page_title: Method of images
section_title: Mass transport in environmental flows with non-infinite domains
hierarchical_section_title: Method of images / Mass transport in environmental flows with non-infinite domains
caption_attribution_description: English: double boundary mass release, method of images treatment
context_page_description: The method of images is a mathematical tool for solving differential equations, in which the domain of the sought function is extended by the addition of its mirror image with respect to a symmetry hyperplane. As a result, certain boundary conditions are satisfied automatically by the presence of a mirror image, greatly facilitating the solution of the original problem. The domain of the function is not extended. The function is made to satisfy given boundary conditions by placing singularities outside the domain of the function. The original singularities are inside the domain of interest. The additional singularities are an artifact needed to satisfy the prescribed but yet unsatisfied boundary conditions.
context_section_description: Environmental engineers are often interested in the reflection (and sometimes the absorption) of a contaminant plume off of an impenetrable (no-flux) boundary. A quick way to model this reflection is with the method of images.
  The reflections, or images, are oriented in space such that they perfectly replace any mass (from the real plume) passing through a given boundary. A single boundary will necessitate a single image. Two or more boundaries produce infinite images. However, for the purposes of modeling mass transport—such as the spread of a contaminant spill in a lake—it may be unnecessary to include an infinite set of images when there are multiple relevant boundaries. For example, to represent the reflection within a certain threshold of physical accuracy, one might choose to include only the primary and secondary images.
  The simplest case is a single boundary in 1-dimensional space. In this case, only one image is possible. If as time elapses, a mass approaches the boundary, then an image can appropriately describe the reflection of that mass back across the boundary.
  Another simple example is a single boundary in 2-dimensional space. Again, since there is only a single boundary, only one image is necessary. This describes a smokestack, whose effluent "reflects" in the atmosphere off of the impenetrable ground, and is otherwise approximately unbounded.
  Finally, we consider a mass release in 1-dimensional space bounded to its left and right by impenetrable boundaries. There are two primary images, each replacing the mass of the original release reflecting through each boundary. There are two secondary images, each replacing the mass of one of the primary images flowing through the opposite boundary. There are also two tertiary images (replacing the mass lost by the secondary images), two quaternary images (replacing the mass lost by the tertiary images), and so on ad infinitum.
  For a given system, once all of the images are carefully oriented, the concentration field is given by summing the mass releases (the true plume in addition to all of the images) within the specified boundaries. This concentration field is only physically accurate within the boundaries; the field outside the boundaries is non-physical and irrelevant for most engineering purposes.
caption: Mass release with two impenetrable boundaries in 1-dimensional space. The vertical axis is expected concentration (of the contaminant), the horizontal axis is the x-direction.
exif: {}