The Airy beam (AiB) has attracted a lot of attention recently because of its intriguing features. We have previously provided a cogent physical explanation for these properties by showing that the AiB is, in fact, a caustic of rays that radiate from the tail of the Airy function aperture distribution. We have also introduced a class of ultra wide band (UWB) Airy pulsed beams (AiPB), where a key step has been the use of a proper frequency scaling of the initial aperture field that ensures that all the frequency components propagate along the same curved trajectory so that the wavepacket of the AiPB does not disperse. An exact closed form solution for the AiPB has been derived using the spectral theory of transients (STT) which is an extension of the well know Cagniard–de Hoop (CdH) method. In this paper we discuss the properties of the AiB and AiPB, and use the present problem to discuss the relation between the CdH method and the STT.