In ultrasonic nondestructive testing, the actual time domain signature of a scattered wave due to a pulsed input is the common observable (via oscilloscope) quantity. Numerically simulated solutions are thus desirable in the time domain. These can be obtained by working directly in the time domain or by inversions of integral transform solutions in the frequency (Fourier) or Laplace domains. Direct time solutions are suitable for short times but generally deteriorate for longer times. Alternately, the transform techniques generally require solutions over a wide spectrum in the transform variable to provide accurate inversions back to the time domain. However, without special provisions, numerical implementations like boundary or finite elements are limited to lower frequencies or Laplace parameter values. Hence, ways to minimize the number of transform variable solutions for canonical problems are important. In this dissertation the Fourier and Laplace transform methods are compared and investigations are made into ways of minimizing the solutions in the spectrum space in a boundary element setting. Three common scattering models--a spherical void and two spherical inclusions--are used in the analysis and the numerical results are compared to analytical solutions and to experimental results.