We investigate the temporal evolution of a simulated gravitational wave spectrum modelled on a cosmological population of transient sources with characteristic rest-frame frequency of 1 kHz. Our purpose is to see what might be learned about the cosmic distribution of sources by the way in which such spectra build up over short spans of observation time. The spectral evolution depends on the history of source rate evolution and the observation time; and for a universal event rate of some tens s(-1) (comparable to the neutron star birth rate) locked to the evolving star formation rate, the spectrum evolves rapidly within the first seconds of observation. A rapid increase in bandwidth occurs because of the large population of sources at moderate to high redshift. Spectra calculated using two observation-based star formation rate models and one simulated model show a relatively stationary low-frequency peak arising from the high-z sources and a time-dependent higher-frequency edge resulting from rarer nearby sources. The spectra converge to a stable form within an observation period of about 20 min for a universal event rate of about 15-30 s(-1). As a supplement, we provide Web-based movie files that highlight the rapid spectral evolution.