Under the assumption of almost constant-Q behavior of solids over a wide range of frequencies, together with some meaningful assumptions about the linear frequency behavior of the attenuation function, we have been able to obtain theoretical expressions describing the waveform distortion of an impulse-excited plane wave as it decays and spreads on passing through large distances of the solid. When typical values for the parameters (as obtained from laboratory model or short range field experiments) for a solid having a mechanical Q of about 50 are used, the resulting waveforms at first glance appear to have a simple and not unexpected behavior. The peak amplitude of the waveform in the time domain varies roughly with the inverse of the square of the travel distance (this includes an inverse first power due to geometrical spreading). Also, a spreading of the waveform occurs that varies roughly linearly with the travel distance. This spreading is such that a positive impulse simply broadens as it travels without developing any zero crossings in its wave shape. However, it turns out that the part of the waveform leading up to the visible onset does not admit of a purely elastic interpretation, so that one cannot relate conceptually the arrival time with a purely elastic-wave velocity. For our solid of Q nearly equal 50, about one-seventh of the arrival time duration is due to a purely inelastic behavior; during this period of inelastic behavior the amplitude is not zero but it is deceptively small. We have designated this portion as a 'pedestal.' On observing seismic records, we are never aware of this conceptual division and consequently attribute to the observed arrival an elastic wave velocity, which in our example happens to be about 15 percent less than the actual elastic wave velocity of the solid. Although the existence of the pedestal may not appear to be significant when we restrict our observations to seismic records limited in bandwidth to the low frequencies encountered in exploration or earthquake seismology, an upward curvature which is not associated with a discontinuity in the slope at the time of visual onset will remain and can be of great importance if very accurate arrival time measurements are made. This part of the pedestal can become even more important when we consider refraction arrivals, where the onset undergoes an additional time integration that further enhances the upward curvature of the pedestal.