In physics, refraction is the change in direction of a wave passing from one medium to another or from a gradual change in the medium. Refraction of light is the most observed phenomenon, but other waves such as sound waves and water waves experience refraction. How much a wave is refracted is determined by the change in wave speed and the initial direction of wave propagation relative to the direction of change in speed. For light, refraction follows Snell's law, which states that, for a given pair of media, the ratio of the sines of the angle of incidence θ1 and angle of refraction θ2 is equal to the ratio of phase velocities in the two media, or equivalently, to the indices of refraction of the two media. Sin ⁡ θ 1 sin ⁡ θ 2 = v 1 v 2 = n 2 n 1 Optical prisms and lenses use refraction to redirect light, as does the human eye; the refractive index of materials varies with the wavelength of light, thus the angle of the refraction varies correspondingly. This is called dispersion and causes prisms and rainbows to divide white light into its constituent spectral colors.

Refraction of light can be seen in many places in our everyday life. It makes objects under a water surface appear closer than they are, it is what optical lenses are based on, allowing for instruments such as glasses, binoculars and the human eye. Refraction is responsible for some natural optical phenomena including rainbows and mirages. A correct explanation of refraction involves two separate parts, both a result of the wave nature of light. Light slows; this is not because of absorption. Rather it is because, as an electromagnetic oscillation, light itself causes other electrically charged particles such as electrons, to oscillate; the oscillating electrons emit their own electromagnetic waves which interact with the original light. The resulting "combined" wave has wave packets; the light has been slowed down. When light returns to a vacuum and there are no electrons nearby, this slowing effect ends and its speed returns to c; when light enters, exits or changes the medium it travels in, at an angle, one side or the other of the wavefront is slowed before the other.

This asymmetrical slowing of the light causes it to change the angle of its travel. Once light is within the new medium with constant properties, it travels in a straight line again; as described above, the speed of light is slower in a medium other than vacuum. This slowing applies to any medium such as air, water, or glass, is responsible for phenomena such as refraction; when light leaves the medium and returns to a vacuum, ignoring any effects of gravity, its speed returns to the usual speed of light in a vacuum, c. Common explanations for this slowing, based upon the idea of light scattering from, or being absorbed and re-emitted by atoms, are both incorrect. Explanations like these would cause a "blurring" effect in the resulting light, as it would no longer be travelling in just one direction, but this effect is not seen in nature. A more correct explanation rests on light's nature as an electromagnetic wave; because light is an oscillating electrical/magnetic wave, light traveling in a medium causes the electrically charged electrons of the material to oscillate..

A moving electrical charge emits electromagnetic waves of its own. The electromagnetic waves emitted by the oscillating electrons, interact with the electromagnetic waves that make up the original light, similar to water waves on a pond, a process known as constructive interference; when two waves interfere in this way, the resulting "combined" wave may have wave packets that pass an observer at a slower rate. The light has been slowed down; when the light leaves the material, this interaction with electrons no longer happens, therefore the wave packet rate return to normal. Consider a wave going from one material to another where its speed is slower as in the figure. If it reaches the interface between the materials at an angle one side of the wave will reach the second material first, therefore slow down earlier. With one side of the wave going slower the whole wave will pivot towards that side; this is why a wave will bend away from the surface or toward the normal when going into a slower material.

In the opposite case of a wave reaching a material where the speed is higher, one side of the wave will speed up and the wave will pivot away from that side. Another way of understanding the same thing is to consider the change in wavelength at the interface; when the wave goes from one material to another where the wave has a different speed v, the frequency f of the wave will stay the same, but the distance between wavefronts or wavelength λ=v/f will change. If the speed is decreased, such as in the figure to the right, the wavelength will decrease. With an angle between the wave fronts and the interface and change in distance between the wave fronts the angle must change over the interface to keep the wave fronts intact. From these considerations the relationship between the angle of incidence θ1, angle of transmission θ2 and the wave speeds v1 and v2 in the two materials can be derived; this is

Chris Stathopoulos

Chris Stathopoulos is a Canadian former soccer player who played in the USL A-League, National Professional Soccer League, the Canadian Professional Soccer League. Stathopoulos began his professional career with Montreal Impact in the USL A-League in 1997. During the 1997 season he helped Montreal clinch the Northeast division title, reach the Division finals where the Long Island Rough Riders defeated them from playoff contention, he played with Montreal during the indoor season in the National Professional Soccer League. His best season was during the 1999 indoor season, he finished as the second highest goalscorer for the club that season. He had a stint with Milwaukee Wave in 1997, won the NPSL Championship with the organization. Stathopoulos next stint at the indoor level was in 2000 this time with the Toronto ThunderHawks. During his tenure with Toronto he finished. On May 16, 2001 the Montreal Dynamites of the Canadian Professional Soccer League announced the signing of Stathopoulos to a contract.

Near the conclusion of the season the organization faced financial difficulties, which resulted in the release of Stathopoulos from his contract after rejecting a pay cut to his salary

Drumming (Reich)

Drumming is a piece by minimalist composer Steve Reich, dating from 1970–1971. Reich began composition of the work after a short visit to Ghana and observing music and musical ensembles there under the Anlo Ewe master drummer Gideon Alorwoyie, his visit was cut short after contracting malaria. K. Robert Schwarz describes the work as "minimalism's first masterpiece." The piece employs Reich's trademark technique of phasing. Phasing is achieved when two players, or one player and a recording, are playing a single repeated pattern in unison on the same kind of instrument. One player changes tempo while the other remains constant, the two players are one or several beats out of sync with each other, they may either phase further, depending on the piece. K. Robert Schwarz characterized Drumming as a "transitional" piece between Reich's early, more austere compositions and his works that use less strict forms and structure. Schwarz has noted that Reich made use of three new techniques, for him, in this work: "the process of substituting beats for rests within a repeating rhythmic cycle", or "rhythmic construction" and "rhythmic reduction" combination of instruments of different timbres at the same time incorporation of human voices in imitation of the sounds of the percussion instruments in the ensemble, including whistling effects In total, the work requires 9 percussionists.

With the additional players, the piece can be performed by 13 players. The work falls into four parts, with the following instrumentation used in each: Part One: 4 pairs of tuned bongo drums, played with double-ended wooden sticks Part Two: 3 marimbas, 2 or 3 female voices Part Three: 3 glockenspiels and piccolo Part Four: complete ensembleThe length of the piece can vary as the number of repeats taken on any given measure is up to the performers. Recordings of the piece span between 84 minutes; the entire piece is structured around one measure of 12/8 long. This rhythm is built up note by note, in the "substitution of beats for rests" technique found in other of Reich's works such as Music for Pieces of Wood, Music for 18 Musicians, others. After the rhythm is built up, two of the players phase to where they are playing the same pattern one quarter-note apart from each other, the other bongo players play resulting patterns that can be heard as a result of the combination of the phased patterns.

The rest of the piece continues to use the techniques of beat/rest substitution and resultant patterns through its four movements. The transitions consist as follows: Movement 2 begins by three marimba players playing the same repeated pattern as the bongo players, fading in while the bongo players fade out. Movement 3 begins similarly. Movement 4 begins after movement 3 reduces its texture to one glockenspiel player, playing a single repeated note from the original pattern. Marimba and bongo players join, build the pattern up again, note by note, until all nine percussionists are playing; the piece ends abruptly, on cue. Choreographers such as Laura Dean, Anne Teresa De Keersmaeker, Ginette Laurin have collaborated on dance performances with Reich on Drumming. 1971 – Gary Burke, Steve Chambers, Ben Harms, Russ Hartenberger, Frank Maefsky, Art Murphy, James Ogden, James Preiss. Recorded on December 16, 1971, live at New York. Duration 1:21:35 1974 – Bob Becker, Cornelius Cardew, Steve Chambers, Tim Ferchen, Ben Harms, Russ Hartenberger, James Preiss, Glen Velez.

1987 – Steve Reich and Musicians. Duration 56:42. 2002 – Ictus Ensemble. Duration 54:49. 2005 – So Percussion. Duration 1:14:02. 2018 – Colin Currie Group, Synergy Vocals.. Duration 55:07. "Analysis of Steve Reich's Drumming and his use of African polyrhythms", blog entry New Music Box, 9 December 1971 article