ABSTRACT. By deriving the appropriate Green's function, a model is developed that allows the interaction of normally-incident, ice-coupled waves with any number of cracks to be studied analytically. For a single crack a simple formula for the reflection and transmission coefficients, R and T, emerges that yields identical results to the computationally intensive work of Barrett and Squire (1996) but is much easier to apply. A crack is found to behave as a steep low-pass filter, allowing long waves through while inhibiting shorter waves, although there is also some fine structure to the response curve. The introduction of more cracks is straightforward.While in that case a formula for R and T is also possible in principle, it is easier to express the result as the solution of a simple matrix equation of order PN, where N is the number of cracks. It is found that perfect transmission (jRj 0) occurs at a set of discrete periods, hereinafter called a comb, for N > 1 and that the comb becomes finer as period decreases. For both periodically distributed cracks and ones that are randomly spaced, the gross shape of the response curve remains similar. The results suggest that it is improbable that waves travelling through the Arctic basin can be used as a remote-sensing agent to determine mean ice thickness. The Green's function technique employed in this paper furnishes solutions to other problems of interest.
The Cvitanović–Feigenbaum (CF) equation arising in the universal scaling theory of iterated maps of the real line has strong links with the classical Schröder and Abel functional equations. This link is exploited to obtain information about the analytic solutions, and specifically the singular solution, of the CF equation, providing an alternative description of the latter to that of Eckmann and Wittwer. We obtain an accurate numerical approximation to this singular solution, using special techniques to handle the divergent series. This accuracy is a substantial improvement on previous estimates of the solution, and of the associated asymptotic feigenvalues α and δ. The solutions of the Feigenbaum–Kadanoff–Shenker equation for universal scaling in circle maps are shown to yield to the same analysis, producing accurate numerical values for the associated α and δ.
Numerous physical systems with two competing frequencies exhibit frequency locking and chaos associated with quasiperiodicity. In this paper we review certain universal aspects of the quasiperiodic route to chaos by making use of the standard circle map. Particular attention is paid to the golden mean and silver mean with a view to comparison with experimental work. (c) 1996 American Institute of Physics.
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