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ACOUSTIC RADIATIONS FROM LIGHTNING 52 winds, nonsteady storm-related horizontal winds, turbulence, aerosol effects, and reflections from irregular terrain produce complications that must be either ignored or examined on a case-by-case basis. Figure 4.9 Comparison of synthetic (upper trace) and real (lower trace) thunder signals (Ribner and Roy, 1982). 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 treatment of this process for application to the thunder problem is the one developed by Otterman (1959). His formulation addressed the lengthening of a Brode-type pulse, such as Figure 4.2, from an initial length (L 0 ) at an initial altitude (H 0 ) down to the surface; his treatment differs from many others that do not include the change of ambient pressure (P 0 ) with altitude. Few (1982) used the Otterman theory to develop an expression 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 R 0 is the distance from the channel to the front of the pulse at the initial state where the fractional overpressure at the pulse front is Î 0 = Î´ P0 /P0 . The angle Î¸ is measured between the acoustic ray path and vertical; Î³ is the ratio of specific heats; and H g is the atmosphere scale height. Equation (4.3) provides the finite-amplitude stretching 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 pressure 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 mentioned 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 R 0 = 10.46 R c , L 0 = 0.53 R c , and Î 0 = 0.03. In addition, if Î³ = 1.4 and H g = 8 Ã 103 m are used in Eq. (4.3), the following equation is obtained: Equation (4.4) has been used to generate the values in Table 4.2. The relaxation radii (R c ) cover the entire range of values for R c in Table 4.1. Three values for Î¸ are represented, as are three heights for the source. In general, the finite-amplitude propagation causes a doubling in the length of the positive pulse within the first kilometer, but beyond this range the wavelength remains approximately constant. The theory developed by Otterman did not include attenuation of the signal; because attenuation reduces wave energy, which in turn