Wave impedances of drill strings and other periodic media

It is commonly known that wave reflections are caused by abrupt spatial variations in the physical parameter called wave impedance. When a material contains a spatially periodic distribution of wave impedances some very interesting and complex wave propagation phenomena will occur. Two examples of such periodic structures immediately come to mind: the first is a sandwiched structure of two types of plates, say for example, identical layers of thin steel plates interspersed with identical thick aluminum plates; and the second is a large number of identical long thin pipes that are connected from end to end with identical short heavy threaded couplings. The pipe assembly is our primary concern here because it represents the drill string, used worldwide to drill for natural energy resources. We want to understand how waves propagate through drill strings because we want to use them as a means of communication. But while the second structure is our primary concern, it is the study of the first structure, composed of layers, that is the truly historical problem and the source of much of our understanding of this rich set of wave physics. Traditionally, wave propagation in periodic media has been studied as an eigenvalue problem. The eigenvalues themselves yield information about phase velocities, group velocities, passbands, and stopbands. Most often the analysis has stopped there and the eigenvectors have been ignored. Here we turn our attention to the eigenvectors, using them to evaluate the impedance of the periodic structure with particular emphasis on the periodic drill string. As you might expect the impedance of the drill string is a complex number, which is evaluated from a very complicated expression. However, we have discovered that the impedance at two physical locations along the length of each piece of drill pipe in the drill string always reduces to a real number. This is immensely important because it allows us to match the impedance of the drill string to our communication devices. We show how this leads to the effective design of repeaters, noise cancelers, wave terminators, and quarter-wave transformers for a drill-string communication system.