Long-range interactions of negative pions with protons resulting in single and double pion production are examined in terms of a meson field theory model in which it is assumed that there is a pion-pion interaction and that the single virtual pion exchange graphs are dominant in large-impact-parameter collisions. This pion exchange leads, in single pion production, to an "excited state," π*, which "decays" into two pions, and, in double pion production, also to an "excited state," N*, which "decays" into a pion and a nucleon. For sufficiently high relative energy of the incident π- and proton, each of these processes can occur for small values of the invariant square of the virtual pion four-momentum, ∆2, in which case the virtual pion carries very little transverse three-momentum. For small ∆2 it is shown that it is reasonable to neglect final state interactions between the "decay products" of the π* and those of the N*. It is found that in the limit ∆2-->-μ2, where μ is the pion rest mass, the virtual pion behaves kinematically, in the π* barycentric system, as an incoming pion which scatters elastically with the incident π-, and also behaves, in the N* barycentric system, as an incoming pion which scatters elastically with the incident proton. Thus, for small ∆2 in the physical region the π* and N* vertices are defined as the corresponding off-the-mass-shell scattering amplitudes. A ratio of appropriately defined double-to-single pion production cross sections is obtained which is independent of the details of the assumed pion-pion interaction, and depends only on the relative strengths of N* and N formation. This ratio is estimated by means of the p-wave static nucleon model, applied in appropriate coordinate systems. For incident 5-Bev/c pions this model leads to double pion production which is important compared to single pion production, both because of the 32-32 pion-nucleon resonance and because of important phase-space factors. Kinematical considerations similar to those described above suggest that this model may also be the theoretical genesis of the "two fireballs" model proposed for ultrarelativistic nucleon-nucleon collisions.