Transient ultrasonic waves in an elastic half-space generated by an ultrasonic transducer of finite size are modeled by the Distributed Point Source Method (DPSM). DPSM which is a Green's function based semi-analytical mesh-free technique is modified to incorporate the transient loading from a finite size acoustic transducer. Fast Fourier transform (FFT) of the transient loading is computed and then DPSM is used to compute the ultrasonic field at different frequencies and then inverse fast Fourier transform (IFFT) is taken to get the transient response of an elastic half-space excited by a bounded acoustic beam. Numerical results are generated for elastic half-space excited with normal incidence of acoustic beam. Then the transient Rayleigh wave in the solid half-space is generated. The modeling is then extended to the transient response of an elastic half-space containing a crack, struck by a bounded acoustic beam. It is discussed in the paper what type of useful information that is hidden in the steady state solution can be obtained from the transient results.