The study is based on the general differential equation of a mechanical system. Driving force and solution are expressed as a series of natural functions of the dissipationless system. Dissipation is taken into account by introducing complex elastic constants. Each Fourier coefficient of the driving force is written as the product of an excitation constant and the total driving force. The excitation constants can then be included in the mode parameters of the system and the solution, written as the product of the total driving force and the “mechanical impedance” of the system. The solution is exact and useful at the lower frequencies. A simple solution for higher frequencies can be derived by replacing the series by an integral and evaluating this integral. Three important parameters can thus be obtained: the driving‐point impedance that describes the velocity of the driving point and the reaction of the system to the driving force, the effective impedance that represents the average velocity amplitude over the system, and the effective dissipation resistance with respect to this average‐velocity amplitude.
The expression derived for the high‐frequency driving‐point impedance then turns out to represent, also, the general background level at lower frequencies. The solution for the driving‐point impedance in the intermediate frequency range is obtained by adding to this background the contribution of the natural mode that has its resonant frequency closest to the frequency of the force; the effective impedance, on the other hand, is given with good approximation as the impedance of the mode whose resonance frequency is closest to the frequency of the force; and the effective dissipation resistance is given by the resistive component of this mode impedance.
The mode parameters are computed for rods, membranes, plates, and shells for three situations—a point force, a linear force distribution, and a force distribution given by a Fourier integral. The results make it possible to handle complicated mechanical systems with almost the same ease as simple mass points, and to compute their sound radiation.