Reply to 'Comment on 'Peculiarities in the energy transfer by waves on strained strings' (Phys. Scr. 88 065402)'

Dr Rowland claims in his 'Comment' [1] that in a transverse traveling wave in a taut string the elastic potential energy is carried with a speed that differs from the speed of the transverse wave. In another place of his 'Comment' [1] Rowland claims that a transverse traveling wave in a taut string carries no potential energy at all. These statements are erroneous. Indeed, in a transverse wave the distortions of the string shape that store the elastic potential energy travel along the string with the speed of this wave, and the potential energy, which is stored and localized in these distortions, travels with the same speed.

A strained string without a wave already stores up the elastic potential energy due to its preliminary stretching, which provides the string tension. A transverse wave in such a string causes additional stretching. Only the potential energy associated with this additional stretching is regarded as the potential energy of the wave. The greater the additional local stretching the higher the density of the wave's potential energy. The elastic potential energy associated with the wave is localized in those string elements, which are subjected to the additional stretching by the wave. Localization of the wave potential energy in the string uniquely depends on the string shape in this wave [2].

In a transverse wave the nonlinear effects lead to some redistribution (within a wavelength) of the additional stretching along the string and, as a consequence, to a spatial redistribution of the elastic potential energy [2]. Specifically, in a sinusoidal transverse traveling wave the additional stretching of the string is uniform due to the nonlinear coupling of transverse and longitudinal distortions (see [3] for details). Hence the potential energy of such a wave is evenly distributed along the string. Nevertheless, this energy propagates along the string together with the transverse wave, and consequently with the speed of this wave.

This constant unidirectional flow of potential energy in the sinusoidal traveling wave is especially obvious in the case of a semi-infinite elastic string, in which a transverse wave is maintained by a source that makes the starting point of the string move in the same manner as in the string with the infinite traveling wave. It is shown in [3] that the work done by this external source per unit time just equals the total unidirectional flow of energy in the semi-infinite string. This total flow consists of the kinetic energy flow that oscillates with double frequency of the wave, and of the constant in time potential energy flow. The flow of potential energy exactly equals the mean value of the oscillating kinetic energy flow.

All other critical remarks in Rowland's 'Comment' [1] refer to transverse pulses and wave trains of finite length and duration. These issues are not discussed in the criticized paper [3]. I emphasize that this paper deals only with solutions to the wave equation (and to the nonlinearly coupled wave equations) that are periodic in space and time, that is, with ordinary traveling and standing sinusoidal waves.