Summary: This paper presents analytical and numerical-analytic approaches to solving the problem of the action of an arbitrarily distributed axisymmetric load applied instantly to the surface of an isotropic elastic half-space. The first approach is built around the Laplace and Hankel integral transforms whose inversion is performed jointly with Cagniard’s technique, and as a result, exact analytical expressions are obtained for computing stresses along an axis of symmetry. The second approach uses the Laplace integral transform and the expansion of sought for values into the Fourier-Bessel series to reduce the problem to a numerical solution of a series of Volterra integral equations. Concrete numerical analysis was performed for cases where the domain of application of a distributed load is fixed or expands in time with both constant and variable velocity.