A detailed theoretical analysis of the problem of elastic second-harmonic generation (SHG) by longitudinal ultrasonic beams of finite dimension propagating in isotroplc media or along certain pure mode axes in crystals has been made. A formulation for optical SHG by Gaussian laser beams, adapted to the ultrasonic case, has allowed calculation of the intensity profile and angular intensity distribution of the second-harmonic beam as a function of distance from the generating fundamental frequency transducer. These calculations were confirmed experimentally, using the beam probing techniques of Bragg diffraction, for both focused and unfocused longitudinal beams, in the frequency range 200-500 MHz, propagating along the Z axis in crystal quartz. Ultrasonic diffraction of the fundamental beam is shown to result in reduction of second-harmonic intensity compared to the intensity normally calculated from plane-wave theory. Consideration of this reduction allowed an accurate value of the third-order elastic constant c333to be determined from the measured second-harmonic intensity. The effect of reflection at a crystal-air interface on SHG has also been investigated.