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Sound Waves: Interference00:53

Sound Waves: Interference

Sound waves can be modeled either as longitudinal waves, wherein the molecules of the medium oscillate around an equilibrium position, or as pressure waves. When two identical waves from the same source superimpose on each other, the combination of two crests or two troughs results in amplitude reinforcement known as constructive interference. If two identical waves, that are initially in phase, become out of phase because of different path lengths, the combination of crests with troughs...
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In the AX proton spin system, proton A can sense the two spin states of a coupled proton X, resulting in a doublet NMR signal with two peaks of equal (1:1) intensity. When proton A is coupled to two equivalent protons (AX2 spin system), the spin states of each X can be aligned with or against the external field, creating three possible scenarios. This results in a 1:2:1  triplet signal, where the central peak corresponds to the chemical shift of A and is twice as large or intense as the others.
¹³C NMR: ¹H–¹³C Decoupling01:04

¹³C NMR: ¹H–¹³C Decoupling

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Related Experiment Video

Updated: Jul 22, 2026

Fabrication And Characterization Of Photonic Crystal Slow Light Waveguides And Cavities
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Published on: November 30, 2012

How an anomalous cusp bifurcates in a weak-noise system

Maier1, Stein

  • 1Mathematics Department, University of Arizona, Tucson, Arizona 85721 and Physics Department, University of Arizona, Tucson, Arizona 85721, USA.

Physical Review Letters
|September 6, 2000
PubMed
Summary

Singularities in double well systems reveal critical phenomena. We found a scaling law governing anomalous cusp bifurcations, extending catastrophe theory for noise-perturbed systems.

Area of Science:

  • Statistical physics
  • Nonlinear dynamics
  • Complex systems

Background:

  • Symmetric double well systems exhibit complex activated trajectories.
  • Absence of detailed balance can lead to singularities like cusps.
  • These phenomena are analogous to optical caustics.

Purpose of the Study:

  • To investigate the nature and bifurcation of singularities in activated trajectories.
  • To derive a scaling law for anomalous cusp formation.
  • To extend classical catastrophe theory to noise-perturbed systems.

Main Methods:

  • Analysis of activated trajectories in a symmetric double well system.
  • Derivation of scaling laws and nonpolynomial equations of state.
  • Examination of system quasipotential to understand bifurcations.

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Main Results:

  • Identified anomalous cusps (coinciding with saddle points) in trajectory patterns.
  • Derived a scaling law governing the bifurcation of anomalous cusps into conventional ones.
  • Demonstrated that these bifurcations are reflected in the system quasipotential, akin to phase transitions.

Conclusions:

  • The study reveals critical phenomena in noise-perturbed systems.
  • Results extend classical catastrophe theory by describing cusp bifurcations.
  • The findings offer insights into nonclassical critical points and system dynamics.