Vibrational sensors can be designed in such a way that the frequency of one of their modes of vibration is a function of temperature, pressure, density, or other parameter, making them the basis of various instrumentation transducers. These can be driven remotely by means of a line carrying bursts of plane waves of stress analogous to sonar or radar. The simple analysis of such a system, where only one mode at a time is considered, is unsatisfactory, and a complete general theory has been developed. This applies to an extended object with an arbitrary pattern of resonant frequencies, internal energy losses, and mechanical couplings to the line. The basic Navier–Stokes equation of motion was solved for this boundary-value problem to give a pole-zero diagram for the system. Computer programs were used to invert the function into the time domain and plot the echo-return traces. The theory has been used to relate experimental observations of the spectra of isotropic disks of a variety of materials to recent theoretical studies. Accurate values of elastic constants and their temperature coefficients have been obtained. The special case of a line resonator, which has a harmonic spectrum, enables the internal friction losses to be evaluated absolutely. To make an automatic system where the drive frequency follows the resonant frequency, a number of control signals are possible. A computer program was prepared to determine the parameters of the various phase control loops available. Subject Classification: 40.24.