Interference and the Law of Energy Conservation

Abstract

Introductory physics textbooks consider interference to be a process of redistribution of energy from the wave sources in the surrounding space resulting in constructive and destructive interferences. As one can expect, the total energy flux is conserved. However, one case of apparent non-conservation energy attracts great attention.1,2 Imagine that a pair of coherent, point-like wave sources (located at the same position) radiates sinusoidal waves of amplitude A, spreading in a uniform medium. Assume also that radiation of the two sources is in phase. Since the energy of oscillation, E, is proportional to amplitude squared, one quickly arrives at an apparent paradox. That is, the energy of oscillation in every point due to only one source is E0=CA2 (C is the coefficient of proportionality), while according to the linear superposition principle, the combined amplitude of oscillations from the two sources is 2A and the energy of oscillations is E=C(2A)2=4CA2=4E0, i.e., four (not two) times greater than the energy of oscillation of one isolated source in the absence of the second. In the general case, superposition of two waves with identical amplitudes and wavelengths produces a wave with an intensity somewhere between zero and four times the intensity of a single wave source (depending on relative phase of the two waves). This leads to the obvious question: how can we account for the extra (or missing) energy that necessarily results from in-phase (or anti-phase) wave interference? This apparent violation of the principle of conservation energy, due to the superposition of waves, is the primary topic of this paper.