On the use of matrices for solving periodic heat flow problems

Summary

The temperature and heat flow on one side of a wall are related to the corresponding quantities on the other side of the wall by linear equations. When these equations are put into matrix form, they lend themselves very readily to the solution of problems in which the wall consists of many parallel layers, for then the matrix representing the composite wall is obtained by multiplying together the matrices for the individual layers. The determinants of these matrices are of particular interest and their values for temperature distributions possessing plane, cylindrical and spherical symmetry are very simple and can be derived directly from the differential equation of heat conduction. An alternative use of matrices, which is useful in one and three dimensions, occurs in problems involving temperature waves in multilayer solids where the reflection and transmission coefficients of the interfaces and the attenuation of the layers can be represented by matrices. Again the determinants of these matrices are shown to have a particularly simple value.