This paper presents an analysis of the error generation mechanisms that affect the accuracy of measurements of ultrasonic wave attenuation coefficient and phase velocity as functions of frequency. In the first stage of the analysis we show that electronic system noise, expressed in the frequency domain, maps into errors in the attenuation and the phase velocity spectra in a highly nonlinear way; the condition for minimum error is when the total measured attenuation is around 1 Neper. The maximum measurable total attenuation has a practical limit of around 6 Nepers and the minimum measurable value is around 0.1 Neper. In the second part of the paper we consider electronic noise as the primary source of measurement error; errors in attenuation result from additive noise whereas errors in phase velocity result from both additive noise and system timing jitter. Quantization noise can be neglected if the amplitude of the additive noise is comparable with the quantization step, and coherent averaging is employed. Experimental results are presented which confirm the relationship between electronic noise and measurement errors. The analytical technique is applicable to the design of ultrasonic spectrometers, formal assessment of the accuracy of ultrasonic measurements, and the optimization of signal processing procedures to achieve a specified accuracy.