Spiral wave wavenumber

At large distances from the center the spiral wave becomes indistinguishable from a plane wave with a certain wavenumber koo, which is the eigenvalue of the nonlinear problem (163), (164) ... 

Fig.21 shows an example of a spiral-wave solution of eq.(159) obtained numerically. The left figure is a snap-shot corresponding to a particular moment of time. The spiral is rotating counterclockwise with constant frequency. One can see that far from the spiral core the wavenumber is constant and so is the amplitude A that is related to the wavenumber by eq.(166). In the core center A = 0. 

It should be remarked that the asymptotic expansions for spirals waves developed here are not regular at the origin. We saw the same problem in Chapter I, see Eq. (1.22). At the moment, the core of a spiral wave still eludes analysis. Another misfortune of this theory is the existence of a continuous family of spiral waves for all wavenumbers k sufficiently small. Only spiral waves of fixed wavenumber (pitch) are observed. An adequate explanation of this fact has also yet to appear. 

With the help of asymptotic analysis and numerical methods, Hagan computed the wavenumber Qs of the spiral wave solution of Equation (1). Hagan could obtain a graph of the function Qs as a function of parameter /3, for the case where a = 0. On the other hand, this author indicated how to compute the spiral wavenumber in the case where o / 0. These indications allow us to express the wavenumber q s(a, / ) in function of the values (O, / ). The principle of the computation relies on a scaling property of the time-periodic solutions of Equation (1), which says that any solution W — cT g r a, /)) of Equation (1) can generate a family of solutions corresponding to other parameters (a, P) of an equation of the same form. A member of this family Has the general form ... 

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