Seismic spectral models for chemical and nuclear explosions are used in many applications including network modeling and yield estimation. Here we compare the models presented in Denny and Johnson (1991) and Mueller and Murphy (1971) with each other and with new results from the Source Physics Experiments (SPE). We demonstrate analytically the two models are in substantial agreement for large and normally buried explosions, consistent with much of the historic data collected during nuclear testing. However for small and/or deeply buried explosions, the spectral predictions of the two models can differ significantly. For example, the predicted yield of a 1--km deep, MW 2 nuclear explosion differs by more than a factor of five, and for the same moment and depth chemical explosion, the difference is greater than a factor of ten. We compare the models with initial data from the Source Physics Experiments (SPE), which include small and over--buried chemical explosions. The corner frequency of the one--ton SPE explosion (SPE--2) is slightly higher than the Mueller and Murphy (1971) The ability to predict the expected seismic amplitudes from an explosion is important in a number of applications including exploration and nuclear test monitoring. An analytical model for P--waves was first developed by Sharpe (1942), based on pressure in a spherical cavity in an elastic medium. Since then a large number of explosion source spectral models have been developed based on empirical data and theoretical considerations (cf. Denny and Johnson, 1991). Such spectral models are used to determine network thresholds for nuclear test monitoring and to estimate source properties, such as yield and depth of burial from seismic data.With the signing of the Comprehensive nuclear Test--Ban Treaty (CTBT) in 1996, and the cessation of testing by the signatories, any future nuclear tests may be conducted outside of standard practices and it is important to predict seismic observables for new regions and testing conditions. The development and validation of physics--based explosion models that extend the range of the current, more empirically, derived models is a focus of the DOE/NNSA funded Source Physics Experiments (SPE) (e.g. NNSA, 2012). The SPE focuses on physics--based model development work supported by a new chemical explosion dataset.We focus here on two of the most widely used far--field explosion P--wave spectral models: Mueller and Murphy (1971), hereafter referred to as MM, and Denny and Johnson (1991), hereafter referred to as DJ. MM developed a spectral model for several different emplacement media as a function of depth and source size. They extended the Sharpe (1942) analytical elastic response and developed a new pressure function applied at the elastic radius (distance from the source where the media begins to respond elastically) that Rougier et al. (2011) showed that historical cavity radius models don't fit numerical modeling for over--buried shots. Hydrodynamic calculations for 1--5 kt explosions in granite pe...