A logarithmic spiral beach is a type of beach which develops in the direction under which it is sheltered by a headland, in an area called the shadow zone. It is characterized as a logarithmic spiral because if one looks at it in plan view or aerially, it represents the same shape that is created from the logarithmic spiral relation. These beaches are also commonly referred to as ‘half heart’ or ‘crenulate’ shaped bays, or ‘headland bays’.
The logarithmic spiral can be determined using the equation (written in polar coordinates):
where:
= the angle of rotation, is located between two lines drawn from the origin to any two points on the spiral.
= the ratio of the lengths between two lines that extend out from the origin. The two lines are given as and . So also equals the ratio .
= the angle between any line from the origin and the line tangent to the spiral which is at the point where line intersects the spiral. is a constant for any given logarithmic spiral.
This type of beach forms due to the refraction of approaching waves and their diffraction by an upcoast headland. The approaching wave front curves as a result of wave diffraction at the headland, which in turn causes the shoreline to bend and yield a log spiral shape. Log spiral beaches are often on swell-dominated coasts where waves generally approach the shoreline from one main direction at an oblique angle. The oblique approaching waves refract and diffract into the "shadow zone" which can be considered a relatively sheltered hook of beach behind the headland. Increase in sediment size, wave height, berm height, and swash zone gradient from the up coast headland generally characterizes the concave seaward curved part of the beach.
Diffraction refers to various phenomena that occur when a wave encounters an obstacle or opening. It is defined as the bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a secondary source of the propagating wave. Italian scientist Francesco Maria Grimaldi coined the word diffraction and was the first to record accurate observations of the phenomenon in 1660.
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line". More than a century later, the curve was discussed by Descartes (1638), and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral".
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider by a factor of φ for every quarter turn it makes.
Angular resolution describes the ability of any image-forming device such as an optical or radio telescope, a microscope, a camera, or an eye, to distinguish small details of an object, thereby making it a major determinant of image resolution. It is used in optics applied to light waves, in antenna theory applied to radio waves, and in acoustics applied to sound waves. The colloquial use of the term "resolution" often causes confusion; when a camera is said to have high resolution because of its good image quality, it actually has a low angular resolution. The closely related term spatial resolution refers to the precision of a measurement with respect to space, which is directly connected to angular resolution in imaging instruments. The Rayleigh criterion shows that the minimum angular spread that can be resolved by an image forming system is limited by diffraction to the ratio of the wavelength of the waves to the aperture width. For this reason, high resolution imaging systems such as astronomical telescopes, long distance telephoto camera lenses and radio telescopes have large apertures.
Longshore drift from longshore current is a geological process that consists of the transportation of sediments along a coast parallel to the shoreline, which is dependent on the angle incoming wave direction. Oblique incoming wind squeezes water along the coast, and so generates a water current which moves parallel to the coast. Longshore drift is simply the sediment moved by the longshore current. This current and sediment movement occur within the surf zone. The process is also known as littoral drift.
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865.
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens. In contrast, the diffraction pattern created near the object is given by the Fresnel diffraction equation.
Etendue or étendue is a property of light in an optical system, which characterizes how "spread out" the light is in area and angle. It corresponds to the beam parameter product (BPP) in Gaussian beam optics. Other names for etendue include acceptance, throughput, light grasp, light-gathering power, optical extent, and the AΩ product. Throughput and AΩ product are especially used in radiometry and radiative transfer where it is related to the view factor. It is a central concept in nonimaging optics.
Coastal geography is the study of the constantly changing region between the ocean and the land, incorporating both the physical geography and the human geography of the coast. It includes understanding coastal weathering processes, particularly wave action, sediment movement and weather, and the ways in which humans interact with the coast
In geometry, a strophoid is a curve generated from a given curve C and points A and O as follows: Let L be a variable line passing through O and intersecting C at K. Now let P1 and P2 be the two points on L whose distance from K is the same as the distance from A to K. The locus of such points P1 and P2 is then the strophoid of C with respect to the pole O and fixed point A. Note that AP1 and AP2 are at right angles in this construction.
A prism coupler is a prism designed to couple a substantial fraction of the power contained in a beam of light into a thin film to be used as a waveguide without the need for precision polishing of the edge of the film, without the need for sub-micrometer alignment precision of the beam and the edge of the film, and without the need for matching the numerical aperture of the beam to the film. Using a prism coupler, a beam coupled into a thin film can have a diameter hundreds of times the thickness of the film. Invention of the coupler contributed to the initiation of a field of study known as integrated optics.
The main trigonometric identities between trigonometric functions are proved, using mainly the geometry of the right triangle. For greater and negative angles, see Trigonometric functions.
In optics, a dispersive prism is an optical prism, usually having the shape of a geometrical triangular prism, used to disperse light, that is, to separate light into its spectral components. Different wavelengths (colors) of light will be deflected by the prism at different angles. This is a result of the prism's material index of refraction varying with wavelength. Generally, longer wavelengths (red) thereby undergo a smaller deviation than shorter wavelengths (blue) where the refractive index is larger.
Acousto-optics is a branch of physics that studies the interactions between sound waves and light waves, especially the diffraction of laser light by ultrasound through an ultrasonic grating.
A blazed grating – also called echelette grating – is a special type of diffraction grating. It is optimized to achieve maximum grating efficiency in a given diffraction order. For this purpose, maximum optical power is concentrated in the desired diffraction order while the residual power in the other orders is minimized. Since this condition can only exactly be achieved for one wavelength, it is specified for which blaze wavelength the grating is optimized. The direction in which maximum efficiency is achieved is called the blaze angle and is the third crucial characteristic of a blazed grating directly depending on blaze wavelength and diffraction order.
The contrast transfer function (CTF) mathematically describes how aberrations in a transmission electron microscope (TEM) modify the image of a sample. This contrast transfer function (CTF) sets the resolution of high-resolution transmission electron microscopy (HRTEM), also known as phase contrast TEM.
For a plane curve C and a given fixed point O, the pedal equation of the curve is a relation between r and p where r is the distance from O to a point on C and p is the perpendicular distance from O to the tangent line to C at the point. The point O is called the pedal point and the values r and p are sometimes called the pedal coordinates of a point relative to the curve and the pedal point. It is also useful to measure the distance of O to the normal even though it is not an independent quantity and it relates to as .
In X-ray diffraction, ideal measurements are done with X-rays of a single wavelength. Practically, the x-rays for a measurement are usually generated in an X-ray tube from a metal's K-alpha line. This generation creates x-rays at a variety of wavelengths, but most of the non K-alpha X-rays can be blocked from reaching the sample by filters. However, the K-alpha line is actually two x-ray lines close together: the stronger K-alpha 1 peak, and the weaker K-alpha 2 peak. Compared to other radiation such as the Bremsstrahlung, the K-alpha two peak is more difficult to filter mechanically. The Rachinger correction is a recursive method suggested by William Albert Rachinger (1927) to eliminate the disturbing peak.
