On wave equations and energy balance for non-linear and linear vibrations of non-uniform and uniform strings and membranes

The energy flux, the energy density and the wave equation are considered for the transverse vibrations of (i) elastic strings and (ii) membranes under isotropic tension. In both cases (i) and (ii) the wave equation with external forces is obtained in general for (a) non-linear vibrations with large slope and (b) mass density and tangential tension arbitrary functions of position and time. The energy equation, including the energy density and flux and power of external forces, exists for (a) non-linear vibrations and (c) mass density and tangential tension independent of time and arbitrary functions of position. In the simplest case of linear vibrations of uniform elastic (i) strings and (ii) membranes are considered for standing modes and propagating waves, confirming the classical results for propagating waves: (a) the equipartition of kinetic and elastic energies; (b) the energy velocity, that is the ratio of energy flux and density, equal to the wave speed. The statements (a) and (b) generally do not hold: (i) for linear waves in non-uniform strings due to wave refraction by the non-uniform wave speed; (ii) for non-linear waves in a uniform string because of the contrast between the kinetic (elastic) energy as a function of velocity (strain) that is quadratic (has higher order terms); (iii) for non-linear waves in non-uniform strings because, as in (i) and (ii) the velocity and strain satisfy different wave equations.