An approximate model is proposed for the solution of the problem of the radiation of transient pulses in an elastic medium. The proposed approximation leads to an explicit integral formula for the elastic displacement vector radiated by an arbitrary source of traction applied to an elastic medium. The radiation integral obtained is shown to be closely related to the well‐known Rayleigh integral for the radiation in fluids. To illustrate the potential of our formulation and to test the accuracy of the approximations made in the derivation, the radiation integral is calculated in the case of a flat disc thickness‐mode transducer directly coupled to an elastic half‐space. The similarity of our formula with the Rayleigh integral allows the use of the impulse‐response approach for the calculation. An analytic expression is obtained for the elastic displacement vector which is directly related to the classical solution for the transient acoustical potential radiated in a fluid, as stated by Weight [J. P. Weight, ‘‘A model for the propagation of short pulses of ultrasound in a solid,’’ J. Acoust. Soc. Am.81(4), 815–826 (1987)]. Results obtained with the present model are compared with exact results. It is shown that the approximate model is suitable for predicting the displacement field generated by ultrasonic transducers typical of those used in pulse‐echo measurements in nondestructive testing. Throughout, particular attention is given to the assumptions and approximations made in deriving the model with clear definition of its domain of validity.