Broadband signals are commonly used in ultrasonic spectroscopy to measure the frequency dependent attenuation characteristics of lossy solid media. Compared to narrowband signals, broadband signals are preferred since they do not require tedious frequency scanning and extensive data reduction efforts. Typically these broadband signals take the form of a pulse. Although the spectral range of a pulse is wide, the spectral resolution is limited by the duration of the signal. By employing signals with large time- bandwidth-products, the overall accuracy and resolution of ultrasonic spectroscopy can be improved. Expressions for the interaction of longitudinal waves, with large time- bandwidth-product, and isotropic materials are developed. The approach is effective for evaluating material with signals optimized for a frequency resolution and range of interest, but can also be used when thin materials (<EQ (lambda) ) are characterized by pulse signals. Using these expressions, the acoustical properties of wave-speed and attenuation can be determined when density and thickness are measured. Explicit account is made for diffraction corrections, multiple echo contributions, and interface scattering losses. The formalism is compared with the traditional analysis approach to illustrate the improved accuracy of the new technique, detailing where diffraction correction and multiple echo effects can become significant. Measured attenuation spectra are presented for common plastic materials as well as for a castable polyurethane commonly used in ultrasonic transducer fabrication.