  We propose a non-perturbative attempt to solve the kinematic equations for
spiral waves in excitable media. From the eikonal equation for the wave front
we derive an implicit analytical relation between rotation frequency $\Omega$
and core radius $R_{0}$. For free, rigidly rotating spiral waves our analytical
prediction is in good agreement with numerical solutions of the linear eikonal
equation not only for very large but also for intermediate and small values of
the core radius. An equivalent $\Omega\left(R_{+}\right)$ dependence improves
the result by Keener and Tyson for spiral waves pinned to a circular defect
with radius $R_{+}$ with Neumann boundaries at the periphery. Simultaneously,
analytical approximations for the shape of free and pinned spirals are given.
We discuss the reasons why the ansatz fails to correctly describe the result
for the dependence of the rotation frequency on the excitability of the medium.
