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ACOUSTIC RADIATIONS FROM LIGHTNING 48 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 4.2 The weak shock formed from the spherical strong shock wave. This is the final pressure profile computed by Brode (1956). For an energy input of 105 J/m (R c = 0.56 m for P0 = 105 Pa) this weak shock wave would be approximately 6 m from the lightning channel. Figure 4.4 presents a graphical summary of the various 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 Ã 103 m/sec) 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 transitions from cylindrical divergence to spherical divergence will occur somewhere beyond X = 0.3 and probably beyond X = 1 depending on the particular geometry of the channel at this point. The family of lines labeled Ï in Figure 4.4 represent transitions occurring at different points. Ï is the effective length, L, of the cylindrical source divided by Rc (Ï = L/R c ); it is approximately equal to the value of X at which the transition to spherical 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 were computed as a function of time for specified current 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 assumed current wave forms used in the models. The recent data obtained with fast-response-time equipment yields current rise times for natural cloud-to-ground lightning in the 35-50 KA/Âµsec range (Weidman and Krider, 1978). These values are considered as representative of normal strokes; extraordinary strokes have been measured with current rise times in the 100-200 KA/Âµsec range. By way of comparison, Hill's (1971) current rise time was 2.5 KA/Âµsec. Laboratory simulations of lightning have been successfully 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 Ã 106 V impulse generator was used to produce 4-m spark discharges in air (Uman et al., 1970). Circuit instrumentation allowed the measurement of the spark-gap voltage and current from which the power deposition can be computed. 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 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 weak shock, these wave shapes are nearly identical. From Few (1969) with permission of the American Geophysical Union.
ACOUSTIC RADIATIONS FROM LIGHTNING 49 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 extensive series of spark measurements. Figure 4.6 is in the same format as Figure 4.4. The center line passing through the scattered points and labeled L = 0.5 m corresponds (using the measured energy input of 5 Ã 103 J/m, which gives R c = 0.126 m) to Ï = 4 in Figure 4.4. The two boundary lines L = 6.25 cm and L = 4.0 m would correspond to Ï 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 (Ï 32) is too large to be depicted in Figure 4.4, where Ï = 4 is the last line shown. The data points of Figure 4.6 corresponding to the larger Ï or L values could represent situations where the shock- wave expansion was following the cylindrical behavior over a long distance, hence large Ï. However, if the expansions were truly cylindrical to that extent, then 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 expansion. (See also Figure 4.5.) 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 cylindrical behavior, but on expanding to distances of the same size as line irregularities they change to spherical expansion following curves similar to the depicted curves. From Few (1969) with permission of the American Geophysical Union. 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. c is assumed to be 4/3. From Few (1969) with permission of the American Geophysical Union. It is obvious from both the spark photographs and wave forms in Uman et al. (1970) that the spark is tortuous and produces multiple pulses. They found that the wave shapes, more distant from the spark where pulsetransit 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 theory. 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 difference 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, continues 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 associated with the higher core temperature the part of the