Standing wave

Summary

In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect to time, and the oscillations at different points throughout the wave are in phase. The locations at which the absolute value of the amplitude is minimum are called nodes, and the locations where the absolute value of the amplitude is maximum are called antinodes. Standing waves were first described scientifically by Michael Faraday in 1831. Faraday observed standing waves on the surface of a liquid in a vibrating container. Franz Melde coined the term "standing wave" (German: stehende Welle or Stehwelle) around 1860 and demonstrated the phenomenon in his classic experiment with vibrating strings. This phenomenon can occur because the medium is moving in the direction opposite to the movement of the wave, or it can arise in a stationary

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Existence And Stability Of High Frequency Standing Waves For A Nonlinear Schrodinger Equation

François Samer Genoud

This article is concerned with the existence and orbital stability of standing waves for a nonlinear Schrodinger equation (NLS) with a nonautonomous nonlinearity. It continues and concludes the series of papers [6, 7, 8]. In [6], the authors make use of a continuation argument to establish the existence in R x H-1(R-N) of a smooth local branch of solutions to the stationary elliptic problem associated with (NLS) and hence the existence of standing wave solutions of (NLS) with small frequencies. Complementary conditions on the nonlinearity are found, under which either stability of the standing waves and bifurcation of the branch of solutions from the point (0, 0) is an element of R x H-1 (R-N) occur, or instability and asymptotic bifurcation occur. The main hypotheses in [6] concern the behaviour of the nonlinearity with respect to the space variable at infinity. The paper [7] extends the results of [6] to (NLS) with more general nonlinearities. In [8], the global continuation of the local branch obtained in [6] is proved under additional hypotheses on the nonlinearity. In particular, spherical symmetry with respect to the space variable is assumed. The aim of the present work is to prove the existence and discuss the orbital stability of standing waves with high frequencies, independently of the results obtained in [6] and [8]. The main hypotheses now concern the behaviour of the nonlinearity with respect to the space variable around the origin. The methods are the same in spirit as that of [6] and permit to discuss the asymptotic behaviour of the global branch of solutions obtained in [8].

2009

Existence and orbital stability of standing waves for some nonlinear Schrodinger equations, perturbation of a model case

François Samer Genoud

The following nonlinear Schrodinger equation is studied

2009

, ,

In this work, we report about optical spectrometry using gold nano-structures printed on a polymer based integrated optical waveguide. The optical waveguide is a single mode buried waveguide, having dimensions of 3x2.2 mu m(2). It is made from a combination of photo-polymerizable materials and is fabricated by photolithography on a glass substrate. To sense the electric field inside the waveguide, a gold nano-coupler array of thin lines (50 nm thick and 8 mu m length) is embedded on top of the aforementioned waveguide. They are produced by E-beam lithography. The array pitch is 2.872 mu m and the number of lines 564, which yields an array of 1.619 mm length. The device is enclosed with a glass superstrate to prevent it from dust and destruction. Both waveguide ports are polished and the output port in particular, is coated with a thin gold layer to assimilate a mirror and hence, it enables the creation of stationary waves inside the structure. The measurement procedure involves light injection using a single mode fiber carrying both visible light (658nm) and infrared light (785nm), used for alignment and measurement purposes respectively. Stationary waves generated inside the guide constitute the spatial interferogram. Locally, light is out-coupled using the nano-couplers and allows measuring the interferogram structure. The resulting pattern is imaged by a vision system involving an optical microscope with a digital camera mounted on-top of it. Signal processing, mainly based on Fast Fourier transform is performed on the captured signal to extract the spectral content of the measured signal.

Spie-Int Soc Optical Engineering2014