1 1 ~1 1 1 1 ~ + + I ~ 1 1 1 1 , 10 ' 30 FIGURE 4.7 Duration of positive part of the shock wave from the long.spark. For the same data of Figure 4.6, we see here the length of the positive pressure pulse for the 5 x 103 J/m sparks at various dis- tances. From Uman et al. ( 1970) . ARTHUR A . FEW, JR. wave following the shock front will form and propagate outward faster than predicted by theory. We expect, therefore, that the elevated core temperature associated with sparks and lightning can reasonably produce the shortened wave forms. The wave shape produced by the shock wave is re- lated to the energy per unit length of the lightning flash; thunder is superposition of many such pulses from the lightning channel; hence, the power spectrum of the thunder, with simplifying assumptions, can be related to the energy per unit length of the channel (Few, 1969~. Other properties (tortuosity and attenuation) that influ- ence the spectrum of thunder are discussed later. The assumptions in this theory all affect the thunder spectrum in the same sense; the peak of the theoretical spectrum will occur at higher frequencies than the peak of the real thunder spectrum (Few, 1982~. The light- ning-channel energy that one estimates from the peak will therefore be an overestimate of the actual lightning- channel energy. Holmes et al. (1971) provided the most complete published thunder spectra to date; these spec- tra show a lot of variation. Most of the spectra are con- sistent with the qualitative expectations of thunder pro- duced by multiple-stroke lightning, but a few of them exhibit very-low-frequency (< 1 Hz) components that are dominant during portions of the record and appear to be totally inconsistent with the thunder-generation theory from the hot explosive channel. Dessler (1973), Bohannon et al. (1978), and Balachandran (1979) sug- gested that these lower-frequency components might be electrostatic in origin; Holmes et al. (1971) also consid- ered that this was a possible explanation. Tortuosity and the Thunder Signature With respect to the effects of lightning-channel tortu- osity on the thunder signal there is almost unanimous agreement among researchers. Lightning channels are undeniably tortuous and are tortuous apparently on all scales (Few et al., 1970~. For convenience in discussing channel tortuosity Few (1969) employed the terms mi- crotortuosity, mesotortuosity, and macrotortuosity rel- ative to the relaxtion radius of the lightning shock wave. For a lightning channel having an internal energy of 105 l/m (see Table 4.1), Rc ~ 1/2 m. The microtortuous' features smaller than Rc, although optically resolvable, are probably not important to the shock wave as mea- sured at a distance because the high-speed internal waves (3 x 103 m/see) are capable of rearranging the distribution of internal energy along the channel while the shock remains in the strong-shock regime. At the me- sotortuous scale ~ ~ Rc) the outward propagating shock wave decouples from the irregular line source because

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ACOUSTIC RADIATIONS FROM LIGHTNING the acoustic waves from the extended line source can no longer catch up with the shock wave. Somewhere in this mesotortuous range the divergence of the shock waves makes the transition from cylindrical to spherical. Whereas the mesotortuous channel segments are im- portant in the formation and shaping of the individual pulses being emitted by the channel the macrotortuous segments are fundamental to the overall organization of the pulses and the amplitude modulation of the resulting thunder signature. Few (1974a) computed that 80 per- cent of the acoustic energy from a short spark was con- fined to within + 30 of the plane perpendicular to the short line source. A macrotortuous segment of a light- ning channel will direct the acoustic radiations from its constituent mesotortuous, pulse-emitting segments into a limited annular zone. An observer located in this zone (near the perpendicular plane bisecting the macrotortu- ous segment) will perceive the group of pulses as a loud clap of thunder, whereas another observer outside the zone will perceive this same source as a lower-amplitude rumbling thunder. This relationship between claps, rumbles, and channel macrotortuosity has been con- firmed by experiment (Few, 1970) and in computer sim- ulations (Ribner and Roy, 1982~. Loud claps of thunder are produced, as mentioned above, near the perpendicular plane of macrotortuous channel segments; there are three contributory effects (Few, 1974a, 1975) to the formation of the thunder claps. The directed acoustic radiation pattern described above is one of the contributing factors, and this effect is distributed roughly between + 30 of the plane. A second effect, which occurs only very close to the plane, is the juxtaposition of several pulses in phase, which increases the pulse amplitude to a greater extent than would a random arrival of the same pulses. The third effect contributing to thunder clap forma- tion is simply the bunching in time of the pulses. In a given period of time more pulses will be received from a nearly perpendicular macrotortuous segment of chan- nel than from an equally long segment that is perceived at a greater angle owing to the overall difference in the travel times of the composite pulses. In this section we have examined the complex nature of the formation of individual pulses from hot lightning channels and how a tortuous line source arranges and directs the pulses to form a thunder signature. The re- sulting thunder signature depends on (l) the number and energy of each rapid channel heating event (leaders and return strokes); (2) the tortuous and branched con- figuration of the individual lightning channel; and (3) the relative position of the observer with respect to the lightning channel. Perhaps the most convincing discussion of thunder 51 generation as described above comes not from analytical evidence but from research using sophisticated com- puter models of thunder. Ribner and Roy (1982) synthe- sized thunderlike acoustic signals utilizing computer- generated waves formed by the superposition of N waveless from tortuous geometric sources. The resulting "thunder" is highly similar to natural thunder (see Fig- ures 4.8 and 4.9~. Where the computer models are used to simulate laboratory experiments, there is also close agreement. PROPAGATION EFFECTS Once generated, the acoustic pulses from the light- ning channel must propagate for long distances through the atmosphere, which is a nonhomogeneous, aniso- tropic, turbulent medium. Some of the propagation ef- fects can be estimated by modeling the propagation us- ing appropriate simplifying assumptions; however, other effects are too unpredictable to be reasonably modeled and must be considered in individual situa- tions. Three of the largest propagation effects finite-am- plitude propagation, attenuation by air, and thermal refraction can be treated with appropriate models to account for average atmosphere effects. Reflections from the flat ground can also be easily treated. Once the horizontal wind structure between the source and the receiver are measured, the refractive effects of wind shear and improved transient times may also be calcu- lated. Beyond these effects, elements such as vertical , O - a. .L~/ u~ ~ _ ~ 4d~ ~ 7b9~ \~- in ~ . > L ~ ~ ~ p'~ -ale _~.0 .lIIl.~lll~lllillIlllllllllllllllllllllllll-~-lJlllllllllllllllllllli 0 ~ ~ taco 4n .~ ~ m . - .o TIME, s FIGURE 4.8 Schematic depiction of the synthetic generation of thunder by computer by the superimposition (upper trace) of N wave- lets from a tortuous line source (Ribber and Roy, 1982); the summed signal is shown on the lower trace.

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52 I.6 ~ a Ill.lll . O ~ 41115111 Ct G1 1.: 1 ~ I ~I I ~ 1.4 1.6 1.8 ~0 TIME, s FIGURE 4.9 Comparison of synthetic (upper trace) and real (lower trace) thunder signals (Ribner and Roy, 1982). winds, nonsteady storm-related horizontal winds, tur- bulence, aerosol effects, and reflections from irregular terrain produce complications that must be either ig- nored or examined on a case-by-case basis. Finite-Amplitude Propagation As large-amplitude acoustic waves propagate through air, theory predicts that the shape of the wave must evolve with time. A single pulse will evolve to the shape of an N wave (see, for example, the spark wave in Figure 4.5~; further propagation of the wave produces a lengthening of this N wave. The best theoretical treat- ment of this process for application to the thunder prob- lem is the one developed by Otterman (1959~. His for- mulation addressed the lengthening of a Brode-type pulse, such as Figure 4.2, from an initial length (Lo) at an initial altitude (Ho) down to the surface; his treat- ment differs from many others that do not include the change of ambient pressure (P0) with altitude. Few ARTHUR A . FEW, JR. (1982) used the Otterman theory to develop an expres- sion for the lengthening of acoustic pulses generated by mesotortuous lightning-channel elements. The result for the length of the positive-pressure pulse at the ground, Lg. is given by 2 (`L 3/2 _ Lo3/2) = + 1 RoLoi/2Ho 4y [ ( Rocose ) 2 H ] (4 3) Ro is the distance from the channel to the front of the pulse at the initial state where the fractional overpres- sure at the pulse front is IIo = bPo/Po. The angle ~ is measured between the acoustic ray path and vertical; By is the ratio of specific heats; and Hg is the atmosphere scale height. Equation (4.3) provides the finite-amplitude stretch- ing that should be applied to the waves predicted by strong-shock theory. Uman et al. (1970) demonstrated that pulse stretching occurred beyond Brode's final pres- sure profile shown in Figure 4.2; we see this clearly in Figure 4.7. Few (1969) used linear propagation beyond the profile of Figure 4.2 to estimate the power spectrum of thunder but commented that nonlinear effects may be important. The need for application of nonlinear or finite-amplitude theory to the thunder signal has been voiced in a number of papers in addition to these men- tioned above (e.g., Holmes et al., 1971; Few, 1975, 1982; Hill, 1977; Bass, 1980~. If the Brode pressure pulse (shown in Figure 4.4) is used as the initial condition for the finite-amplitude propagation effect, the following values for input to Eq. (4.3) are R0 = 10.46RC, Lo = 0.53RC, andII0 = 0.03. In addition, if By = 1.4 and Hg = 8 x 103 m are used in Eq. (4.3), the following equation is obtained: L. = RCLO.386 + 0.147~1n: g ~ 10.46RC cos 16 X 103 ]3 Equation (4.4) has been used to generate the values in Table 4.2. The relaxation radii (Rc) cover the entire range of values for Rc in Table 4.1. Three values for ~ are represented, as are three heights for the source. In gen- eral, the finite-amplitude propagation causes a dou- bling in the length of the positive pulse within the first kilometer, but beyond this range the wavelength re- mains approximately constant. The theory developed by Otterman did not include attenuation of the signal; because attenuation reduces wave energy, which in turn

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ACOUSTIC RADIATIONS FROM LIGHTNING TABLE 4.2 Finite-Amplitude Stretching of a Positive Pulse (Length, Lo) for a Range of Cylindrical Relaxation Radii (Rc), Source Heights (Ho), and Angles (id, See Eq. (4.4) 53 Rc (m) 0.20 0.40 0.60 0.80 1.00 1.50 2.00 2.50 3.00 3.50 L,, (m) 0.11 0.21 0.32 0.42 0.53 0.80 1.06 1.33 1.59 1.86 = Do Lg(m),H(,= lkm 0.24 0.45 0.65 0.84 1.03 1.49 1.93 2.35 2.77 3.17 L~(m),HO = 4km 0.26 0.49 0.71 0.93 1.14 1.66 2.16 2.65 3.12 3.59 L~(m),H`, = 8km 0.26 0.51 0.74 0.96 1.18 1.72 2.24 2.75 3.25 3.74 = 45 L~(m),H,,= lkm 0.24 0.46 0.67 0.87 1.06 1.54 1.99 2.44 2.87 3.30 L(m), H., = 4km 0.26 0.50 0.73 0.96 1.18 1.71 2.22 2.73 3.22 3.71 L~(m),H,' = 8km 0.27 0 52 0.76 0.99 1.22 1.77 2.31 2.83 3.35 3.85 = 60 Lg(m),H,,= lkm 0.25 0.47 0.69 0.89 1.10 1.59 2.06 2.52 2.98 3.42 L~(m),H,, = 4km 0.27 0.51 0.75 0.98 1.21 1.76 2.29 2.81 3.32 3.82 Lg(m),H,,= 8km 0.28 0.53 0.77 1.01 1.25 1.81 2.37 2.91 3.44 3.97 reduces the wave stretching, this theory should be viewed as a maximum estimator of the pulse length. The finite-amplitude propagation effect does, how- ever, help to resolve the overestimate of lightning-chan- nel energy made by acoustic power-spectra measure- ments. Few (1969) noted that the thunder-spectrum method yielded a value for E' that was an order of mag- nitude greater than an optical measurement by Krider et al. (1968~. By assuming a doubling in wavelength by the finite-amplitude propagation, the energy estimate is reduced by a factor of 4, bringing the two measurements into a range of natural variations and measurement pre . . clslon. Attenuation There are three processes on the molecular scale that attenuate the signal by actual energy dissipation; the wave energy is transferred to heat. Viscosity and heat conduction, called classical attenuation, represent the molecular diffusion of wave momentum and wave in- ternal energy from the condensation to the rarifaction parts of the wave. The so-called molecular attenuation results from the transfer of part of the wave energy from the translational motion of molecules to their internal molecular rotational and vibrational energy during the condensation part of the wave and back out during the rarifaction part of the wave. The phase lag of the energy transfer relative to the wave causes some of the internal energy being retrieved from the molecules to appear at an inappropriate phase; thus it goes into heat rather than the wave. These three processes can be treated the- oretically within a common framework (Kinsler and Frey, 1962; Pierce, 1981~. The amplitude of a plane wave, ~ P. as a function of the distance, x, from the coor- dinate origin is given by hP= bPOe~~X, (4.5) where TYPO is the wave amplitude at the origin. The coef- ficient of attenuation, cat, can be shown in the low-fre- quency regime to be <,,2 2e (4.6) In Eq. (4.6), ~ is the angular frequency and T iS the re- laxation time (or e-folding time) for the molecular pro- cess being considered; c is the speed of sound. The low- frequency condition above assumes that cur < 1. The expressions for depend on the particular molecular pro- cesses under consideration; it is important to note, how- ever, that cat is proportional to w2 for the assumed condi- tions; hence, attenuation alters the spectral shape of the propagating signals. For thunder at frequencies below 100 Hz it can be shown (Few, 1982) that the total attenuation is insignifi- cant. However, for the many small branches having much lower energy than the main channel, the frequen- cies will be much higher and attenuation is important. Because of lower initial acoustic energies, spherical di- vergence, and attenuation it is unlikely that acoustics emitted by the smaller branches and channels can be easily detected over longer distances (see also Bass, 1980; Arnold, 1982~. Scattering andAerosol Effects The scattering of acoustic waves from the cloud parti- cles is similar to the scattering of radar waves from the particles; both are strongly dependent on wavelength. The intensity of the scattered sound waves from a plane acoustic wave of wavelength, )\' incident on a hard sta

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54 tionarv.snhere of rarli'~.s ra , ~ is proportional to (brat (alkyd; this is the same relationship that appears in the radar cross-section expression for these parameters. For thun- der wavelengths ~ ~ 1 m3 and cloud particles ~ ~ 10 - 3 m) the ratio (alkyd is 10- ]2. The cloud is, therefore, trans- parent to low-frequency thunder just as it is to meter- wavelength electromagnetic radiation, although insig- nificant fractions of the radiation do get scattered. There are, however, eddies in the same size range as low-frequency thunder wavelengths, and these fea- tures, owing to small thermal changes and flow shears, produce a distortion of wave fronts and scattering-type effects. For the part of the turbulent spectrum having wavelengths smaller than the acoustic wavelengths of interest, the turbulence can be treated statistically by scattering theory. Larger-scale turbulence must be de- scribed with geometric acoustics. For the low-frequency thunder, turbulent scattering will attenuate the high- amplitude beamed parts of the thunder signal; this in- creases the rumbles at the expense of claps. In the first part of this subsection we discussed the cloud particles as sources of acoustic scattering; there are other and probably more important ways in which these aerosol components interact with the acoustic waves. First, the surface area of the cloud particles within a volume provides preferred sites for enhanced viscosity and heat conduction; hence, the presence of particles increases the classical attenuation coefficient. Another totally different process produces attenuation by changing the thermodynamic parameters associated with the acoustic wave over the surfaces of cloud parti- cles; this changes the local vapor-to-liquid or vapor-to- solid conversion rates. For example, during the com- pressional part of the wave the air temperature is increased and the relative humidity is decreased relative to equilibrium; the droplets partially evaporate in re- sponse and withdraw some energy from the wave to ac- complish it. The opposite situation occurs during the ex- pansion part of the wave. Because the phase-change energy is ideally 180 out of phase with the acoustic-wave energy this process produces attenuation. Landau and Lifshitz (1959) included this effect in their "second vis- cosity" term. This attenuation process differs from the other microscopic processes in that it can be effective at the lower frequencies. The magnitude of this effect plus the enhanced attenuation by viscous and heat conduc- tion at the surface exceed that of particle-free air by a factor of 10 or greater depending on the type, size, and concentration of the cloud particles (Kinsler and Frey, 1962). Finally, there is a mass-loading effect with respect to the cloud particles that must be considered. The ampli- tude of the fluid displacement, (, produced by an acous ARTHUR A. FEW, JR. tic wave of pressure amplitude lip and angular fre- quency ~ is (Kinsler and Frey, 1962) it= hip . po c cd (4.7) Using 50 Pa as a representative value of lip for thunder inside a cloud we find for a 100-Hz frequency that ~ = 100 ,um. The part of the cloud particle population whose diameter is much smaller than this, say 10 ,um, should, owing to viscous drag, come into dynamic equilibrium with the wave flow. [Dessler (1973) computed the re- sponse time for a ~ 10-,um droplet to re-establish dy- namic equilibrium with drag forces; only 10 - 3 see is re- quired. ~ These cloud particles, which participate in the wave motion, add their mass to the effective mass of the air; this effects both the speed of sound and the impe- dence of the medium. For higher-frequency waves, fewer cloud particles participate, so the effect is re- duced; whereas lower-frequency waves include greater percentages of the population and are more strongly af- fected. Clouds are, therefore, dispersive with respect to low-frequency waves. Also, the cloud boundary acts as a partial reflector of the low-frequency acoustic signals because of the impedence change at the boundary. As- suming a total water content of order 5 g/m3, we esti- mate that the order of magnitude of the effect on sound speed and impedence is 10 ~ 3; this is not large, but it may be detectable. The cloud aerosols interact with the acoustic waves in three different ways depending on their size relative to the amplitude of air motion of the sound. The smallest fraction "ride with the wave" altering the wave-prop- agation parameters. The largest particles are stationary and act as scatterers of the acoustic waves. The particles in the middle range provide a transition scale for the above effects but are primarily responsible for enhanced viscous attenuation. In summary, there are several processes that can ef- fectively attenuate higher-frequency components of thunder; this is in support of the conclusions of the pre- vious section. We have, in addition, found three pro- cesses that affect the low-frequency components. Low frequencies can be attenuated by turbulent scattering and, in the cloud, by coupling wave energy to phase changes. We have also found that low frequencies inter- act with the cloud population dynamically; as a result, cloud boundaries may act as partial reflectors and in- cloud propagation may be dispersive. Refraction There is a wide range of refractive effects in the envi- ronment of thunderstorms. In the preceding section we

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ACOUS TIC RADIA TIONS FROM LIGHTNING found that turbulence on the scale of the acoustic wave- length and smaller could be treated with scattering the- ory. Turbulence larger than acoustic wavelengths, up to and including storm-scale motions, should be describ- able by geometric acoustics or ray theory. To actually do this is impractical because it requires detailed informa- tion (down to the turbulent scale) of temperature and velocity of the air everywhere along the path between the source and the observer. Since the thunder sources are widely distributed we would require complete knowledge of the storm environment down to the meter scale to trace accurately the path of an individual acous- tic ray. These requirements can be relieved if we relax somewhat our expectations regarding the accuracy of our ray path. The three fluid properties that cause an acoustic ray to change its direction of propagation are the components of thermal gradient, velocity gradient, and velocity that are perpendicular to the direction of propagation. Beyond the overall thermal structure of the environment, which will be approximately adia- batic, we do not expect that the thermal perturbations due to turbulence will be systematic. In fact, the turbu- lent thermal perturbations should be random with a zero average value; hence, an acoustic ray propagating through turbulence should not deviate markedly owing to thermal gradients associated with the turbulence from the path predicted by the overall thermal structure of the environment. Similarly, velocity and velocity gra- dients should produce a zero net effect on the acoustic ray propagating through the turbulence. This argument of compensating effects is not valid for large eddies whose dimensions are equal to or greater than the path length of the ray because the ray path is over a region containing a systematic component of the gradients associated with the large eddy. We can obtain a worst-case estimate of these effects by examining a horizontal ray propagating from a source at the center of an updraft of 30 m/see through 2 km to the cloud boundary where the vertical velocity is assumed to be zero; we also assume a linear decrease in vertical veloc- ity between the center and boundary. The ray will be "adverted" by 90 m upward during this transit, which requires approximately 6 see, while the direction of propagation of the ray will be rotated through 5 down- ward (maximum angle ~ tan- ~ /\ VIC). Owing to this rotation, which is a maximum computation, the "ap- parent" source by straight-ray path would be 180 m above the real source. These two effects have been esti- mated independently when, in fact, they are coupled and are to some extent compensatory; when we merely add them the result is an overestimate of the apparent source shift, which in this example is 270 m. If this worst case is the total error in propagation to the receiver at 5 55 km then this error represents 5 percent of the range; over the length in which it occurs, 2 km, it represents 13 per- cent error. Now we turn our attention to the large-scale refrac- tion effects that can be incorporated in an atmospheric model that employs horizontal stratification. The two strongest refractive effects of the atmosphere the ver- tical thermal gradient and boundary-layer wind shears fall into this category along with other winds and wind shears of less importance. The nearly adiabatic thermal structure of the atmo- sphere during thunderstorm conditions has been recog- nized for a long time as a strong influence on thunder propagation (Fleagle, 1949~. This thermal gradient is effective because it is spatially persistent and unavoid- able. Even though the temperature in updrafts and downdrafts inside and outside the cloud may differ (sometimes significantly), the thermal gradients in all parts of the system will be near the adiabatic limit (or pseudoadiabatic in some cases) because of the vertical motion. Hence, the acoustic rays propagate in this strong thermal gradient throughout its existence. We can employ a simplified version of ray theory to illustrate some of the consequences of this thermal struc- ture. If we assume no wind, a constant lapse rate (F = - ~ Tl~z), and is Tl To << 1 (A T is the change in temper- ature and To is the maximum temperature along the path), then the ray path may be described as a segment of a parabola 2 = 4To h. (4.8, In Eq. (4.8), To also corresponds to the vertex of the parabola where the ray slope passes through zero and starts climbing. h and I are, respectively, the height above the vertex and the horizontal displacement from the vertex. To apply Eq. (4.8) to all rays it is necessary to ignore (mathematically) the presence of the ground be- cause the vertices of rays reflecting from the ground are mathematically below ground. In addition, we must in other cases visualize rays extending backward beyond the source to locate their mathematical vertices. If To is set equal to the surface temperature, a special acoustic ray that is tangent to the surface when it reaches the surface is defined; this is depicted in Figure 4.10. This same ray is applicable to any source, such as So, S2, or S3, that lies on this ray path. For the conditions assumed in this approximation it is not possible for rays from a point source to cross one another (except those that reflect from the surface). The other acoustic rays emanating from S2 must pass over the point on the ground where the tangent ray makes contact; this is also true for rays reflecting from the surface inside the tan

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56 FIGURE 4.10 Parabolic acoustic ray from sources So, S. or S3 tangent to the surface at P. This ray was generated utilizing Eq. (4.8) with T.`, = 30C and ~ = 9.8 K/km. Observ- ers on the surface to the right of P cannot de- tect sound from sources So, S. S3, or S.'; an observer at P can only detect sound originat- ing on or above the parabolic ray shown. ARTHUR A . FEW, JR. l - o Ss SO - ~ - ~. . . . . . , , . , , , , ,,77717 S 0 5 10 40 3S 30 gent point. The shaded zone in Figure 4.10 corresponds to a shadow zone that receives no sound from any point source on the tangent ray beyond the tangent point. Point sources below the tangent ray, such as source S4 in Figure 4.10, have their tangent ray shifted to the left in this representation and similarly cannot be detected in the shadow zone. However, sources above the tangent ray, S5 for example, can be detected in some parts of the shadow zone. For each observation point on the ground one can de- fine a paraboloid of revolution about the vertical gener- ated by the tangent ray through the observation point; the observer can only detect sounds originating above this parabolic surface. For this reason we usually hear only the thunder from the higher parts of the lightning channel unless we are close to the point of a ground strike. For evening storms, which can often be seen at long distances, it is common to observe copious li~ht- ning activity but hear no thunder at all; thermal refrac- tion is the probable cause of this phenomenon. For To = 30C, 1~ = 9.8K/km,andh = 5kmwefindthatl= 25 km; as noted by Fleagle (1949) thunder is seldom heard beyond 25 km. (See also the discussion in Ribner and Roy, 1982.) Winds and wind shears also produce curved-ray paths but are more difficult to describe because they af- fect the rays in a vectorial manner, whereas the temper- ature was a scalar effect. If you are downwind of a source and the wind has positive vertical shear (bu/3z > 0), the rays will be curved downward by the shear; on the upwind side, the rays are curved upward. Wind shears are very strong close to the surface and can effec- tively bend the acoustic rays that propagate nearly par- allel to the surface. The combined effects of tempera- ture gradients, winds, and wind shears can best be handled with a ray-tracing program on a computer. With such a program one can accurately trace ray paths through a multilevel atmosphere with many variations in the parameters; it is usually necessary in these pro- grams to assume horizontal stratification of the atmo- sphere. The accuracy of the ray tracing by these tech 2S 20 t 5 tO Distance in km A ,0; i_ s ~ ._ - niques can be very high, usually exceeding the accuracy with which temperature and wind profiles can be deter- mined. MEASUREMENTS AND APPLICATIONS A number of the experimental and theoretical re- search papers dealing with thunder generation have been discussed in earlier sections and will not be re- peated here. In this section we describe additional results, techniques, and papers that deal with thunder measurements. Propagation Effects Evaluation The reader should have, at this point, an appreciation for the difficulty in quantitatively dealing with the propagation effects on both the spectral distribution of thunder and the amplitude of the signal. If, however, we are willing to forfeit the information content in the higher-frequency ~ > 100 lIz) portion of the thunder sig- nal, which is most strongly affected by propagation, we can recover some of the original acoustic properties from the low-frequency thunder signal. If the peak in the original power spectrum of thunder is assumed to be below 100 Hz, then the "2-attenuation effects deplete the higher frequencies without shifting the position of the peak. Most spectral peaks of thunder tend to be around or below 50 Hz; therefore, this as- sumption appears to be safe even with finite-amplitude stretching effects considered. Further assume that the spectra are not substantially altered by turbulent scat- tering and cloud aerosols. To the extent that these as- sumptions are valid, the finite-amplitude stretching can be removed from the thunder signal and its peak fre- quency at the source can be estimated. This technique enables a rough estimate of the energy per unit length of the stroke to be made; the result is corrected for first- order propagation effects. Holmes et al. (1971) found that the spectral peak overestimated the channel energy using Few's (1969) method; if corrected for stretching

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A CO US TIC RADIA TIONS FR OM LIGH TNING these measurements are in closer agreement, with the exception of those events containing other lower-fre- quency acoustic sources. There are a number of experiments that could and should be done to evaluate the propagation effects. Us- ing thunder as the acoustic source, several widely sepa- rated arrays of microphones could compare signals from the same source at several distances. If carefully exe- cuted this experiment could quantify some of the propa- gation effects. Another approach would be to employ a combination of active and passive experiments such as point-source explosions inside clouds from either bal- loons or rockets. This experiment provides an additional controllable factor that can yield more precise data; it also involves greater cost and hazard. Acoustic Reconstruction of Lightning Channels In the section on refraction we mentioned the utility of ray-tracing computer programs that could accurately calculate the curved path of an acoustic ray from its source to a receiver; the accuracy is limited to the preci- sion with which we are able to define the atmosphere. An obvious application of thunder measurements is to invert this process; one measures thunder then traces it backward from the point of observation along the ap- propriate ray to its position at the time of the flash. Few (1970) showed that by performing this reverse-ray prop- agation for many sources in a thunder record it was pos- sible to reconstruct in three dimensions the lightning channel producing the thunder signal. The sources in this case were defined by dividing the thunder record into short ~ ~ 1/2 see) intervals and associating the acous- tics in a given time interval with a source on the channel. Within each time interval the direction of propaga- tion of the acoustic rays are found by cross correlating the signals recorded by an array of microphones. The position of the peak in the cross-correlation fraction gives the difference in time of arrival of the wave fronts at the microphones; from this and the geometry of the array, one calculates the direction of propagation. At least three noncollinear microphones are required. Close spacing of the microphones produces higher corre- lations and shorter intervals thus more sources; how- ever, the pointing accuracy of a small array is less than that of a large array. Based on experiences with several array shapes and sizes, 50 m2 has been adopted as the optimum by the Rice University Group (see Few, 1974a). The reconstruction of lightning channels by ray trac- ing was described by Few (1970) and Nakano (1973~. A discussion of the accuracy and problems of the tech- nique is given in Few and Teer (1974) in which acousti 57 cally reconstructed channels were found to agree closely with photographs of the channels below the clouds. The point is dramatically made in these comparisons that the visual part of the lightning channel is merely the "tip of the iceberg." Nakano (1973) reconstructed, with only a few points per channel, 14 events from a single storm. Teer and Few (1974) reconstructed all events during an active pe- riod of a thunderstorm cell. MacGorman et al. (1981) similarly performed whole-storm analyses by acoustic channel reconstruction and compared statistics from several different storm systems. Reconstructed lightning channels by ray tracing have been used to support other electric observations of thunderstorms at the Langmuir Laboratory by Weber et al. (1982) and Winn et al. (1978~. A second technique for reconstructing lightning channels has been developed that is called thunder rang- ing. This technique was developed to provide a quick coarse view of channels (within minutes after lightning if necessary) as opposed to the ray-tracing technique, which is slow and time consuming. Thunder ranging requires thunder data from at least three noncollinear microphones separated distances on the order of kilome- ters. Experience with cross-correlation analysis of thun- der signals has shown that the signals become spatially incoherent at separations greater than 100 m owing to differences in perspective and propagation path. How- ever, the envelope of the thunder signals and the gross features such as claps remain coherent for distances on the order of kilometers. As discussed earlier these gross features are produced by the large-scale tortuous sec- tions of the lightning channel. Thunder ranging works as follows: (1) The investigator identifies features in the signals (such as claps) that are common to three thunder signatures on an oscillograph. (2) The time lags between the flash and the arrival of each thunder feature at each measurement point are determined. (3) The ranges to the lightning channel segments producing each thunder feature are computed. (4) The three ranges from the three separated observation points for each thunder fea- ture define three spheres, which should have a unique point in space that is common to all of them. (5) The set of points gives the locations of the channel segments pro- ducing the thunder features (see Few, 1974b; Uman et al., 1978~. The basic criticism of the thunder-ranging technique is that the selection of thunder features is the subjective judgment of the researcher; for many features the selec- tion is unambiguous; other features, which are close to- gether, may appear separated at one location and merged at another. The program developed by Bohan- non (1978) included these uncertainties in the estima

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58 lion of errors associated with such points. Most of the recent thunder research has used a combination of rang- ing and ray tracing. The whole-storm studies in which an extended series of channels are reconstructed have proven to be the most valuable use of thunder data to date. They define the volume of the cloud actually producing lightning, the evolution of the lightning-producing volume with time, and the relationship of individual channels with other cloud observations such as radar reflectivity and envi- ronmental winds (Nakano, 1973; Few, 1974b; Teer and Few, 1974; Few et al., 1977, 1978; MacGorman and Few, 1978; MacGorman et al., 1981~. ELECTROSTATICALLY PRODUCED ACOUSTIC EMISSIONS The concept of electrostatically produced acoustic waves from thunderclouds goes all the way back to the ARTHUR A . FEW, JR. writings of Benjamin Franklin in the eighteenth cen- tury; Wilson (1920) provided a rough quantitative esti- mate of the magnitude of the electrostatically produced pressure wave. McGehee (1964) and Dessler (1973) de- veloped quantitative models for this phenomenon McGehee for spherical symmetry and Dessler for spheri- cal, cylindrical, and disk symmetries. The theory developed by Dessler is of particular importance be- cause it made specific predictions regarding the direc- tivity and shape of the wave. The predictions were sub- sequently verified in part by Bohannon et al. (1977) and Balachandran (1979, 1983~. The charge in a thundercloud resides principally on the cloud drops and droplets. In a region of the cloud where the charge is concentrated producing an electric field E, the charged particles will experience an electric force, which is directed outward with respect to the charge center, in addition to the other forces expressed on them. These particles quickly (on the order of milli ,._. , _ ~, ~'a ~ ~7 ~ ~' T i ! ~ ~ S !. ~t' ~' j j ~ ' ~ ~' ~ ~ T ~' i i ' ~ ! ~ i ; ~, __ ~,! r ' i ~ ~ ~' ~~~~ ~ ~ ~ 1 ~ __ - ~ = {_ ~ FIGURE 4.11 Low-frequency acoustic pulse thought to have been generated by an electrostatic pressure change inside the cloud during a light- ning flash. The higher-frequency signals from thunder have been removed from this record. From Balachandran (1979) with permission of the American Geophysical Union.

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ACOUSTIC RADIATIONS FROM LIGHTNING seconds) come into dynamic equilibrium where the hy- drodynamic drag force associated with their motion is balanced by the sum of all the externally expressed forces. When the electric field is quickly reduced by a lightning flash the cloud particles readjust to a new dy- namic equilibrium. The change in the hydrodynamic drag force requires a change in the pressure distributions surrounding all the charged cloud particles, hence, the pressure in the volume continuing the cloud particles is altered by the sudden reduction of the electric field. Since the electric force from a charge concentration is outward, the pressure inside the charged volume will be slightly lower than the surrounding air. When E is re- duced by the lightning flash the charged volume pro- duces a slight implosion; this radiates a negative wave. Few (1982) derived a general expression for the internal pressure gradient produced by the electrostatic force; when integrated the result is p p ~ 1y co(Eo - E2) (49) In Eq. (4.9) the parameter n takes the value 0 for plane geometry, 1 for cylindrical geometry, and 2 for spherical geometry; P0 and En are the values at the edge of the charged volume. The amplitude of this pressure signal is related to the electric field, the wavelength to the thickness of the charged region, and the directivity of the wave to the geometry to the source (Dessler, 1973~. If the theory can be quantitatively verified, the signal can be used to determine remotely internal cloud electric parameters. The experimental search for electrostatic pressure waves has been difficult. The wave is low frequency (~ 1 Hz), small amplitude (~ 1 Pa), and buried in large background pressure variations produced by wind, tur- bulence, and thunder. Prior to Dessler's prediction of beaming, one wondered why the signal was not more frequently seen in thunder measurements. Holmes et al. (1971) measured a low-frequency component in a few of their power spectra of thunder but found these compo- nents completely missing in others. Dessler showed that that signal would be beamed for cylindrical and disk geometry; the disk case would require that the detectors be placed directly underneath the charged volume for observation. This relationship has been observed by Bo- hannon et al. (1977) and by Balachandran (1979, 1983~. The electrostatic pressure wave predicted by the the- orv discussed above is a negative pulse. The measured acoustic signature thought to be the verification of the prediction actually exhibits a positive pulse followed by a negative pulse (see Figure 4.11~. The negative pulse appears to fit the theory, but the theory is deficient in 59 that the positive component of the wave is not de scribed. Recently, Few (1984) suggested that the dia- batic heating of the air in the charged volume by posi- tive streamers may be the source of the positive pulse. Colgate and McKee (1969) described theoretically an electrostatic pressure pulse using this same mechanism but applied to a volume of charged air surrounding a stepped leader. This particular signature has not been experimentally verified because it has the regular thun- der signal, which is 300 times more energetic, superim- posed on it. ACKNOWLEDGMENT The author's research into the acoustic radiations from lightning has been supported under various grants and contracts from the Meteorology Program, Division of Atmospheric Sciences, National Science Foundation, and the Atmospheric Sciences Program, Office of Naval Research; their support is gratefully acknowledged. REFERENCES Arnold, R. T. (1982). Storm acoustics, in Instruments and Techniques for Thunderstorm Observation and Analysis, E. Kessler, ea., U.S. Department of Commerce, Washington, D.C., pp. 99-116. Balachandran, N. K. (1979~. Infrasonic signals from thunder, J. Geophys. Res. 84,1735-1745. Balachandran, N. K. (1983). Acoustic and electric signals from light- ning, J. Geophys. Res. 88, 3879-3884. Bass, H. E. (1980~. The propagation of thunder through the atmo- sphere, J. Acoust. Soc. Am. 67, 1959-1966. Bohannon,J.L.(1978).Infrasonicpulsesfromthunderstorms,M.S. thesis, Rice Univ., Houston, Tex. Bohannon, J. L., A. A. Few, and A. J. Dessler (1977). Detection of infrasonic pulses from thunderclouds, Geophys. Res. Lett. 4, 49-52. Brode, H. L. (1955). Numerical solutions of spherical blast waves, J. Appl. Phys. 26, 766. Brode, H. L. (1956). The blast wave in air resulting from a high tem- perature, high pressure sphere of air, Rand Corp. Res. Memoran- dum RM-1825-AEC. Colgate, S. A., and C. McKee (1969). Electrostatic sound in clouds and lightning, J. Geophys. Res. 74, 5379-5389. Dessler, A. J. (1973). Infrasonic thunder, J. Geophys. Res. 78, 1889- 1896. Few, A. A. (1969). Power spectrum of thunder, J. Geophys. Res. 74, 6926-6934. Few, A. A. (1970). Lightning channel reconstruction from thunder measurements, J. Geo phys. Re.s. 75, 7517-7523. Few, A. A. (1974a). Thunder signatures, EOS 55, 508-514. Few, A. A. (1974b). Lightning sources in severe thunderstorms, in Conference on Cloud Physics (Preprint volume), American Mete- oroloaical Society, Boston, Mass., pp. 387-390. Few, A. A. (1975). Thunder, Sci. Am. 233(1), 80-90. Few, A. A. (1982). Acoustic radiations from lightning, in Handbook of Atmo.spheric.s, Vol. 2, H. Volland, ed.- CRC Press, Inc., Boca Ra- ton, Fla., pp. 257-289. Few, A. A. (1984). Lightning-associated infrasonic acoustic sources, in

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