The exact analytical solution of fully nonlinear non-breaking shallow-water waves on a uniformly sloping beach was derived by Carrier and Greenspan in 1958. Because the derivation involves nonlinear and hodograph-type transformation, the Carrier-Greenspan solution is not in a convenient form to be converted to presentations in real time and space domains. For this reason, only a limited number of the applications have been reported. To improve this deficiency, we re-derive the Carrier-Greenspan equation (the form of a linear cylindrical wave equation) with different non-dimensionalization and different transformation. To solve the problem with arbitrary initial conditions, we apply the Fourier-Bessel transform, and inversion of the transform leads to an exact integral representation of the solution. However, this integral is not convenient for computation due to its highly oscillatory behavior of the integrand. The integral is further manipulated to yield the exact Green function representation, which involves the complete elliptic integral of the first kind. This solution form is convenient for numerical integration to obtain the solutions in the physical time and space domains. With this semi-analytic solution technique, several examples of tsunami runup and drawdown motions are presented. In particular, detailed shoreline motion, velocity field, and inundation depth on the shore are closely examined. It was found that the maximum flow velocity occurs at the moving shoreline and the maximum momentum flux occurs in the vicinity of the extreme drawdown location. The direction of both the flow velocity and the maximum momentum flux depend on the initial waveform: it is in the inshore direction when the initial waveform is predominantly depression; in the offshore direction, when the initial waves have a dominant elevation characteristic. Lastly, we present an envelop-curve of the maximum momentum flux in space, which might be used as a design criterion for the tsunami resistant building code. Mathematical derivation presented in this talk was reported by Carrier, Wu and Yeh (2003 JFM 475).