1.6: Transfer Relations and Continuum Dynamics of Linear Systems

Fields, flows and deformations in systems that are uniform in one or more "longitudinal" directions can have the dependence on the associated coordinate represented by complex amplitudes, Fourier series, Fourier transforms, or the appropriate extension of these in various coordinate systems. Typically, configurations are nonuniform in the remaining "transverse" coordinate. The dependence of variables on this direction is represented by "transfer relations." They are first introduced in Chapter 2 as flux-potential relations that encapsulate Laplacian fields in coordinate systems for which Laplace's equation is variable separable.

At the risk of having a forbidding appearance, most chapters include summaries of transfer relations in the three common coordinate systems. This is done so that they can be a resource, helping t obviate tedious manipulations that tend to obscure what is essential in the derivation of a model. The transfer relations help in organizing a development. Once the way in which they represent the spacetime dynamics of a given medium is appreciated, they are also a way of quickly communicating the physical nature of a continuum.

Applications in Chapter 4 begin to exemplify how the transfer relations can help to organize the representation of configurations involving piece-wise uniform media. The systems considered there are spatially periodic in the "longitudinal" direction.

With each of the subsequent chapters, the application of the transfer relations is broadened. In Chapter 5, the temporal transient response is described in terms of the temporal modes. Then, spatial transients for systems in the temporal sinusoidal steady state are considered. In Chapter 6, magnetic diffusion processes are represented in terms of transfer relations, which take a form equally applicable to thermal and particle diffusion.

Much of the summary of fluid mechanics given in Chapter 7 is couched in terms of transfer relations. There, the variables are velocities and stresses. In a wealth of electromechanical examples, coupling between fields and media can be represented as occurring at boundaries and interfaces, where there are discontinuities in properties. Thus, in Chapter 8, the purely mechanical relations of Chapter 7 are combined with the electrical relations from Chapter 2 to represent electromechanical systems. More specialized are electromechanical transfer relations representing charged fluids, electron beams, hydromagnetic systems and the like, derived in Chaps. 8-11.

A feature of many of the examples in Chapter 8 is instability, so that again the temporal modes come to the fore. But with effects of streaming brought into play in Chapter 11, there is a question of whether the instability is absolute in the sense that the response becomes unbounded with time at a given point in space, or convective (amplifying) in that a sinusoidal steady state can be established but with a response that becomes unbounded in space. These issues are taken up in Chapter 11.