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ACOUSTIC RADIATIONS FROM LIGHTNING 53 reduces the wave stretching, this theory should be viewed as a maximum estimator of the pulse length. TABLE 4.2 Finite-Amplitude Stetching of a Positive Pulse (Length, L 0) for a Range of Cylindrical Relaxation Radii (R c), Source Heights (H 0), and Angles (Î¸), See Eq. (4.4) R c (m) 0.20 0.40 0.60 0.80 1.00 1.50 2.00 2.50 3.00 3.50 L 0 (m) 0.11 0.21 0.32 0.42 0.53 0.80 1.06 1.33 1.59 1.86 Î¸ = 0Â° L g (m), H 0 = 1 km 0.24 0.45 0.65 0.84 1.03 1.49 1.93 2.35 2.77 3.17 L g (m), H 0 = 4 km 0.26 0.49 0.71 0.93 1.14 1.66 2.16 2.65 3.12 3.59 L g (m), H 0 = 8 km 0.26 0.51 0.74 0.96 1.18 1.72 2.24 2.75 3.25 3.74 Î¸ = 45Â° L g (m), H 0 = 1 km 0.24 0.46 0.67 0.87 1.06 1.54 1.99 2.44 2.87 3.30 L g (m), H 0 = 4 km 0.26 0.50 0.73 0.96 1.18 1.71 2.22 2.73 3.22 3.71 L g (m), H 0 = 8 km 0.27 0.52 0.76 0.99 1.22 1.77 2.31 2.83 3.35 3.85 Î¸ = 60Â° L g (m), H 0 = 1 km 0.25 0.47 0.69 0.89 1.10 1.59 2.06 2.52 2.98 3.42 L g (m), H 0 = 4 km 0.27 0.51 0.75 0.98 1.21 1.76 2.29 2.81 3.32 3.82 L g (m), H 0 = 8 km 0.28 0.53 0.77 1.01 1.25 1.81 2.37 2.91 3.44 3.97 The finite-amplitude propagation effect does, however, help to resolve the overestimate of lightning-channel energy made by acoustic power-spectra measurements. Few (1969) noted that the thunder-spectrum method yielded a value for E l that was an order of magnitude 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 precision. 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 internal 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 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 theoretically within a common framework (Kinsler and Frey, 1962; Pierce, 1981). The amplitude of a plane wave, Î´ P, as a function of the distance, x, from the coordinate origin is given by where Î´P0 is the wave amplitude at the origin. The coefficient of attenuation, Î±, can be shown in the low- frequency regime to be In Eq. (4.6), Ï is the angular frequency and Ï is the relaxation time (or e-folding time) for the molecular process being considered; c is the speed of sound. The low-frequency condition above assumes that wt < 1. The expressions for depend on the particular molecular processes under consideration; it is important to note, however, that Î± is proportional to Ï2 for the assumed conditions; 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 insignificant. However, for the many small branches having much lower energy than the main channel, the frequencies will be much higher and attenuation is important. Because of lower initial acoustic energies, spherical divergence, 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 and Aerosol Effects The scattering of acoustic waves from the cloud particles 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, Î» , incident on a hard sta