The transmission bottleneck describes the number of viral particles that initiate an infection in a new host. Previous studies have used genome sequence data to suggest that transmission bottlenecks for influenza and SARS-CoV-2 involve few viral particles, but the general principles of virus transmission are not fully understood. Here we show that, across a broad range of circumstances, tight transmission bottlenecks are a simple consequence of the physical process of airborne viral transmission. We use mathematical modelling to describe the physical process of the emission and inhalation of infectious particles, deriving the result that that the great majority of transmission bottlenecks involve few viral particles. While exceptions to this rule exist, the circumstances needed to create these exceptions are likely very rare. We thus provide a physical explanation for previous inferences of bottleneck size, while predicting that tight transmission bottlenecks prevail more generally in respiratory virus transmission.