A.1.1 Fourier Transform. Spectrum of Finite Harmonic Wave Train

Consider an oscillation of finite length, a chirp, such that F(t) = A cos( ? ₀ t) between t = ? t ₀ and t = t ₀ and zero elsewhere. According to Eq. 2.67, the Fourier amplitude of this function is

where ? = 2 ?? and X ₊ = ( ? + ? ₀) t ₀ and X _? = ( ? ? ? ₀) t ₀, which, for computational purposes, we express as X = ( ? 1)2 ?t ₀ /T ₀, where T ₀ = 2 ?/ ? ₀ is the period and ? = ?/ ? ₀. With reference to Section 2.6.3, the energy spectrum density is E( ?) = 2 F( ?) ² and it should be recalled that only positive frequencies are involved in this expression. In Fig. A.1 are shown the energy spectra for signal lengths from 0.5 to 8 periods. The 0.5 period signal covers the central maximum of the signal and, unlike the other cases, has a time average different from zero. This is the reason why the corresponding spectrum is quite different from the others; it has a maximum at zero frequency. Such a pulse is typical for explosive events which are...