The rupture process at the near-front zone of a propagating crack, being strongly nonlinear and irreversible, a detailed Fourier analysis becomes complicated, often impractical. Meanwhile, a simplified approach, employing formally derived dispersion equation, may serve for obtaining meaningful information. Data on the rupture velocity, the size of the fracture process zone (FPZ), the possibility of exponential growth in time, and also on decaying with the distance from the rupture surface become available. The approach actually presumes that the size of the FPZ defines a dominant wavelength in the Fourier spectrum of a propagating rupture pulse. We verify this suggestion by revisiting the classical problem of a propagating shear crack having a cohesive FPZ with linear softening. It appears that in this particular problem, the simplified approach provides results which remarkably agree with the exact solution. This suggests using the approach for the analysis of the unique data on a rupture pulse obtained in the Chi-Chi (Taiwan, 20 September 1999) earthquake. We conclude that the approach allows easy interpretation of the observed phenomena. The importance of the softening modulus as an independent characteristic of initiation and propagation of rupture is emphasized.