Convolution Approach to the Solution for the Dissolved Oxygen Balance Equation in a Stream

In terms of its response to biochemical oxygen demand (BOD) and dissolved oxygen (DO) inputs, a natural waterway may be treated as a system governed by the BOD and DO balance differential equations. Using the response of the DO balance differential equation to impulse inputs of DO and BOD, one can compute by convolution the response of the system to arbitrary BOD and DO inputs. The convolution technique requires numerical integration, but it is not a numerical solution to a differential equation; it therefore avoids the stability problems inherent in such solutions. The convolution technique permits consideration of longitudinal dispersion in systems that have time varying DO and BOD inputs, a situation that could previously be investigated only by numerical solutions to the basic differential equations. Examples show the capability of the convolution technique to reproduce field data and to match previously developed analytical and numerical techniques.