Analytical solutions are developed for the flow induced by a vertically distributed turbulent plume in an otherwise quiescent uniform environment. The plume considered is that which forms adjacent to a vertical wall source that emits a flux of buoyancy uniformly over its entire area. Two cases are considered: a plume from an elevated source that is offset vertically a distance a > 0 from a horizontal boundary, and a source with zero offset a = 0. We adopt a solution technique for the induced flow inspired by Taylor [22], with the model of a vertically distributed plume developed by Cooper and Hunt [5] used to represent the boundary condition that induces the flow. The solution, developed in terms of the stream function, indicates that the induced flow approaches the plume perimeter along an upwardly inclined and continuously steepening path. Speeds in the induced flow increase with horizontal distance from the plume perimeter. This occurs as a result of the increasing plume entrainment demand with height. Analysing the flow in a Lagrangian framework we show that fluid parcels in the induced flow do not simply accelerate towards the plume but, in fact, fluid moving along streamlines decelerates to a minimum speed before accelerating towards the plume. For the plume with zero offset, the local minimum in speed is predicted to occur once fluid parcels cross the locus at θ = 7π/8 radians (≡ 157.5 °), where increases anticlockwise from the (negative) vertical θ = 0. Finally, the solution derived is applied in the context of the built environment to describe the plume induced flow adjacent to the wall of a room heated by the sun. The solution indicates that a typical thermal wall plume has the ability to draw air laterally over significant horizontal distances and we consider the implications for the spread of airborne contaminants.