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American Journal of Physics

Superposition and energy conservation for small amplitude mechanical waves

W. N. Mathews

Published Mar 1, 1986• 13 citations
Publisher

Abstract

The energy that would be carried by a wave produced by the superposition of two small amplitude mechanical waves of the same type would generally differ from the sum of the energies of the component waves in the superposition. Thus energy conservation appears to be violated. In this paper we address the question of the compatibility of the principles of superposition and energy conservation for small amplitude mechanical waves. We show that in the course of producing a pair of waves that interfere constructively, a second pair of waves that interfere destructively is naturally and unavoidably produced. Moreover, we show that the combination of constructive and destructive intereference is just that necessary for energy conservation. Thus we conclude that the principles of superposition and energy conservation are fully compatible.

DOI
10.1119/1.14661

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Interaction between sources and the superposition principle in interference phenomena

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Abstract In the explanations of the double slit experiment it is usually assumed that the superposition of the diffracted waves by two separate slits (one open and another closed) is the same as superimposing the diffracted waves when both slits are open at the same time. This naïve use of the superposition principle is generally not valid in either classical electromagnetism or quantum mechanics. As we will see, the diffraction diagram of one of the slits is altered when the other is open and when the slits are closer, the interaction between them is greater. In this work we will analyse the interaction between different types of sources (antennas, loudspeakers and, of course, slits) and we will show how this interaction explains the apparent anomalies in energy conservation and also the recent results of Young’s interference that show that the total diffracted intensity is reduced or increased depending on the distance between the slits. We also make a brief comment on the contributions of our work to the supposed looped trajectories of energy in interference phenomena.

Journal of Optics•Oct 15, 2021•2 citations

Wave Diffraction

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When two waves overlap, the resultant wave function is the sum ( superposition ) of the two individual wave functions. This overlap is also called interference , which is not necessarily a negative term as might be anticipated from its everyday use. Constructive interference occurs when the resultant wave has an amplitude larger than the individual waves, and destructive interference occurs when the resultant wave amplitude is smaller. The term diffraction refers to the interference of “many” waves originating from many sources, but where the term “many” is ill-defined.

CRC Press eBooks•Mar 13, 2025•0 citations

Classical sources of non-classical physics: the case of linear superposition

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Classical linear wave superposition produces the appearance of interference. This observation can be interpreted in two equivalent ways: one can assume that interference is an illusion because input components remain unperturbed, or that interference is real and input components undergo energy redistribution. Both interpretations entail the same observable consequences at the macroscopic level, but the first approach is considerably more popular. This preference was established before the emergence of quantum mechanics. Unfortunately, it requires a non-classical underlying mechanism and fails to explain well-known microscopic observations. Classical physics appears to collapse at the quantum level. On the other hand, quantum superposition can be described as a classical process if the second alternative is adopted. The gap between classical mechanics and quantum mechanics is an interpretive problem.

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Five Different Superposition Principles With/without Test Charge, Retarded Waves/advanced Waves Applied to Electromagnetic Fields or the Photons

Shuang-Ren Zhao

In electromagnetic theory and quantum theory, there are superposition principle. Traditionally there are only one kind superposition principle. However, in this author's theory the retarded wave and advanced wave are all involved, the superposition become multiple. For example this author need to deal the problem how to superpose the retarded wave and the advanced wave. This author found that there are 5 different kinds of superpositions. The superposition principles have some differences. The research about these differences is a key to open the door of many physical difficulties. For example the particle and wave duality problem, and to judge which interpretation of the quantum mechanics is a correct one. The first two superposition principles are the superpositions with and without the test charges. The slight different superposition principles are the superposition with a retarded wave alone and the superposition with the advanced wave alone. According to theory of this author, the emitter sends the retarded wave, the absorber sends the advance wave. Hence, normal electromagnetic field actually is consist of retarded wave and advanced wave. This two waves together become the traditional electromagnetic fields. This kind of electromagnetic fields can be seen approximately as retarded wave only, this kind wave also has its own superposition. This kind of superposition is also different with the superposition when we consider the retarded waves alone and also the advanced waves alone. In this article this author will discuss the differences of these different superpositions. This author will also discuss the different physical result with a few different superposition principles. In this article this author will prove the superposition with test charge and the superposition with retarded waves alone or advance wave alone can be see one kind of superposition. This superposition is correct and can be derived from the mutual energy principle. This will be referred as first kind of superposition. The superposition without test charge and the superposition with the retarded waves and advanced waves can be seen as one kind of superposition. For this kind of superpositions are not correct naturally. This kind of super position are applied to N-charge's Poynting theorem. In order to make this superposition work, the self-energy principle have to be applied. Without self-energy principle this kind of superpositions will violate the energy conservation law. Considering the self-energy conservation this kind of superposition become the superposition of first kind. This is referred as second kind of superposition. The third kind of superposition is for the traditional electromagnetic field. Here the traditional electromagnetic field means the electromagnetic field where the advanced waves are omit. When the advanced waves are omit doesn't mean the advanced waves do not exist, that is only means the retarded wave and the advanced wave are with nearly same intensity and can be treated together in some situation, for example, in wave guide, or in the free space where the absorber are uniformly distributed at infinite big sphere. Abstract Only when the self-energy principle is accept, all kinds of superposition can be accept. However different superposition has different physical meanings. Otherwise only the superposition with test charge or the superposition with only one kind wave either retarded waves or advanced waves can be accepted. Hence, the discussion about the superposition also support the concept of the self-energy principle which means there must exist the time reversal waves. That also means the waves do not collapse but collapse back. Wave collapse means collapse to its target, for example, the retarded wave will collapse to an absorber and the advanced wave will collapse to an emitter. Wave collapse back means the retarded wave sent from emitter will collapse back to an emitter; The advanced wave sent from the absorber will collapse back to an absorber. Hence, one purpose of this article is to clarify the superposition principles, and another purpose is to support this author's electromagnetic field theory which is started from two new axioms the self-energy principle and the mutual energy principle.

•Nov 1, 2018•0 citations

Wave-Mechanical Account of Quantum Reality and Gravity: Ontology, Mechanism, and Experimental Programme

YOUNG HO GOH

Paper 74 — No Quantum Superposition This paper argues, by reductio ad absurdum, that the physical realisation of superposition — simultaneous multi-location existence — is incompatible with seven independent and well-established physical principles: (1) the observational fact that no infinite physical quantity has ever been measured; (2) conservation of energy; (3) the mass-energy equivalence E = mc²; (4) the Bekenstein entropy bound; (5) the second law of thermodynamics; (6) the Heisenberg uncertainty principle; and (7) causality. The foundational identification is that information, energy, and mass stand in proportional existential relation (Bekenstein bound: I ∝ M; Landauer principle: ΔE ∝ ΔI; mass-energy equivalence: E = mc²). Complete superposition requires infinite information processing, and therefore infinite energy and infinite mass — quantities universally absent from the observational record of physics. Finite N-state superposition is excluded by a complementary argument: maintaining N states simultaneously requires N times the information-processing load of a single undetermined state, violating the principle that nature selects minimum-load configurations. The positive alternative proposed is the undetermined state: during matter-wave propagation, the matter wave is real and physically present, but matter has not yet appeared at any location. This is not a hidden-variable state but an ontological absence of localisation prior to the formation of a constructive interference maximum. Paper 75 — The Double-Slit Experiment Reinterpreted This paper applies the undetermined-state ontology to the paradigmatic demonstration of quantum mechanics. The de Broglie matter wave — a real physical wave — passes through both slits, while the particle remains in an undetermined state until a constructive interference maximum is formed at the detection screen. The disappearance of the interference pattern upon which-path measurement is explained without wavefunction collapse: the measurement interaction physically disrupts the matter wave's coherent two-slit propagation. Three experimentally testable predictions distinguish this framework: (1) weak-measurement trajectory reconstruction should reveal single-path trajectories inconsistent with simultaneous two-slit passage; (2) diffraction fringe spacing should depend on the mass density of the slit material, because denser slit edges generate a steeper refractive-index gradient; (3) the fringe pattern should exhibit a density-dependent phase shift measurable with current matter-wave interferometry technology. The Schrödinger equation and Born's rule are retained without modification. Paper 76 — Discriminating Experiments for the Constructive-Interference Account of Gravity This paper derives the specific numerical values predicted by Paper 29 — n(r) from the Schwarzschild metric, Δφ = k₀∫[n₁(r)−n₂(r)]dr from the medium-structure integral, and Δx_max = f(∇n, geometry, λ₀) from the interference structure — and assesses whether current experimental technology can measure the gap between Paper 29's predictions and standard QM's predictions. Three formal conditions for a discriminating experiment are established: structural absence of the variable from standard QM, signal exceeding current resolution, and impossibility of alternative explanation. Two experimental classes satisfy all three conditions with existing apparatus: COW path-geometry (predicted δ(Δφ) ~ 10⁻³ rad vs. resolution ~10⁻⁴ rad, path shape structurally absent from the standard COW formula) and slit-material density (predicted Δx_max ~ tens of nm vs. resolution 1–5 nm, material density structurally absent from Δy ≈ λL/a). Cumulative agreement across both systems constitutes genuine inductive support for the proposed mechanism. Paper 77 — From Electrons to Fullerenes This paper addresses three questions that Paper 76 does not: (1) why the slit-material density experiment has never been performed in the hundred years since matter-wave interferometry began — the answer being that standard QM contains no density variable in the slit formula, so experimenters had no reason to vary it, with the Copenhagen prohibition on mechanism questions as the deeper structural reason; (2) why confirmation in exactly two systems — electrons and fullerenes C₆₀ — is sufficient to establish the theoretical outline across six orders of magnitude in mass, because these systems are maximally independent in particle type, mass, apparatus, energy scale, and decoherence environment; (3) what confirmation across this mass range would establish for physics: the first experimentally supported identification of the physical mechanism of gravity, the first experimentally verifiable quantum-gravity connection at accessible energy scales, a physical grounding of Born's rule, and a mechanistic account of the measurement problem. A complete three-layer control protocol for excluding the slit-width confound is specified. Paper 78 — A Three-Dimensional Experimental Matrix This paper extends the experimental programme by introducing slit thickness as a third independent variable alongside matter-wave species and slit-material density. The wave-mechanical account predicts that phase shift scales linearly with slit thickness d: Δφ(ρ,d) = (2π/λ)·[n(ρ)−1]·d — a prediction structurally absent from standard QM, which assigns no physical role to slit thickness. The resulting 5×5×3 matrix of 75 independent measurements has a random coincidence probability of approximately 1/1,728,000, rendering chance agreement with theoretical prediction physically impossible. Three-layer control protocols for fabrication, TEM cross-section verification, and linearity testing are specified for the thickness dimension. Together, the five papers constitute a complete programme: ontological foundation (P74), phenomenological reinterpretation (P75), formal discriminating analysis (P76), strategic and historical argument (P77), and three-dimensional experimental design (P78).

•Apr 28, 2026•0 citations
The energy that would be carried by a wave produced by the superposition of two small amplitude mechanical waves of the same type would generally differ from the sum of the energies of the component waves in the superposition. Thus energy conservation appears to be violated. In this paper we address the question of the compatibility of the principles of superposition and energy conservation for small amplitude mechanical waves. We show that in the course of producing a pair of waves that interfere constructively, a second pair of waves that interfere destructively is naturally and unavoidably produced. Moreover, we show that the combination of constructive and destructive intereference is just that necessary for energy conservation. Thus we conclude that the principles of superposition and energy conservation are fully compatible.
Abstract In the explanations of the double slit experiment it is usually assumed that the superposition of the diffracted waves by two separate slits (one open and another closed) is the same as superimposing the diffracted waves when both slits are open at the same time. This naïve use of the superposition principle is generally not valid in either classical electromagnetism or quantum mechanics. As we will see, the diffraction diagram of one of the slits is altered when the other is open and when the slits are closer, the interaction between them is greater. In this work we will analyse the interaction between different types of sources (antennas, loudspeakers and, of course, slits) and we will show how this interaction explains the apparent anomalies in energy conservation and also the recent results of Young’s interference that show that the total diffracted intensity is reduced or increased depending on the distance between the slits. We also make a brief comment on the contributions of our work to the supposed looped trajectories of energy in interference phenomena.
When two waves overlap, the resultant wave function is the sum ( superposition ) of the two individual wave functions. This overlap is also called interference , which is not necessarily a negative term as might be anticipated from its everyday use. Constructive interference occurs when the resultant wave has an amplitude larger than the individual waves, and destructive interference occurs when the resultant wave amplitude is smaller. The term diffraction refers to the interference of “many” waves originating from many sources, but where the term “many” is ill-defined.
Classical linear wave superposition produces the appearance of interference. This observation can be interpreted in two equivalent ways: one can assume that interference is an illusion because input components remain unperturbed, or that interference is real and input components undergo energy redistribution. Both interpretations entail the same observable consequences at the macroscopic level, but the first approach is considerably more popular. This preference was established before the emergence of quantum mechanics. Unfortunately, it requires a non-classical underlying mechanism and fails to explain well-known microscopic observations. Classical physics appears to collapse at the quantum level. On the other hand, quantum superposition can be described as a classical process if the second alternative is adopted. The gap between classical mechanics and quantum mechanics is an interpretive problem.
In electromagnetic theory and quantum theory, there are superposition principle. Traditionally there are only one kind superposition principle. However, in this author's theory the retarded wave and advanced wave are all involved, the superposition become multiple. For example this author need to deal the problem how to superpose the retarded wave and the advanced wave. This author found that there are 5 different kinds of superpositions. The superposition principles have some differences. The research about these differences is a key to open the door of many physical difficulties. For example the particle and wave duality problem, and to judge which interpretation of the quantum mechanics is a correct one. The first two superposition principles are the superpositions with and without the test charges. The slight different superposition principles are the superposition with a retarded wave alone and the superposition with the advanced wave alone. According to theory of this author, the emitter sends the retarded wave, the absorber sends the advance wave. Hence, normal electromagnetic field actually is consist of retarded wave and advanced wave. This two waves together become the traditional electromagnetic fields. This kind of electromagnetic fields can be seen approximately as retarded wave only, this kind wave also has its own superposition. This kind of superposition is also different with the superposition when we consider the retarded waves alone and also the advanced waves alone. In this article this author will discuss the differences of these different superpositions. This author will also discuss the different physical result with a few different superposition principles. In this article this author will prove the superposition with test charge and the superposition with retarded waves alone or advance wave alone can be see one kind of superposition. This superposition is correct and can be derived from the mutual energy principle. This will be referred as first kind of superposition. The superposition without test charge and the superposition with the retarded waves and advanced waves can be seen as one kind of superposition. For this kind of superpositions are not correct naturally. This kind of super position are applied to N-charge's Poynting theorem. In order to make this superposition work, the self-energy principle have to be applied. Without self-energy principle this kind of superpositions will violate the energy conservation law. Considering the self-energy conservation this kind of superposition become the superposition of first kind. This is referred as second kind of superposition. The third kind of superposition is for the traditional electromagnetic field. Here the traditional electromagnetic field means the electromagnetic field where the advanced waves are omit. When the advanced waves are omit doesn't mean the advanced waves do not exist, that is only means the retarded wave and the advanced wave are with nearly same intensity and can be treated together in some situation, for example, in wave guide, or in the free space where the absorber are uniformly distributed at infinite big sphere. Abstract Only when the self-energy principle is accept, all kinds of superposition can be accept. However different superposition has different physical meanings. Otherwise only the superposition with test charge or the superposition with only one kind wave either retarded waves or advanced waves can be accepted. Hence, the discussion about the superposition also support the concept of the self-energy principle which means there must exist the time reversal waves. That also means the waves do not collapse but collapse back. Wave collapse means collapse to its target, for example, the retarded wave will collapse to an absorber and the advanced wave will collapse to an emitter. Wave collapse back means the retarded wave sent from emitter will collapse back to an emitter; The advanced wave sent from the absorber will collapse back to an absorber. Hence, one purpose of this article is to clarify the superposition principles, and another purpose is to support this author's electromagnetic field theory which is started from two new axioms the self-energy principle and the mutual energy principle.
Paper 74 — No Quantum Superposition This paper argues, by reductio ad absurdum, that the physical realisation of superposition — simultaneous multi-location existence — is incompatible with seven independent and well-established physical principles: (1) the observational fact that no infinite physical quantity has ever been measured; (2) conservation of energy; (3) the mass-energy equivalence E = mc²; (4) the Bekenstein entropy bound; (5) the second law of thermodynamics; (6) the Heisenberg uncertainty principle; and (7) causality. The foundational identification is that information, energy, and mass stand in proportional existential relation (Bekenstein bound: I ∝ M; Landauer principle: ΔE ∝ ΔI; mass-energy equivalence: E = mc²). Complete superposition requires infinite information processing, and therefore infinite energy and infinite mass — quantities universally absent from the observational record of physics. Finite N-state superposition is excluded by a complementary argument: maintaining N states simultaneously requires N times the information-processing load of a single undetermined state, violating the principle that nature selects minimum-load configurations. The positive alternative proposed is the undetermined state: during matter-wave propagation, the matter wave is real and physically present, but matter has not yet appeared at any location. This is not a hidden-variable state but an ontological absence of localisation prior to the formation of a constructive interference maximum. Paper 75 — The Double-Slit Experiment Reinterpreted This paper applies the undetermined-state ontology to the paradigmatic demonstration of quantum mechanics. The de Broglie matter wave — a real physical wave — passes through both slits, while the particle remains in an undetermined state until a constructive interference maximum is formed at the detection screen. The disappearance of the interference pattern upon which-path measurement is explained without wavefunction collapse: the measurement interaction physically disrupts the matter wave's coherent two-slit propagation. Three experimentally testable predictions distinguish this framework: (1) weak-measurement trajectory reconstruction should reveal single-path trajectories inconsistent with simultaneous two-slit passage; (2) diffraction fringe spacing should depend on the mass density of the slit material, because denser slit edges generate a steeper refractive-index gradient; (3) the fringe pattern should exhibit a density-dependent phase shift measurable with current matter-wave interferometry technology. The Schrödinger equation and Born's rule are retained without modification. Paper 76 — Discriminating Experiments for the Constructive-Interference Account of Gravity This paper derives the specific numerical values predicted by Paper 29 — n(r) from the Schwarzschild metric, Δφ = k₀∫[n₁(r)−n₂(r)]dr from the medium-structure integral, and Δx_max = f(∇n, geometry, λ₀) from the interference structure — and assesses whether current experimental technology can measure the gap between Paper 29's predictions and standard QM's predictions. Three formal conditions for a discriminating experiment are established: structural absence of the variable from standard QM, signal exceeding current resolution, and impossibility of alternative explanation. Two experimental classes satisfy all three conditions with existing apparatus: COW path-geometry (predicted δ(Δφ) ~ 10⁻³ rad vs. resolution ~10⁻⁴ rad, path shape structurally absent from the standard COW formula) and slit-material density (predicted Δx_max ~ tens of nm vs. resolution 1–5 nm, material density structurally absent from Δy ≈ λL/a). Cumulative agreement across both systems constitutes genuine inductive support for the proposed mechanism. Paper 77 — From Electrons to Fullerenes This paper addresses three questions that Paper 76 does not: (1) why the slit-material density experiment has never been performed in the hundred years since matter-wave interferometry began — the answer being that standard QM contains no density variable in the slit formula, so experimenters had no reason to vary it, with the Copenhagen prohibition on mechanism questions as the deeper structural reason; (2) why confirmation in exactly two systems — electrons and fullerenes C₆₀ — is sufficient to establish the theoretical outline across six orders of magnitude in mass, because these systems are maximally independent in particle type, mass, apparatus, energy scale, and decoherence environment; (3) what confirmation across this mass range would establish for physics: the first experimentally supported identification of the physical mechanism of gravity, the first experimentally verifiable quantum-gravity connection at accessible energy scales, a physical grounding of Born's rule, and a mechanistic account of the measurement problem. A complete three-layer control protocol for excluding the slit-width confound is specified. Paper 78 — A Three-Dimensional Experimental Matrix This paper extends the experimental programme by introducing slit thickness as a third independent variable alongside matter-wave species and slit-material density. The wave-mechanical account predicts that phase shift scales linearly with slit thickness d: Δφ(ρ,d) = (2π/λ)·[n(ρ)−1]·d — a prediction structurally absent from standard QM, which assigns no physical role to slit thickness. The resulting 5×5×3 matrix of 75 independent measurements has a random coincidence probability of approximately 1/1,728,000, rendering chance agreement with theoretical prediction physically impossible. Three-layer control protocols for fabrication, TEM cross-section verification, and linearity testing are specified for the thickness dimension. Together, the five papers constitute a complete programme: ontological foundation (P74), phenomenological reinterpretation (P75), formal discriminating analysis (P76), strategic and historical argument (P77), and three-dimensional experimental design (P78).