A Transfer Matrix Analysis of the Energetics of Structural Wave Motion and Harmonic Vibration

The concept of the transfer matrix has been used extensively in the dynamic analysis of engineering structures. It is shown here that the transfer matrix which governs harmonic motion in a conservative system is K-unitary. This property is then used to derive a number of disparate new results regarding the energetics of wave motion and harmonic vibration. Most notably: (i) the conditions under which two wave components can interact to transmit energy are derived and a succinct expression is developed for the resulting power; (ii) it is shown that the time-averaged kinetic energy associated with harmonic vibration can be expressed in terms of the transfer matrix and its frequency derivative; (iii) a general proof is given of the fact that the energy flow velocity is equal to the group velocity for a periodic structure. The effect of damping on the energetics of wave motion and harmonic vibration is also considered.