An apparatus and method are disclosed for characterizing semiconductor wafers or other test objects that can support acoustic waves. Source and receiving transducers are configured in various arrangements to respectively excite and detect acoustic waves (e.g., Lamb waves) in a wafer to be characterized. Signals representing the detected waves are digitally processed and used to compute a measurement set correlated with the waves' velocity in the wafer. A characterization sensitivity is provided that describes how different wafer characteristics of interest vary with changes in the propagation of the acoustic waves. Using the characterization sensitivity and measurement sets computed at a setup time when all wafer characteristics are known and one or more process times when at least one of the characteristics is not known the perturbation in wafer characteristics between the setup and the process times can be determined. Characterization accuracy is improved by a wafer calibration procedure wherein measurement offsets from known conditions are determined for each wafer being characterized. An apparatus and technique are disclosed for correcting for anisotropy of acoustic wave velocity due to the direction of wave propagation with respect to a preferred crystallographic axis of the wafer. An apparatus and technique are also described for measuring wafer temperature using a single transducer whose temperature is related to the temperature of the wafer and, optionally, resonator structures. For characterization steps that occur when the wafer is chucked, a chuck structure is described that reduces the likelihood of the chuck interfering with the waves in the wafer.