Keywords: Sturm-Liouville equation, homogeneous trees, sub-treeing, transcendental eigenvalue problem, Wittrick-Williams algorithm.

Consideration is given to the mathematical problem of calculating the eigenvalues on homogeneous trees, for which there is much current interest. The tree in question comprises a single trunk that divides into b branches at its tip, with each branch dividing into b sub-branches ad infinitum. A second order Sturm-Liouville equation is then used to describe each branch and these equations define a matrix that can be used to describe the tree.

This is accomplished by developing the Sturm-Liouville equation into an 'edge' matrix, which describes each branch of the tree in a way that enables the branches to be linked together at vertices to form the tree. This linking is analogous to the way elements are joined together at nodes to form structures when using the stiffness technique. The structural mechanics analogy is further enhanced by noting the correspondence between the 'edge' matrix and the exact dynamic stiffness matrix of an axially vibrating, uniform bar. This means that the eigenvalues of the mathematical problem correspond precisely to natural frequencies when the branches are replaced by axially vibrating bars. This gives considerable insight into the problem when associating a physical relevance to the results determined.

The theory presented yields exact solutions to the Sturm-Liouville problem on homogeneous trees and necessitates the solution of a transcendental eigenvalue problem. This is achieved using the Wittrick-Williams algorithm, which guarantees that the required eigenvalues are converged upon to any desired accuracy with the certain knowledge that none have been missed. However, the application of the Wittrick-Williams algorithm in this case is somewhat unusual due to the presence of deeply nested sub-structures that are used to describe the trees, which can easily have in excess of 10¹² branches.

The theory is accompanied by an extremely efficient and compact FORTRAN 77 computer program, comprising only 42 statements, that implements the theory and builds the required tree structure. The program is fully annotated and explained for those who might wish to extend its capability. In addition, the use of the program as a 'black box' is fully described.

The program has been validated against existing results in the literature, which usually have boundary conditions that generate a band-gap frequency spectrum for a tree. A parametric study was therefore undertaken to investigate both parameter variations and the effect of boundary conditions. The former showed that the parameters associated with the Sturm-Liouville equation have considerable influence on the shape of the spectrum, although the fundamental band-gap structure was unaffected for the specific boundary condition considered. On the other hand, consideration of the possible combinations of boundary conditions revealed a more fundamental issue. Namely, that for certain boundary conditions the band gap structure of the spectrum disintegrates, leaving a continuous spectrum.

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