The Mathematics of Bubbles in Beer :: Math Carbonation

deflexionRefraction is what happens to light when it passes from one medium to another. For example, things appear distinguishablely from the bottom of a swimming kitten than on the top. Simplistically, refraction is the bending of light. The explanation for this phenomenon, however, can be described with light as beams and light as waves. No matter the case, it is important to echo that the speed of light is constant in every homogenised medium, regardless of shape, size or form.The index of refractionLight travels ( in certain substances ) at a fraction of the velocity if it travelled in a vacuum. The index of refraction is the inverse of this fraction. Thus, this number is greater than or equal to 1. This index is too specific to light, so unlike light in different mediums have different indices. For example, here is a table of indicesMaterial IndexVacuum 1.00000 style at STP 1.00029Ice 1.31Water at 20 C 1.33Acetone 1.36Ethyl alcohol 1.36Sugar solution(30%)1.38Diamond 2.417So lets observe the consummation of refraction in terms of rays. A ray strikes the get up between substance I and substance R. The move i between the incoming ray and the normal vector at the boundary is called the tilt of incidence, and the angle r between the refracted ray and the normal vector on the opposite side is called the angle of refraction.This is related in the following law, called Snells Law ni loathsomeness i = nr nefariousness r. For red light in air hitting water this gives sin r = sin i/1.33Solving for Snells Law for r gives the relation r = arcsin (sin i/n)Explicit CalculationIn simple ray tracing, a ray originates at a point P at a directional vector v, which is of unit length. This is the set of all points P+tv where t is a non-negative scalar. When the ray hits the boundary between two different substances, it will refract, and begin a new ray.The Mathematics of Bubbl es in Beer Math CarbonationRefractionRefraction is what happens to light when it passes from one medium to another. For example, things appear differently from the bottom of a swimming pool than on the top. Simplistically, refraction is the bending of light. The explanation for this phenomenon, however, can be described with light as rays and light as waves. No matter the case, it is important to remember that the speed of light is constant in every homogeneous medium, regardless of shape, size or form.The index of refractionLight travels ( in certain substances ) at a fraction of the velocity if it travelled in a vacuum. The index of refraction is the inverse of this fraction. Thus, this number is greater than or equal to 1. This index is also specific to light, so different light in different mediums have different indices. For example, here is a table of indicesMaterial IndexVacuum 1.00000Air at STP 1.00029Ice 1.31Water at 20 C 1.33Acetone 1.36Ethyl alcohol 1.36Sugar solution(30%)1.38Diamond 2.417So lets observe the effect of refraction in terms of rays. A ray strikes the surface between substance I and substance R. The angle i between the incoming ray and the normal vector at the boundary is called the angle of incidence, and the angle r between the refracted ray and the normal vector on the opposite side is called the angle of refraction.This is related in the following law, called Snells Law ni sin i = nr sin r. For red light in air hitting water this gives sin r = sin i/1.33Solving for Snells Law for r gives the relation r = arcsin (sin i/n)Explicit CalculationIn simple ray tracing, a ray originates at a point P at a directional vector v, which is of unit length. This is the set of all points P+tv where t is a non-negative scalar. When the ray hits the boundary between two different substances, it will refract, and begin a new ray.