The Gaussian plume model describes the dispersion of a pollutant from a point source. The region at risk is where the concentration c exceeds a critical concentration c*, and we call this region the critical plume. In a steady-state windless situation, the critical plume is spherically symmetric. When the wind is blowing steadily, the critical plume is reduced in size in all directions except downwind, where the wind has no effect, contrary to physical intuition. Thus the wind reduces the risk of all the points not directly downwind, but has no effect on points directly downwind. In a transient situation with an instantaneous point source, the instantaneous critical plume is always spherical in shape, and the union of all the instantaneous critical plumes with time forms a shape similar to that of the steady-state critical plume. The overall exposure at a fixed point downwind, i.e. the integral in time of the concentration at that point, is independent of the wind speed, again contrary to physical intuition but in agreement with the steady-state results.