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Physical Review

Two Energy Types in Wave Motion and Their Relation to Group and Wave Velocity

Lewi Tonks

Published Feb 1, 1929• 10 citations

Abstract

The energy of an element in a wave motion can be of two kinds: first, non-interactive like the energy of a set of independent pendulum bobs, and second, interactive like the potential energy of an element of a stretched string which is dependent only on the relative position of the neighboring elements. In the former case the group velocity is zero, and in the latter it equals the wave velocity. The slowing down of energy transmission in intermediate cases is discussed qualitatively and the ratio of group to wave velocity is calculated quantitatively in a number of cases on the assumption that the energy propagation is given not only by the product of energy density and group velocity, but also by the product of twice the interactive energy density and wave velocity. A general relation is suggested without proof for the connection between energy types and the ratio of group to wave velocity. The same ideas are applied to de Broglie's phase waves, the group velocity of which is the particle velocity.

DOI: 10.1103/physrev.33.239

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The energy of an element in a wave motion can be of two kinds: first, non-interactive like the energy of a set of independent pendulum bobs, and second, interactive like the potential energy of an element of a stretched string which is dependent only on the relative position of the neighboring elements. In the former case the group velocity is zero, and in the latter it equals the wave velocity. The slowing down of energy transmission in intermediate cases is discussed qualitatively and the ratio of group to wave velocity is calculated quantitatively in a number of cases on the assumption that the energy propagation is given not only by the product of energy density and group velocity, but also by the product of twice the interactive energy density and wave velocity. A general relation is suggested without proof for the connection between energy types and the ratio of group to wave velocity. The same ideas are applied to de Broglie's phase waves, the group velocity of which is the particle velocity.

It is noted that for a monochromatic de Broglie wavelength corresponding to the free inertial particle wave group consists of a single wave, so the group velocity is identically equal to the phase. Justified by the presence in the expression for the wave kinetic energy factor ½, which is the same as in classical case which is due to the inertia of the particle. The representation of particle as a set of sub-particles, of which N is infinite, and therefore receive a confirmation of the equality group and phase velocities is described. It is shown that with the ratio for wave kinetic energy, expression for the total energy of the wave, the wave function and the Schrodinger equation for a free particle are inextricably linked

In the first section of this paper I derive the formulas for the phase velocity and group velocity as a function of the total relativistic energy and the momentum of a particle. In the second section I derive similar formulas as a function of the de Broglie and the Compton wavelengths of the particle. In the third section, I derive similar formulas as a function of the angular frequency and the wave number. Finally, two additional meanings of the Compton wavelength are derived. The first one is derived from the equation of the group velocity in terms of the de Broglie and the Compton wavelengths and the second one from the range of the weak nuclear force.

In quantum mechanics the center of a wave packet is precisely defined as the center of probability. The center-of-probability velocity describes the entire motion of the wave packet. In classical physics there is no precise counterpart to the center-of-probability velocity of quantum mechanics, in spite of the fact that there exist in the literature at least eight different velocities for the electromagnetic wave. We propose a center-of-energy velocity to describe the entire motion of general wave packets in classical physical systems. It is a measurable quantity, and is well defined for both continuous and discrete systems. For electromagnetic wave packets it is a generalization of the velocity of energy transport. General wave packets in several classical systems are studied and the center-of-energy velocity is calculated and expressed in terms of the dispersion relation and the Fourier coefficients. These systems include string subject to an external force, monatomic chain and diatomic chain in one dimension, and classical Heisenberg model in one dimension. In most cases the center-of-energy velocity reduces to the group velocity for quasi-monochromatic wave packets. Thus it also appears to be the generalization of the group velocity. Wave packets of the relativistic Dirac equation are discussed briefly.

For the wave-particle duality of microscopic particles,the relationship of the velocity and the velocity of phase and the velocity of group for the microscopic particles is analyzed,it is pointed out that the velocity relating to the frequency and wavelength of two De Broglie expressions is not the velocity of group but the velocity of phase.On the contrary,the velocity in the defining expression of momentum is the real speed of particles,which doesn't equal to the velocity of phase but equals to the velocity of group.